I hope that most of you have either asked yourselves this question explicitly, or at least felt a vague sense of unease about how the definitions I gave in lectures, namely

and

relate to things like the opposite, adjacent and hypotenuse. Using the power-series definitions, we proved several facts about trigonometric functions, such as the addition formulae, their derivatives, and the fact that they are periodic. But we didn’t quite get to the stage of proving that if and is the angle that the line from to makes with the line from to , then and . So how does one establish that? How does one even define the angle? In this post, I will give one possible answer to these questions.

A couple of possible approaches that I won’t attempt to use

A cheating and not wholly satisfactory method would be to define the angle to be . Then it would be trivial that and we could use facts we know to prove that . (Or could we? Wouldn’t we just get that it was ? The fact that many angles have the same and creates annoying difficulties for this approach, though ones that could in principle be circumvented.) But if we did this, how could we be confident that the notion of angle we had just defined coincided with what we think angle should be? The problem has not been fully solved.

Another approach might be to define trigonometric functions geometrically, prove that they have the basic properties that we established using the power series definitions, and prove that these properties characterize the trigonometric functions (meaning that any two functions and that have the properties must be and ). However, this still requires us to make sense of the notion of angle somehow, and we might also feel slightly worried about whether the geometric arguments we used to justify the addition formulae and the like were truly rigorous. (I’m not saying it can’t be done satisfactorily — just that I don’t immediately see a good way of doing it, and I have a different approach to present.)

Defining angle

How are radians defined? You take a line L starting at the origin, and it hits the unit circle at some point P. Then the angle that line makes with the horizontal (or rather, the horizontal heading out to the right) is defined to be the length of the circular arc that goes anticlockwise round the unit circle from to P. (This defines a number between 0 and , but we can worry about numbers outside this range later.)

Calculating the length of a circular arc

There is nothing wrong with this definition, except that it requires us to make rigorous sense of the length of a circular arc. How are we to do this?

For simplicity, let’s assume that our point P is and that both and are positive. So P is in the top right quadrant of the unit circle. How can we define and then calculate the length of the arc from to , or equivalently from to ?

One non-rigorous but informative way of thinking about this is that for each between and , we should take an interval , work out the length of the bit of the circle vertically above this interval, and sum up all those lengths. The bit of the circle in question is a straight line (since is infinitesimally small) and by similar triangles its length is .

How did I write that down? Well, the big triangle I was thinking of was one with vertices , and the point on the circle directly above , which is , by Pythagoras’s theorem. The little triangle has one side of length , which corresponds to the side in the big triangle of length . So the hypotenuse of the little triangle is , as I claimed.

Adding all these little lengths up, we get , so it remains to evaluate this integral.

This is of course a very standard integral, usually solved by substituting or for . If you do that, you find that the length works out as , which is just what we hoped. However, we haven’t discussed integration by substitution in this course, so let us see it in a more elementary way (not that proving an appropriate form of the integration-by-substitution rule is especially hard).

Using the rules for differentiating inverses, we find that

and since , this gives us . So the integrand has as an antiderivative, and therefore, by the fundamental theorem of calculus,

So the angle between the horizontal and the line joining the origin to is (by definition) the length of the arc from to , which we have calculated to be . Therefore, .

How close was that to being rigorous?

The process I just went through, of saying “Let’s add up a whole lot of infinitesimal lengths; that says we should write down the following integral; calculating the integral gives us L, so the length is L,” is a process that one often goes through when calculating similar quantities. Why are we so confident that it is OK?

I sometimes realize with mathematical questions like this that I have been a mathematician for many years and never bothered to worry about them. It’s just sort of obvious that if a function is reasonably nice, then writing something down that’s approximately true with and turning into and writing a nice sign in front gives you a correct expression for the quantity in question. But let’s try to think a bit about how we might define length rigorously.

Curves

First, we should say what a curve is. There are various definitions, according to how much niceness one wants to assume, but let me take a basic definition: a curve is a continuous function from an interval to . (I haven’t defined continuous functions to , but it simply means that if , then and are both continuous functions from to .)

This is an example of a curious habit of mathematicians of defining objects as things that they clearly aren’t. Surely a curve is not a function — it’s a special sort of subset of the plane. In fact, shouldn’t a curve be defined as the image of a continuous function from to ? It’s true that that corresponds more closely to what we are thinking of when we use the word “curve”, but the definition I’ve just given turns out to be more convenient, though it’s important to add that two curves (as I’ve defined them) and are equivalent if there is a strictly increasing continuous bijection such that for every . In this situation, we think of and as different ways of representing the same curve.

Incidentally, if you want a reason not to identify curves with their images, then one quite good reason is the existence of objects called space-filling curves. These are continuous functions from intervals of reals to that fill up entire two-dimensional sets. Here’s a picture of one, lifted from Wikipedia.

It shows the first few iterations of a process that gives you a sequence of functions that converge to a continuous limit that fills up an entire square.

Lengths of curves

Going back to lengths, let’s think about how one might define them. The one thing we know how to define is the length of a line segment. (Strictly speaking, I’m not allowed to say that, since a line segment isn’t a function, but let’s understand it as a particularly simple function from an interval to a line segment in the plane.) Given that, a reasonable definition of length would seem to be to approximate a given curve by a whole lot of little line segments. That leads to the following idea for at least approximating the length of a curve . We take a dissection and add up all the little distances . Here I am defining the distance between two points in in the normal way by Pythagoras’s theorem. This gives us the expression

for the approximate length given by the dissection. We then hope that as the differences get smaller and smaller, these estimates will tend to a limit. It isn’t hard to see that if you refine a dissection, then the estimate increases (you are replacing the length of a line segment that joins two points by the length of a path that consists of line segments and joins the same two points).

Actually, that hope is not always fulfilled: sometimes the estimates tend to infinity. Indeed, for space-filling curves, or fractal-like curves such as the Koch snowflake, the estimates do tend to infinity. In this case, we say that they have infinite length. But if the estimates tend to a limit as the maximum of the differences tends to zero, we call that limit the length of the curve. A curve that has a finite length defined this way is called rectifiable.

Suppose now that we have a curve given by and that the two functions and are continuously differentiable. Then both and are bounded on , so let’s suppose that is an upper bound for and . Then by the mean value theorem,

Therefore, for every dissection, which implies that the curve is rectifiable. (Remark: I didn’t really use the continuity of the derivatives there — just their boundedness.)

We can say slightly more than this, however. The differentiability of tells us that for some . And similarly for with some . Therefore, the estimate for the length can be written

This looks very similar to the kind of thing we write down when doing Riemann integration, so let’s see whether we can find a precise connection. We are concerned with the function . If we now do use the continuity of and , then is continuous too, so it can be integrated. Now since and belong to the interval , and both lie between the lower and upper sums given by the dissection. That implies the same for

Since is integrable, the limit of as the largest (which is often called the mesh of the dissection) tends to zero is .

We have shown that the length of the curve is given by the formula

Now, finally, let’s see whether we can justify our calculation of the length of the arc of the unit circle between and . It would be nice to parametrize the circle as , but we can’t do that, since we are defining using length, so we would end up with a circular definition (in more than one sense). [Actually, we can do something very close to this. See the final section of the post for details.] So let’s parametrize it as follows. We’ll define on the interval and we’ll send to . Then and , so

So the length is , which is exactly the expression we wrote down earlier.

Let me make two quick remarks about that. First, you might argue that although I have shown that the final expression is indeed correct, I haven’t shown that the informal argument is (essentially) correct. But I more or less have, since what I have effectively done is calculate the lengths of the hypotenuses of the little triangles in a slightly different way. Before, I used the fact that one side was and used similar triangles. Here I’ve used the fact that one side is and another side is and used Pythagoras.

A slightly more serious objection is that for this calculation I used a general result that depended on the assumption that both and are continuously differentiable, but didn’t check that the appropriate conditions held, which they don’t. The problem is that , so , which tends to infinity as and is undefined at .

However, it is easy to get round this problem. What we do is integrate from to , in which case the argument is valid, and then let tend to zero. The integral between and is , and that tends to .

One final remark is that this length calculation explains why the usual substitution of for in an integral of the form is not a piece of unmotivated magic. It is just a way of switching from one parametrization of a circular arc (using the x-coordinate) to another (using the angle, or equivalently the distance along the circular arc) that one expects to be simpler.

An easier argument

Thanks to a comment of Jason Fordham below, I now realize that we can after all parametrize the circle as . However, this is not the I’m trying to calculate, so let’s call it . I’m just taking to be an ordinary real number, and I’m defining and using the power-series definition. Then the arc of the unit circle that goes from to can be defined as the curve defined on the interval by the formula . The general formula for the length of a curve then gives us

So the length of the arc satisfies .

Ben Lillie:

This week I finally realized what bugged me about the talk I was hearing about the science of science communication: Nothing. The issue is what I wasn’t hearing.

This was catalyzed by a short news item Lauren Rugani linked on twitter. A scientist had run a study where they discovered that sometimes a punchline is funnier if words from the punchline had been mentioned several minutes earlier. From the abstract:

“These findings also show that pre-exposing a punchline, which in common knowledge should spoil a joke, can actually increase funniness under certain conditions.”

This is shocking. Not the conclusion, which is clearly correct. The problem is that the conclusion has been known to comedians for at least the last several thousand years. When I trained in improv comedy the third class was on callbacks, the jargon term for that technique. The entire structure of an improv comedy set is based around variations on the idea that things are funnier if they’re repeated. And yet to the authors it was “common knowledge” that this will spoil a joke. There is a long tradition of people who know, from experience, how this works, and yet the idea of asking them is not evident anywhere in the paper. This is the problem — the sense that the only valid answers come from inside science and the research world.

Yes!  Everybody knows by now that when mathematicians try to do mathematical biology alone, without people with domain knowledge of biology in the room, they do crappy mathematical biology.  ”Digital humanities” or “neuroaesthetics” or “culturomics” &c are just the same.  New techniques drawn from science and mathematics are fantastic research tools now, and they’re only getting better, but it seems like a terrible idea to study cultural objects from scratch, without domain experts in the room.

Guest post by Dimitri Zaganidis

First of all, I would like to thank Emily for organizing the Kan extension seminar. It is a pleasure to be part of it. I want also to thank my advisor Kathryn Hess and my office mate Martina Rovelli for their revisions.

In the fifth installment of the Kan Extension Seminar we read the paper “Review of the Elements of 2-categories” by G.M Kelly and Ross Street. This article was published in the Proceedings of the Sydney Category Theory Seminar, and its purpose is to “serve as a common introduction to the authors’ paper in this volume”.

The article has three main parts, the first of them being definitions in elementary terms of double categories and 2-categories, together with the notion of pasting. In a second chapter, they review adjunctions in 2-categories with a nice expression of the naturality of the bijection given by mates using double categories. The last part of the article introduces monads in 2-categories, and specializing to 2-monads towards the end.

Double categories and 2-categories

The article starts with the definition of a double category as a category object in the (not locally small) category of categories $\mathbf{CAT}$. (I think that there might be some set theoretic issues with such a category, but you can add small everywhere if you want to stay safe.)

The authors then switch to a description of such an object in terms of objects, horizontal arrows, vertical arrows, and squares, with various compositions and units. I will explain a bit how to go from one description to the other.

A category object is constituted of a category of objects, a category of morphisms, target and source functors, identity functor and a composition.

The category of objects is the category whose morphisms are “the objects” and whose morphisms are the vertical arrows. The category of morphisms is the category whose objects are the horizontal morphisms and whose morphisms are the squares, with vertical composition.

Since the functors $\mathrm{Obj}, \mathrm{Mor}: \mathbf{CAT} \longrightarrow \mathbf{SET}$ preserve pullbacks, by applying them to a double category seen as a category object, we get actual categories. Applying $\mathrm{Obj}$ to the double category, we get the category whose objects are “the objects” and whose morphisms are the horizontal arrows. Applying $\mathrm{Mor}$, we get the category whose objects are the vertical morphisms and whose morphisms are the squares, but this time with horizontal composition.

An interesting thing to notice is that the symmetry of the explicit description of a double category is much more apparent than the symmetry of its description as a category object.

One can define a $2$-category as a double category with a discrete category of objects, or as a $\mathbf{CAT}$-enriched category, exactly as one can define a simplicially enriched small category as either a category enriched over $\mathbf{sSet}$ or as a category object in $\mathbf{sSet}$ with a discrete simplicial set of objects.

The second viewpoint on 2-categories leads to definitions of 2-functors and 2-natural transformations and also to modifications, once one makes clear what enrichment a category of 2-functors inherits.

It is also worthwhile mentioning that the pasting operation makes computations easier to make, because they are more visual. The proof of proposition 2.1 of this paper is a good illustration of this.

The basic example of a 2-category is $\mathbf{CAT}$ itself, with natural transformations as 2-cells (squares).

As category theory describes set-like constructions, 2-category theory describes category-like constructions. You can usually build up categories with as objects sets with extra structure. In the same way, small V-categories, V-functors, and V-natural transformations form a 2-category.

My first motivation to learn about 2-categories was the 2-category of quasi-categories defined by Joyal and which has been studied by Emily Riehl and Dominic Verity in the article The 2-category theory of quasi-categories in particular the category-like constructions one can make with quasi-categories, such as adjunctions and limits.

Adjunctions and mates in 2-categories

It is not a surprise that 2-categories are the right framework in which to define adjunctions. To build the general definition from the usual one, you just need to replace categories by objects in a 2-category, functors by 1-cells of the 2-category, and natural transformations by its 2-cells.

Adjunctions in a 2-category $\mathcal{C}$ compose (as in $\mathbf{CAT}$), and one can form two, a priori distinct double categories of adjunctions. Both of them will have the objects of $\mathcal{C}$ as objects and the horizontal morphisms being the morphisms of $\mathcal{C}$, while their vertical morphisms are the adjunctions (going in the same direction as the right adjoint, by convention). The two double categories differ on the squares. Given adjunctions $f \dashv u$ and $f' \dashv u'$ together with 1-cells $a:A \longrightarrow A'$ (between the domains of $u$ and $u'$) and $b:B \longrightarrow B'$ (between the codomains of $u$ and $u'$), the squares of the first double category are 2-cells $b u \Rightarrow u'a$ while the squares of the second are 2-cells $f'b \Rightarrow a f$.

Now, the bijective correspondence between these kind of 2-cells given by mates induces an isomorphism of double categories. This means in particular that the horizontal (or vertical) composite of mates is equal to the mate of the corresponding composite.

This is a very beautiful way to express the naturality of the mate correspondence, and it provides a one-line proof of the fact that two 1-cells that are left adjoints to a same 1-cell are naturally isomorphic.

Monads in 2-categories

2-categories are also the right framework to define monads. A monad in a 2-category $\mathcal{C}$ and on an object $B$ is a 1-cell $t:B \longrightarrow B$ together with 2-cells $\mu: t^2 \Rightarrow t$ and $\eta: 1_B \Rightarrow t$, verifying the usual equations $\mu \circ (t\mu)= \mu \circ (\mu t)$ and $\mu \circ(t\eta) = 1_B = \mu \circ(\eta t)$. Since 2-functors preserve both horizontal and vertical compositions, for all objects $X$ of $\mathcal{C}$, $t$ induces a monad on $\mathcal{C}(X,B)$, given by post-composition $(t_{\ast},\mu_{\ast},\eta_{\ast})$. The authors call * an action of $t$ on $s:X \longrightarrow B$* a $t_\ast$ algebra structure on $s$.

The definition of monad morphism in this article is quite surprising for someone who has read the previous article by Ross Street, The formal theory of monads, which we read for the second meeting of the Kan extension seminar.

In Ross Street’s original paper, a monad morphism $(B,t,\mu, \eta) \longrightarrow (B',t',\mu', \eta')$ is a 1-cell $f: B \longrightarrow B'$ together with a $2$-cell $\phi: t'f \Rightarrow f t$ verifying certain conditions.

In this paper, morphisms of monads are defined only for monads on the same object, letting the $1$-cell part of a monad transformation of the previous article be the identity. This leads the authors to reverse the direction of the morphism, since the $2$-cell seems to go in the reverse direction of the $1$-cell!

One might think that fixing $f=1$ is needed by the result which explains that there is a bijection between monad morphisms $t \Rightarrow t'$ and actions of $t$ on $t'$ making $t'$ a “$(t,t')$-bimodule”. In fact, in the case where $f$ is not necessarily the identity, there is a bijection between 2-cells $\phi:t f \Rightarrow f t'$ such that $(f,\phi)$ is a monad functor and actions of $t$ on $ft'$ making $ft'$ a “$(t,t')$-bimodule”. A statement of the same kind can be also made for monad functor transformations (in the sense of the formal theory of monads). A 2-cell $\sigma : f \Rightarrow f'$ is a monad functor transformation $(f,\phi) \longrightarrow (f', \phi')$ if and only if $\sigma t': f t' \Rightarrow f' t'$ is a morphism of “$(t,t')$-bimodules”.

A 2-category admits the construction of algebras if for every monad $(B,t,\mu, \eta)$, the 2-functor $X \mapsto \mathcal {C}(X,B)^{(t_\ast, \mu_\ast, \eta_\ast)}$ is representable. The representing object is called the object of $t$-algebras. By Yoneda, the free-forgetful adjunction can be made internal in this case.

The terminology is justified, because in the $2$-category $\mathbf{CAT}$, it specializes to the usual notions of the category of $t$-algebras and the corresponding free-forgetful adjunction.

A monad in $\mathcal{C}$ is the same as a 2-functor $\mathbf{Mnd} \longrightarrow \mathcal{C}$, where $\mathbf{Mnd}$ is the 2-category with one object and $\Delta_+$, the algebraist’s simplicial category as monoidal hom-category (with ordinal sum). Since moreover, $$\mathcal {C}(X,B)^{(t_\ast, \mu_\ast, \eta_\ast)} \cong [\mathbf{Mnd}, \CAT]( \Delta_{+\infty}, \mathcal{C}(X,-)),$$ (where $\Delta_{+\infty}$ is the subcategory of maps of $\Delta$ preserving maxima, which is acted on by $\Delta_+$ via ordinal sum) one can see that the object of t-algebras can be expressed as a weighted limit.

As a consequence, it is not surprising that a 2-category admits the construction of algebras under some completeness assumptions.

Doctrines

In the last part of the article, the authors review the notion of a doctrine, which is a 2-monad in 2-$\mathbf{CAT}$, i.e., a 2-functor $D: \mathcal {C} \longrightarrow \mathcal{C}$, where $\mathcal{C}$ is a 2-category, and 2-natural transformations $m$ and $j$, which are respectively the multiplication and the unit, verifying the usual identities. The fact that it is both a monad on a 2-category and in another one can be a bit disturbing at first.

If $(D,m,j)$ is a doctrine over a 2-category $\mathcal{C}$, then its algebras will be objects $X$ of $\mathcal{C}$ together with an action $DX \longrightarrow X$, exactly as in the case of algebras over a usual monad.

Already with morphisms, we can take advantage of the fact that a 2-category $\mathcal{C}$ has 2-cells, and define $D$-morphisms to be lax in the sense that the diagram $$\begin{matrix} DX & \longrightarrow & DY \\ \downarrow & & \downarrow \\ X & \longrightarrow & Y \end{matrix}$$ is not supposed to be commutative, but is rather filled by a 2-cell with some coherence properties.

As one might expect, we can actually form a 2-category of such $D$-algebras by adding 2-cells, using again the $2$-cells existing in $\mathcal{C}$.

If we keep only the $D$-morphisms that are strict, we obtain the object of algebras (which should be a $2$-category) that we discussed before.

One example of a doctrine is $\Delta_+ \times - : \mathbf{CAT} \longrightarrow \mathbf{CAT}$ together with the multiplication induced by the ordinal sum, and unit given on $\mathcal {D}$ by the functor $\mathcal{D} \longrightarrow \Delta_+ \times \mathcal{D}$ that sends $d$ to $(\emptyset,d)$.

The algebras for this doctrine will be categories equipped with a monad acting on them, while the $D$-morphisms are transformations of monads, and the $D$-2-cells are exactly the monad functor transformations of Street’s article.

Here, since we have two different 2-categories of algebras (with strict $D$-morphisms or with all of them), one can wonder if monad morphisms $D \longrightarrow D'$ will induce $2$-functors $D'$-$\mathbf{Alg} \longrightarrow D$-$\mathbf{Alg}$ on the level of these $2$-categories.

This is indeed the case, and one can actually go even one step further and define monad modifications, using the fact that 2-$\mathbf{CAT}$ is in fact a 3-category! These modifications between two given monad morphisms are in fact in bijective correspondence with the 2-natural transformations between the $2$-functors induced by these monad morphisms on the level of algebras (with lax D-morphisms). Note that they are not the same as monad morphisms transformations of Street’s article.

This bijection is nice because it implies that you can compare 2-categories of algebras by only looking at the doctrines: if they are equivalent, so are the 2-categories of algebras.

The fact that this bijection does not hold when we restrict only to strict morphism was really surprising to me, but I guess this is the price to pay to use the 3-category structure.

During the last days of April, the Kan extension seminar will be reading the article “Two dimensional monad theory”, by Blackwell, Kelly and Powell. We will then have more to say about these 2-monads!

One of the endlessly recurring topics around here is the use of PowerPoint and comparable presentation software. Usually because of some ill-informed rant against the use of PowerPoint.

It’s come around yet again in a particularly ironic fashion, via an online slideshow at Slate, the only medium more consistently exasperating than a bad PowerPoint presentation. In keeping with modern tastes, this has been re-shared so many times that I finally went to look at it, but it’s more of the usual, in a more annoying medium. This is a bit of a shame, as there’s actually some good presentation advice buried in there, but between the format and the usual raft of misconceptions, it’s kind of hard to find.

The biggest problem I have with this is the blanket declaration that PowerPoint– or, more precisely, a particular style of PowerPoint– is “destroying higher education.” Which is an ill-formed claim on a lot of levels, but mostly in the claim that “higher education” is a single coherent thing. Which it’s not. Standards for what counts as good and effective teaching vary dramatically across disciplines and even within them. What’s well suited to one area is slow death in another, but that doesn’t mean that either of those groups is wrong.

So, yes, it’s absolutely true that text-heavy PowerPoint presentations that are made available online are deadly for classes where the discussion in class is the whole point of the course. But that’s not all of higher education. And if you’re working in a subject that involves more direct transfer of knowledge than mutual construction thereof, text-heavy PowerPoints made available online are the right tool for the job.

For the umpteenth time, then: PowerPoint is a tool, nothing more and nothing less. It can be used in many different ways, some of them good, some of them bad. The key to using it effectively is knowing your audience. Or, more precisely, your audiences, plural.

If I’m giving a public lecture or a TED-type talk, I’ll use slides with big splashy images and very few words. If I’m putting together a colloquium talk, I’ll use a bit more text, but still try to emphasize images. If I’m teaching an intro physics class, where I expect students will need to go back over the material multiple times while doing homework, I’ll use slides with a lot more text and equations, because I’m going to make those slides available on the course website after class, so students can go back and check stuff. If I’m teaching a seminar-type class, I’ll use a mix of those– on days when discussion is the main point, it’ll be mostly images, but on days when I need to convey factual information, there will be more text.

Even the better bits of that slideshow’s advice aren’t universal. I’m generally against giving an outline of your talk up front, because it mostly just wastes time. But there are topics where the path to be followed is sufficiently twisty that an outline is genuinely helpful, as a road map to the talk. (And that’s without even getting into fields where a clear outline at the beginning is an absolute cultural norm.) There’s no rule that applies to absolutely every presentation in absolutely every circumstance.

PowerPoint is a tool, but it’s not a single blunt instrument like a hammer. It’s very versatile, and can be used in lots of different ways. The key is knowing which of those ways to use in different sets of circumstances, and adjusting your style to match the audiences. Many of the alleged misuses of it are really “This isn’t how it ought to be used in my field,” which manages to be both perfectly true and perfectly useless.

Despite the old canard about nobody understanding quantum mechanics, physicists do understand it.  With all of the interpretations ever conceived for quantum mechanics[1], this claim may seem a bit of a stretch, but like the proverbial ostrich with its head in the sand, many physicists prefer to claim they do not understand quantum mechanics, rather than just admit that it is what it is and move on.

What is it about quantum mechanics that generates so much controversy and even had Albert Einstein (1879 – 1955) refusing to accept it? There are three points about quantum mechanics that generate controversy. It is probabilistic, eschews realism, and is local. Let us look at these three points in more detail.

1. Quantum mechanics is probabilistic, not determinist. Consider a radioactive atom. It is impossible, within the confines of quantum mechanics, to predict when an individual atom will decay. There is no measurement or series of measurements that can be made on a given atom to allow me to predict when it will decay. I can calculate the probability of when it will decay or the time it takes half of a sample to decay but not the exact time a given atom will decay. This lack of ability to predict exact outcomes, but only probabilities, permeates all of quantum mechanics. No possible set of measurements on the initial state of a system allows one to predict precisely the result of all possible experiments on that state.
2. Quantum mechanics eschews realism[2]. This is a corollary of the first point. A quantum mechanical system does not have well defined values for properties that have not been directly measured. This has been compared to the moon only existing when someone is looking at it. For deterministic systems one can always safely infer back from a measurement what the system was like before the measurement. Hence if I measure a particle’s position and motion I can infer not only where it will go but where it has come from. The probabilistic nature of quantum mechanics prevents this backward looking inference. If I measure the spin of an atom, there is no certainty that is had only that value before the measurement. It is this aspect of quantum mechanics that most disturbs people, but quantum mechanics is what it is.
3. Quantum mechanics is local. To be precise, no action at point A will have an observable effect at point B that is instantaneous, or non-causal.  Note the word observable. Locality is often denied in an attempt to circumvent Point 2, but when restricted to what is observable, locality holds. Despite the Pentagon’s best efforts, no messages have been sent using quantum non-locality.

Realism, at least, is a common aspect of the macroscopic world. Even a baby quickly learns that the ball is behind the box even when he cannot see it. But much about the microscopic world is not obviously determinist, the weather in Vancouver for example (it is snowing as I write this). Nevertheless, we cling to determinism and realism like a child to his security blanket. It seems to me that determinism or realism, if they exist, would be at least as hard to understand as their lack. There is no theorem that states the universe should be deterministic and not probabilistic or vice versa. Perhaps god, contrary to Einstein’s assertion, does indeed like a good game of craps[3].

So quantum mechanics, at least at the surface level, has features many do not like. What has the response been? They have followed the example set by Philip Gosse (1810 – 1888) with the Omphalos hypothesis[4]. Gosse, being a literal Christian, had trouble with the geological evidence that the world was older than 6,000, so he came up with an interpretation of history that the world was created only 6,000 years ago but in such a manner that it appeared much older. This can be called an interpretation of history because it leaves all predictions for observations intact but changes the internal aspects of the model so that they match his preconceived ideas. To some extent, Tycho Brahe (1546 – 1601) used the same technique to keep the earth at the center of the universe. He had the earth fixed and the sun circle the earth and the other planets the sun. With the information available at the time, this was consistent with all observations.

The general technique is to adjust those aspects of the model that are not constrained by observation to make it conform to one’s ideas of how the universe should behave. In quantum mechanics these efforts are called interpretations. Hugh Everett (1930 – 1982) proposed many worlds in an attempt to make quantum mechanics deterministic and realistic. But it was only in the unobservable parts of the interpretation that this was achieved and the results of experiments in this world are still unpredictable. Louis de Broglie (1892 – 1987) and later David Bohm (1917 – 1992) introduced pilot waves in an effort to restore realism and determinism. In doing do they gave up locality. Like Gosse’s work, theirs was nice proof in principle that, with sufficient ingenuity, the universe could be made to conform to almost any preconceived ideas, or at least appear to do so. Reassuring I guess, but like Gosse it was done by introducing non-observable aspects to the model: not just unobserved but in principle unobservable. The observable aspects of the universe, at least as far as quantum mechanics is correct, are as stated in the three points above: probabilistic, nonrealistic and local.

Me, I am not convinced that there is anything to understand about quantum mechanics beyond the rules for its use given in standard quantum mechanics text books. However, interpretations of quantum mechanics might, possibly might, suggest different ways to tackle unsolved problems like quantum gravity and they do give one something to discuss after one has had a few beers (or is that a few too many beers).

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One of my dreams these days is to get people to apply modern math to ecology and biology, to help us design technologies that work with nature instead of against it. I call this dream ‘green mathematics’. But this will take some time to reach, since living systems are subtle, and most mathematicians are more familiar with physics.

So, I’ve been warming up by studying the mathematics of chemistry, evolutionary game theory, electrical engineering, control theory and information theory. There are a lot of ideas in common to all these fields, but making them clear requires some category theory. I call this project ‘network theory’. I’m giving some talks about it at Oxford.

(This diagram is written in Systems Biology Graphical Notation.)

Here’s the plan:

Network Theory

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks:

I. Electrical circuits and signal-flow graphs.

II. Stochastic Petri nets, chemical reaction networks and Feynman diagrams.

III. Bayesian networks, information and entropy.

All these talks will be in Lecture Theatre B of the Computer Science Department—you can see a map here, but the entrance is on Keble Road. Here are the times:

• Friday 21 February 2014, 2 pm: Network Theory: overview. Also available on YouTube.

• Tuesday 25 February, 3:30 pm: Network Theory I: electrical circuits and signal-flow graphs. Also available on YouTube.

• Tuesday 4 March, 3:30 pm: Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams. Also available on YouTube.

• Tuesday 11 March, 3:30 pm: Network Theory III: Bayesian networks, information and entropy.

I thank Samson Abramsky, Bob Coecke and Jamie Vicary of the Computer Science Department for inviting me, and Ulrike Tillmann and Minhyong Kim of the Mathematical Institute for helping me get set up. I also thank all the people who helped do the work I’ll be talking about, most notably Jacob Biamonte, Jason Erbele, Brendan Fong, Tobias Fritz, Tom Leinster, Tu Pham, and Franciscus Rebro.

Ulrike Tillmann has also kindly invited me to give a topology seminar:

Operads and the Tree of Life

Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely “phylogenetic tree” describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann–Vogt “W construction” to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. It also shows up in tropical algebra. This talk is based on work in progress with Nina Otter.

I’m not sure exactly where this will take place, but probably somewhere in the Mathematical Institute, shown on this map. Here’s the time:

• Monday 24 February, 3:30 pm, Operads and the Tree of Life.

If you’re nearby, I hope you can come to some of these talks — and say hi!

(This diagram was drawn by Darwin.)

I’m giving some talks at Oxford:

Network Theory

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks:

I. Electrical circuits and signal-flow graphs.

II. Stochastic Petri nets, chemical reaction networks and Feynman diagrams.

III. Bayesian networks, information and entropy.

If you’re nearby I hope you can come! All these talks will take place in Lecture Theatre B in the Computer Science Department—see the map below. Here are the times:

• Friday 21 February 2014, 2 pm: Network Theory: overview. See the slides or watch a video.

• Tuesday 25 February, 3:30 pm: Network Theory I: electrical circuits and signal-flow graphs. See the slides or watch a video.

• Tuesday 4 March, 3:30 pm: Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams. See the slides or watch a video.

• Tuesday 11 March, 3:30 pm: Network Theory III: Bayesian networks, information and entropy. See the slides.

The first talk will be part of the OASIS series, meaning the “Oxford Advanced Seminar on Informatic Structures”.

I thank Samson Abramsky, Bob Coecke and Jamie Vicary of the Computer Science Department for inviting me, and Ulrike Tillmann and Minhyong Kim of the Mathematical Institute for helping me get set up. I also thank all the people who helped do the work I’ll be talking about, most notably Jacob Biamonte, Jason Erbele, Brendan Fong, Tobias Fritz, Tom Leinster, Tu Pham, and Franciscus Rebro.

Ulrike Tillmann has also kindly invited me to give a topology seminar:

Operads and the Tree of Life

Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely “phylogenetic tree” describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann–Vogt “W construction” to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. It also shows up in tropical algebra. This talk is based on work in progress with Nina Otter.

I’m not sure exactly where this will take place, but surely somewhere in the Mathematical Institute building:

• Monday 24 February, 3:30 pm, Operads and the Tree of Life. See the slides.

The Computer Science Department is shown in the map here:

The Mathematical Institute is a bit to the west:

In a day of talks—I had to leave early a talk by Ruth Angus (Oxford) on stellar ages from Kepler to see a talk in Computer Science by Alekh Agarwal (Microsoft) on distributed and clever machine-learning algorithms and engineering—Bekki Dawson (Berkeley) showed us results on the statistics of exoplanet populations and tests of planetary migration scenarios. She showed that the continuity of tidal circularization models (conservative exoplanet flow, in some sense) makes a prediction for the distribution of planets in the period–eccentricity plane, and that the prediction is falsified strongly by Kepler. There is not yet any good model for the formation and migration of exoplanets that explains the main features of the data, but there are many possible effects, and it is possible that all of them are acting at some level. Her talk suggested scores of other projects that could and should be done. On a side note, she showed convincingly that you can measure eccentricities just with Kepler data alone, and that there are strong asymmetries that make it much more likely that you will see a faster-than-circular transit than a slower-than-circular transit when the transiting-planet orbit is eccentric. She also showed some transit timing work by our own Foreman-Mackey.

The hunt for dark matter has been heating up once again, driven (as usual) by tantalizing experimental hints. This time the hints are coming mainly from outer space rather than underground laboratories, which makes them harder to check independently, but there’s a chance something real is going on. We need more data to be sure, as scientists have been saying since the time Eratosthenes measured the circumference of the Earth.

As I mentioned briefly last week, Katherine Freese of the University of Michigan has a new book coming out, The Cosmic Cocktail, that deals precisely with the mysteries of dark matter. Katie was also recently at the UCLA Dark Matter Meeting, and has agreed to share some of her impressions with us. (She also insisted on using the photo on the right, as a way of reminding us that this is supposed to be fun.)

Dark Matter Everywhere (at the biannual UCLA Dark Matter Meeting)

The UCLA Dark Matter Meeting is my favorite meeting, period. It takes place every other year, usually at the Marriott Marina del Rey right near Venice Beach, but this year on UCLA campus. Last week almost two hundred people congregated, both theorists and experimentalists, to discuss our latest attempts to solve the dark matter problem. Most of the mass in galaxies, including our Milky Way, is not comprised of ordinary atomic material, but instead of as yet unidentified dark matter. The goal of dark matter hunters is to resolve this puzzle. Experimentalist Dave Cline of the UCLA Physics Department runs the dark matter meeting, with talks often running from dawn till midnight. Every session goes way over, but somehow the disorganization leads everybody to have lots of discussion, interaction between theorists and experimentalists, and even more cocktails. It is, quite simply, the best meeting. I am usually on the organizing committee, and cannot resist sending in lots of names of people who will give great talks and add to the fun.

Last week at the meeting we were treated to multiple hints of potential dark matter signals. To me the most interesting were the talks by Dan Hooper and Tim Linden on the observations of excess high-energy photons — gamma-rays — coming from the Central Milky Way, possibly produced by annihilating WIMP dark matter particles. (See this arxiv paper.) Weakly Interacting Massive Particles (WIMPs) are to my mind the best dark matter candidates. Since they are their own antiparticles, they annihilate among themselves whenever they encounter one another. The Center of the Milky Way has a large concentration of dark matter, so that a lot of this annihilation could be going on. The end products of the annihilation would include exactly the gamma-rays found by Hooper and his collaborators. They searched the data from the FERMI satellite, the premier gamma-ray mission (funded by NASA and DoE as well as various European agencies), for hints of excess gamma-rays. They found a clear excess extending to about 10 angular degrees from the Galactic Center. This excess could be caused by WIMPs weighing about 30 GeV, or 30 proton masses. Their paper called these results “a compelling case for annihilating dark matter.” After the talk, Dave Cline decided to put out a press release from the meeting, and asked the opinion of us organizers. Most significantly, Elliott Bloom, a leader of the FERMI satellite that obtained the data, had no objection, though the FERMI team itself has as yet issued no statement.

Many putative dark matter signals have come and gone, and we will have to see if this one holds up. Two years ago the 130 GeV line was all the rage — gamma-rays of 130 GeV energy that were tentatively observed in the FERMI data towards the Galactic Center. (Slides from Andrea Albert’s talk.) This line, originally proposed by Stockholm’s Lars Bergstrom, would have been the expectation if two WIMPs annihilated directly to photons. People puzzled over some anomalies of the data, but with improved statistics there isn’t much evidence left for the line. The question is, will the 30 GeV WIMP suffer the same fate? As further data come in from the FERMI satellite we will find out.

What about direct detection of WIMPs? Laboratory experiments deep underground, in abandoned mines or underneath mountains, have been searching for direct signals of astrophysical WIMPs striking nuclei in the detectors. At the meeting the SuperCDMS experiment hammered on light WIMP dark matter with negative results. The possibility of light dark matter, that was so popular recently, remains puzzling. 10 GeV dark matter seemed to be detected in many underground laboratory experiments: DAMA, CoGeNT, CRESST, and in April 2013 even CDMS in their silicon detectors. Yet other experiments, XENON and LUX, saw no events, in drastic tension with the positive signals. (I told Rick Gaitskell, a leader of the LUX experiment, that I was very unhappy with him for these results, but as he pointed out, we can’t argue with nature.) Last week at the conference, SuperCMDS, the most recent incarnation of the CDMS experiment, looked to much lower energies and again saw nothing. (Slides from Lauren Hsu’s talk.) The question remains: are we comparing apples and oranges? These detectors are made of a wide variety of types of nuclei and we don’t know how to relate the results. Wick Haxton’s talk surprised me by discussion of nuclear physics uncertainties I hadn’t been aware of, that in principle could reconcile all the disagreements between experiments, even DAMA and LUX. Most people think that the experimental claims of 10 GeV dark matter are wrong, but I am taking a wait and see attitude.

We also heard about the hints of detection of a completely different dark matter candidate: sterile neutrinos. (Slides from George Fuller’s talk.) In addition to the three known neutrinos of the Standard Model of Particle Physics, there could be another one that doesn’t interact with the standard model. Yet its decay could lead to x-ray lines. Two separate groups found indications of lines in data from the Chandra and XMM-Newton space satellites that would be consistent with a 7 keV neutrino (7 millionths of a proton mass). Could it be that there is more than one type of dark matter particle? Sure, why not?

On the last evening of the meeting, a number of us went to the Baja Cantina, our favorite spot for margaritas. Rick Gaitskell was smart: he talked us into the $60.00 pitchers, high enough quality that the 6AM alarm clocks the next day (that got many of us out of bed and headed to flights leaving from LAX) didn’t kill us completely. We have such a fun community of dark matter enthusiasts. May we find the stuff soon!

We’ve returned from the lovely city of New Orleans, where within a short period of time I was able to sample shrimp and grits, bread pudding soufflé, turtle soup, chicken gumbo, soft-shelled crab with crawfish étouffée, and of course beignets. Oh yes, also participated in the Greer-Heard Forum, where I debated William Lane Craig, and then continued the discussion the next day along with Alex Rosenberg, Tim Maudlin, James Sinclair, and Robin Collins. The whole event was recorded, and will be released on the internet soon — hopefully within a couple of days.

[Update: Here is the video.]

In the meantime I thought I’d provide some quick post-debate reflections. Overall I think it went pretty well, although I certainly could have done better. Then again I’m biased, both by being hard on myself in terms of the debate performance, but understandably of the opinion that my actual ideas are correct. I think I mostly reached my primary goal of explaining why many of us think theism is undermined by modern science, and in particular why there is no support to be found for it in modern cosmology. For other perspectives see Rational Skepticism or the Reasonable Faith forums.

Clockwise from top left: William Lane Craig, Alex Rosenberg, Sean Carroll, James Sinclair, Robert Stewart (Greer-Heard organizer), Tim Maudlin, and Robin Collins. Screenshot by Maryanne Spikes.

Short version: I think it went well, although I can easily think of several ways I could have done better. On the substance, my major points were that the demand for “causes” and “explanations” is completely inappropriate for modern fundamental physics/cosmology, and that theism is not taken seriously in professional cosmological circles because it is hopelessly ill-defined (no matter what happens in the universe, you can argue that God would have wanted it that way). He defended two of his favorite arguments, the “cosmological argument” and the fine-tuning argument; no real surprises there. In terms of style, from my perspective things got a bit frustrating, because the following pattern repeated multiple times: Craig would make an argument, I would reply, and Craig would just repeat the original argument. For example, he said that Boltzmann Brains were a problem for the multiverse; I said that they were a problem for certain multiverse models but not others, which is actually good because they help us to distinguish viable from non-viable models; and his response was the multiverse was not a viable theory because of the Boltzmann Brain problem. Or, he said that if the universe began to exist there must be a transcendent cause; I said that everyday notions of causation don’t apply to the beginning of the universe and explained why they might apply approximately inside the universe but not to it; and his response was that if the universe could just pop into existence, why not bicycles? I was honestly a bit surprised at the lack of real-time interaction, since one of Craig’s supporters’ biggest complaints is that his opponents don’t ever directly respond to his points, and I tried hard to do exactly that. To be fair, I bypassed some of his arguments (see below) because I thought they were irrelevant, and wanted to focus on the important issues; he might feel differently. I’m sure that others will have their own opinions, but soon enough the videos will allow all to judge for themselves. Overall I was moderately satisfied that I made the responses I had hoped to make, clarified some points, and gave folks something to think about.

Longer version (much longer, sorry): the format was 20-minute opening talks by each speaker (Craig going first), followed by 12-minute rebuttals, and then 8-minute closing statements. Among the pre-debate advice I was given was “make it a discussion, not a debate” and “don’t let WLC speak first,” both of which I intentionally ignored. I wanted all along to play by his rules, in front of his crowd, and do the best job I could do without any excuses.

In his opening speech Craig gave two arguments: the Kalam Cosmological Argument (the universe must be caused, and the cause is God), and the teleological/fine-tuning argument (the parameters of the universe appear designed for the existence of life). To his great credit, WLC actually stuck to arguments concerning physical cosmology, where presumably my expertise would be most valuable; he didn’t hide behind primarily metaphysical arguments like the ontological argument or the denial of realized infinities. The two he used were familiar from his repertoire, and they were the two that I was primarily interested in talking about myself, so we were off. (I’ll try to reconstruct the logic rather than doing a point-by-point recap, since I’m mostly working from memory. Naturally, my memory of my own parts will be sharper than my memory of WLC’s, so I’ll happily accept factual corrections.)

The cosmological argument has two premises: (1) If the universe had a beginning, it has a transcendent cause; and (2) The universe had a beginning. He took (1) as perfectly obvious, and put his effort into establishing (2). Partly he used the celebrated (by theologians) Borde-Guth-Vilenkin theorem, which says that a universe with an average expansion rate greater than zero must be geodesically incomplete in the past. But he also used an argument I hadn’t heard before: from the Second Law of Thermodynamics (entropy in a closed system doesn’t decrease). I think the argument was basically that the Second Law implies that we approach equilibrium, and in an infinitely-old universe we should therefore have reached equilibrium long ago, which we haven’t, so the universe began at some finite time in the past.

My attitude toward the above two premises is that (2) is completely uncertain, while the “obvious” one (1) is flat-out false. Or not even false, as I put it, because the notion of a “cause” isn’t part of an appropriate vocabulary to use for discussing fundamental physics. Rather, modern physical models take the form of unbreakable patterns — laws of Nature — that persist without any external causes. The Aristotelian analysis of causes is outdated when it comes to modern fundamental physics; what matters is whether you can find a formal mathematical model that accounts for the data. The Hartle-Hawking “no-boundary proposal” for the wave function of the universe, for example, is completely self-contained, not requiring any external cause.

Mostly Craig ignored this argument, which to me was the most important part of the debate. In the first rebuttal he said that the Hartle-Hawking model was indeed lacking something — a reason why the universe exists at all. To me this looks like confusing the cosmological argument with the argument from contingency, but since my objection applied to that case as well I didn’t raise that as an rebuttal. Rather, I pointed out that this response sailed right by my actual argument, which was that a self-contained physical model is all you need, and asking for anything more is completely unwarranted. To drive the point home, I elaborated on why things like “causes” and “explanations” make perfect sense for parts of the universe, but not for the universe itself: namely, that we live in a world with unbreakable patterns (laws of physics) and an arrow of time, but the universe itself (or the multiverse) is not one element of a much bigger pattern, it’s all there is. Finally in the closing speech WLC finally offered arguments in favor of the idea that the beginning of the universe implies a transcendental cause: (1) it’s a metaphysical principle; (2) if universe could pop into existence, why not bicycles?; and (3) there’s no reason to treat the universe differently than things inside the universe. To me, (1) isn’t actually an argument, just a restatement; and I had already explained why (2) and (3) were not true, and he didn’t actually respond to my explanation. So by the time my rebuttal came around I didn’t have much more new to say. Craig spent some time mocking the very idea that the universe could just “pop into existence.” I explained that this isn’t the right way to think about these models, which are better understood as “the universe has an earliest moment of time,” which doesn’t misleadingly appeal to our intuitions of temporal sequence; but my explanation seemed to have no effect.

The second premise of the Kalam argument is that the universe began to exist. Which may even be true! But we certainly don’t know, or even have strong reasons to think one way or the other. Craig thinks we do have a strong reason, the Borde-Guth-Vilenkin theorem. So I explained what every physicist who has thought about the issue understands: that the real world is governed by quantum mechanics, and the BGV theorem assumes a classical spacetime, so it says nothing definitive about what actually happens in the universe; it is only a guideline to when our classical description breaks down. Indeed, I quoted a stronger theorem, the “Quantum Eternity Theorem” (QET) — under conventional quantum mechanics, any universe with a non-zero energy and a time-independent Hamiltonian will necessarily last forever toward both the past and the future. For convenience I quoted my own paper as a reference, although I’m surely not the first to figure it out; it’s a fairly trivial result once you think about it. (The Hartle-Hawking model is not eternal to the past, which is fine because they imagine a universe with zero energy. In that situation time is an approximation rather than fundamental in any case — that’s the “problem of time” in quantum gravity.)

Sadly, Craig never responded to my point about the QET. Instead, he emphasized another “theorem” in a paper by Aron Wall. This is a great paper, well worth reading — but it doesn’t say what Craig wants it to say, which I was only able to check after the debate. Wall (like BGV) proves theorems that apply to semiclassical gravity (classical spacetime with propagating quantum fields — see comment from Aron below), and then speculates “the results may hold in full quantum gravity” and “there is a reasonable possibility that the Penrose singularity theorem can be proven even in the context of full quantum gravity.” As good as the paper is, proving a theorem in the semiclassical case and then opining that it is probably extendable to the full quantum gravity case does not actually represent a “theorem” about the quantum case. And in fact I think it’s highly unlikely to be extendable in the sense Craig wants it to be, since the QET says that’s impossible (unless the universe has zero energy or a time-dependent Hamiltonian, in which case it’s easy to avoid eternity). But I had never seen Wall’s paper before, and Craig didn’t give a precise statement of the purported theorem, only the above quote about “reasonable possibility”; as a result I didn’t know the range of applicability of the “theorem” or its assumptions, so chose not to talk about it rather than making guesses. That was probably a strategic mistake on my part.

While I’m lingering over my mistakes, I made a related one, when Craig emphasized a recent paper by Anthony Aguirre and John Kehayias. They examined the “emergent universe” scenario of George Ellis and Roy Maartens, in which the universe is in a quasi-static pre-Big-Bang state infinitely far into the past. Aguirre and Kehayias showed that such behavior is unstable; you can’t last in a quasi-static state for half of eternity and then start evolving. Personally, I didn’t think this was worth talking about; I completely agree that it’s unstable, I never promoted or defended that particular model, and I just didn’t see the relevance. But he kept bringing it up. Only after the debate did it dawn on me that he takes the specific behavior of that model as representative of any model that has a quantum-gravity regime (the easiest way out of the “beginning” supposedly predicted by the BGV theorem). That’s completely false. Most models with a quantum-gravity phase are nothing like the emergent universe; typically the quantum part of the evolution is temporary, and is surrounded on both sides by classical spacetime. But that’s so false that I didn’t even pick up that WLC was presuming it, so I never responded. Bad debater.

The other argument from Craig in favor of the universe having a beginning comes from the fact that entropy is increasing, in accordance with the Second Law. This is another case where it took me a while to understand the point he was trying to get at. To me, it is perfectly obvious and well-understood that the Second Law comes about because of the configuration of matter in our local region of universe, not because of some ironclad fundamental law. (Otherwise Ludwig Boltzmann lived in vain — and I have his tombstone right up there on my blog header.) A theory like my model with Jennifer Chen tries to explain how the Second Law emerges in local regions of the universe, by showing how a universe with no equilibrium state can evolve forever (rather than settling down), and entropy will naturally increase both in the far past and the far future. Craig seems to think that the Second Law should be absolute, so that models like ours are ruled out because entropy doesn’t increase monotonically — i.e. they “violate” the Second Law. (Unless I’m still misunderstanding his point — his presentation was uncharacteristically muddled here.) This is a pretty straightforward misunderstanding of the origin of the Second Law and the point of our model, although to be fair I caught on too late to present a strong counterargument.

It was interesting that Craig spent so much time talking about the Carroll-Chen model, when I wouldn’t have brought it up at all if left to my own devices. I think the model is very useful as an illustration of an interesting fact: you can derive a natural dynamical origin of the Second Law in a universe that creates new entropy without bound by creating regions of space that look like our Big Bang. But I’m the first to admit that it’s speculative, and especially that the process of baby-universe creation is ill-understood, if it’s possible at all. So I probably wouldn’t have dwelt on it, but Craig really went to town on the model. Unfortunately, nearly everything he said about it was just wrong. First, he tried to claim that having a moment of time in the history of the universe when entropy was lowest counts as a “thermodynamic beginning,” even if there is more universe in both direction of time around that moment. That’s quite an innovative definition (to be polite), but more importantly that kind of “beginning” has nothing to do with the kind of “beginning” where God would create the universe. I made this point, but it wasn’t answered. Relatedly, he seemed to think it was a glaring mistake (or perhaps intentionally subterfuge?) that on one picture of our model I had the “time” axis have only one arrow, while on another version of the picture I put arrows pointing in both directions. I apologize for being sloppy, but it’s neither a mistake nor a dastardly plot; either version is acceptable, because you have a time coordinate that runs monotonically from -infinity to +infinity, but the direction of entropy increase (that defines the arrow of time) is not monotonic. Next, he tried to claim that our model violated unitarity (conservation of quantum information), which is flatly wrong. He supplied two pieces of evidence, in the form of quotes from Stephen Hawking and Chris Weaver. But the Hawking quote was completely out of context; he was talking about the fact that he no longer thought that wormholes would lead to violation of unitarity in black-hole evaporation, nothing to do with cosmology. And the Weaver quote that Craig read had nothing to do with unitarity at all; it merely pointed out that the process of baby-universe creation is speculative and not well-understood, which I’m the first to admit. Again — I didn’t actually hold up this model as a solution to anything, but he felt the need to attack it, so I had to defend its honor a little bit.

In contrast, I wanted to talk about a model developed by Anthony Aguirre and Stephen Gratton. They have a very simple and physically transparent model that (unlike my theory with Chen) imposes a low-entropy boundary condition at a mid-universe “bounce.” It’s a straightforward example of a perfectly well-defined theory that is clearly eternal, one that doesn’t have a beginning, and does so without invoking any hand-waving about quantum gravity. I challenged Craig to explain why this wasn’t a sensible example of an eternal universe, one that was in perfect accord with the BGV theorem, but he didn’t respond. It wasn’t until the talks on the following day that Craig’s teammate James Sinclair admitted that it seemed like a perfectly good model to him.

But again — my main point was not to push this or that specific model, but to argue that it’s the models that matter, not some general theorem in a regime we don’t pretend to understand. So I listed a bunch of plausible-looking eternal cosmologies. The point is not that all or some of these models is perfect; it’s that they’re eternal. So we should judge them on their merits, rather than claiming to have general arguments that there are no such things. (It’s as if WLC has a powerful general theorem against heavier-than-air flying machines, while airplanes keep buzzing overhead.) On occasion it would be as if Craig admitted that there were indeed eternal cosmologies, but they were all ruled out for various unspecified other problems. That sounds like a suspiciously far-reaching claim, but one that is hard to directly dispute without any details being presented. More importantly, it’s beside the point. Which, in case it isn’t yet clear, is that it’s the models that matter, not any general theorems. If there are some cosmological models that are eternal but have other problems, there’s no reason to stop looking for other models that are also eternal but don’t have those problems. This is a case where working scientists are quick to admit that we don’t know the answer, so we shouldn’t stop considering all legitimate possibilities.

The fine-tuning argument proceeded in a more straightforward way. Unlike the cosmological argument, where Craig presented a few twists I hadn’t heard before (though he may have used them in other debates, I don’t know), the fine-tuning presentation was pretty standard. I acknowledged that, unlike the cosmological argument that is based on outdated metaphysics, the fine-tuning argument is a respectable scientific claim: two models trying to account for some data. But I gave five reasons why it was nevertheless not a good argument for theism:

1. We don’t really know that the universe is tuned specifically for life, since we don’t know the conditions under which life is possible.
2. Fine-tuning for life would only potentially be relevant if we already accepted naturalism; God could create life under arbitrary physical conditions.
3. Apparent fine-tunings may be explained by dynamical mechanisms or improved notions of probability.
4. The multiverse is a perfectly viable naturalistic explanation.
5. If God had finely-tuned the universe for life, it would look very different indeed.

Craig didn’t respond to 2. or 3. here. To counter 1. he simply noted that other physicists disagreed with me, which again really isn’t an argument; he didn’t offer any suggestion that we actually do know the conditions under which life can and cannot form. Against 5. he invoked an argument by Robin Collins that the universe is optimized for “discoverability,” at least when we consider the known physical parameters. To me this argument is completely implausible right on the face of it, since it’s trivially easy to imagine ways to make it easier to discover the universe (just make the Higgs boson lighter!). But I knew Collins was going to give a full discussion of that argument the next day, so I saved my response until then.

In my first speech I used 5. above as a launching pad to make a bigger point: the real reason theism isn’t taken seriously is because it’s completely ill-defined. If we would presume to contemplate theism from an intellectually honest perspective, we would try to decide what kind of universe we would expect to live in if theism were true; then we would do the same for naturalism; and finally we would compare those expectations to the real world. But when we do that we find theistic expectations failing to match reality over and over again. Now, I know perfectly well (from experience as well as from cogitation) that you can never make headway with theists by claiming “If God existed, He would do X, and He doesn’t” (where X is “prevent needless suffering,” “make His existence obvious,” “reveal useful non-trivial information to us,” “spread religious messages uniformly over the world,” etc.) Because they have always thought through these, and can come up with an explanation why God would never have done that. (According to Alvin Plantinga, our world — you know, the one with the Black Death, the Holocaust, AIDS, Hurricane Katrina, and so on — is “so good that no world could be appreciably better.”) But these apologetic moves come at a price: they imply a notion of theism so flexible that it becomes completely ill-defined. That’s the real problem. Craig’s way of putting it is to suggest that God is “like the cosmic artist who wants to splash his canvas with extravagance of design.” That’s precisely why naturalism has pulled so far ahead of theism in the intellectual race to best model our world: because it plays by rules and provides real explanations for why the world is this way rather than that way.

Against the multiverse, Craig’s major argument (surprisingly) was the Boltzmann Brain problem. I say “surprisingly” because it’s such an easy argument to rebut. Sure, Boltzmann Brains are a problem — for those models with a Boltzmann Brain problem. Not all models have them! And a good modern multiverse cosmologist focuses on those models that avoid them. In this sense, the BB problem is a good thing; it helps us distinguish viable models from non-viable ones. As far as I can tell, this straightforward response was completely ignored by Craig. He just kept repeating that Boltzmann Brains were really bad things. He aimed this criticism particularly at the Carroll-Chen model, which I would say is very bad aim; it’s much less likely that BB’s are a problem in our scenario than in most other multiverse theories, since you actually produce baby universes (with potentially billions of observers) more frequently than you produce individual Boltzmann Brains. But I didn’t emphasize that point, since my goal wasn’t to defend that particular model.

As an aside, the Boltzmann Brain discussion illustrated a problem with the debate as a whole: it was too technical (and I think mostly on WLC’s side, although I deserved some of the blame). I had prepared a single slide about the topic of BB’s, but upon reflection I figured that many people wouldn’t be familiar with the term, so I replaced it with “observers as random fluctuations.” But Craig just plowed right into the technicalities, without trying to explain what the Boltzmann Brain problem really was. When my term came I gave a one-sentence definition, but in retrospect that wasn’t really sufficient, so on Saturday (where we both gave five-minute responses to a series of 40-minute talks by the four other speakers) I took a few minutes to give a more careful explanation. It’s probably an inherent flaw of the format; if a speaker takes a minute to do a bit of pedagogy rather than argumentation, the audience benefits but the speaker suffers. So the game theory tells you to be obscure, to the general detriment.

A couple of words about the rebuttal speeches in particular. Each of us pulled one “gotcha” move in the rebuttals; mine worked, his didn’t (in my opinion, of course). His referred to fine-tuning: he found a quote by me saying that the low entropy of the early universe seems finely-tuned. Which is true! But he claimed that I had said we didn’t know whether fine-tuning was real. That part is completely false. I had said we didn’t know that fine-tuning for life was real; indeed, the words “for life” were italicized on my slide. And it’s perfectly obvious that the fine-tuning of our initial entropy isn’t necessary for life; the entropy could have been fantastically larger than it actually was and life still could have arisen. (For example, you could imagine a universe with our Solar System as it essentially is, but otherwise in thermal equilibrium.)

On my part, I knew that WLC liked to glide from the BGV theorem (which says that classical spacetime description fails in the past) to the stronger statement that the universe probably had a beginning, even though the latter is not implied by the former. And his favorite weapon is to use quotes from Alex Vilenkin, one of the authors of the BGV theorem. So I talked to Alan Guth, and he was gracious enough to agree to let me take pictures of him holding up signs with his perspective: namely, that the universe probably didn’t have a beginning, and is very likely eternal. Now, why would an author of the BGV theorem say such a thing? For exactly the reasons I was giving all along: the theorem says nothing definitive about the real universe, it is only a constraint on the classical regime. What matters are models, not theorems, and different scientists will quite naturally have different opinions about which types of models are most likely to prove fruitful once we understand things better. In Vilenkin’s opinion, the best models (in terms of being well-defined and accounting for the data) are ones with a beginning. In Guth’s opinion, the best models are ones that are eternal. And they are welcome to disagree, because we don’t know the answer! Not knowing the answer is perfectly fine. What’s not fine is pretending that we do know the answer, and using that pretend-knowledge to draw premature theological conclusions. (Chatter on Twitter reveals theists scrambling to find previous examples of Guth saying the universe probably had a beginning. As far as I can tell Alan was there talking about inflation beginning, not the universe, which is completely different. But it doesn’t matter; good scientists, it turns out, will actually change their minds in response to thinking about things.)

I very much hope that I hammered these points home enough to help clarify issues in the minds of listeners/readers. But from Craig’s (lack of) reaction, and from the online discussion from his supporters, I doubt it will make any difference. He will continue to quote Vilenkin saying the universe probably had a beginning, which is fine because that’s what Vilenkin actually thinks. He will not start adding in the fact that Guth thinks the universe is probably eternal, nor will he take the even more respectable position of not relying on people’s individual opinions at all and simply admitting that we don’t have good scientific reasons to think one way or the other at the moment. But we’ll see. (And to reiterate: I think the whole discussion is enormously less important than the bigger point that a “cause” is completely unnecessary even if the universe did have a beginning.)

For my closing statement, I couldn’t think of many responses to Craig’s closing statement that wouldn’t have simply be me reiterating points from my first two speeches. So I took the opportunity to pull back a little and look at the bigger picture. Namely: we’re talking about “God and Cosmology,” but nobody really becomes a believer in God because it provides the best cosmology. They become theists for other reasons, and the cosmology comes later. That’s because religion is enormously more than theism. Most people become religious for other (non-epistemic) reasons: it provides meaning and purpose, or a sense of community, or a way to be in contact with something transcendent, or simply because it’s an important part of their culture. The problem is that theism, while not identical to religion, forms its basis, at least in most Western religions. So — maybe, I suggested, tentatively — that could change. I give theists a hard time for not accepting the implications of modern science, but I am also happy to give naturalists a hard time when they don’t appreciate the enormous task we face in answering all of the questions that we used to think were answered by God. We don’t have final answers to the deep questions of meaning and fulfillment and what it means to lead a good life. Religion doesn’t have the final answers, either; but maybe it has learned something interesting over the course of thousands of years of thinking about these issues. Maybe there is some wisdom to be mined from religious traditions, even for naturalists (which everyone should be).

More than once of the course of the weekend I spoke conciliatory-sounding words about how we’re really all in this together, theists and naturalists, trying to understand the deep questions in a confusing world. And I meant all of it, in complete sincerity. I will be absolutely uncompromising about what I think the truth is concerning questions of substance; but I don’t ever want to start thinking of people who disagree with me about those questions as my enemies. Many times in New Orleans, people on “the other side” came up after my presentations (in which I said that their most deeply held beliefs had been definitively refuted by the progress of modern science) and thanked me, saying how much they appreciated my ideas and that I had given them something to think about, even if they remained quite resolute in their beliefs. Not every single person was so gracious, but the vast majority. I admire those folks, and I hope I can be as sincere and open when people who I disagree with are speaking in good faith.

To me, Craig’s best moment of the weekend came at the very end, as part of the summary panel discussion. Earlier in the day, Tim Maudlin (who gave an great pro-naturalism talk, explaining that God’s existence wouldn’t have any moral consequences even if it were true) had grumped a little bit about the format. His point was that formal point-counterpoint debates aren’t really the way philosophy is done, which would be closer to a Socratic discussion where issues can be clarified and extended more efficiently. And I agree with that, as far as it goes. But Craig had a robust response, which I also agree with: yes, a debate like this isn’t how philosophy is done, but there are things worth doing other than philosophy, or even teaching philosophy. He said, candidly, that the advantage of the debate format is that it brings out audiences, who find a bit of give-and-take more exciting than a lecture or series of lectures. It’s hard to teach subtle and tricky concepts in such a format, but that’s always a hard thing to do; the point is that if you get the audience there in the first place, a good debater can at least plant a few new ideas in their heads, and hopefully inspire them to take the initiative and learn more on their own.

I completely agree. Events like this are valuable, not because they are efficient ways to find the truth, nor even because there is any reasonable chance of changing the minds of people who are relatively secure in their beliefs (on either side). It’s because there are a lot of people who are not secure in their beliefs, or at least are curious and willing to listen to a variety of ideas. If we think we have good ideas, we should do everything we can to bring them to as many people as possible. I think science and naturalism include some pretty awesome ideas, and I’m happy to share them with as many different people as I can.

p.s. Sorry I didn’t talk about Saturday’s talks, in which Tim Maudlin discussed the relationship between theism and morality (claiming there isn’t any), Alex Rosenberg drew connections between thermodynamics and natural selection to argue that theism is incompatible with Darwinism, Robin Collins argued that the discoverability of the universe is evidence for theism, and James Sinclair talked about notions of time and the origin of the universe. But I’m kind of all talked out on this topic for right now.

Not that long ago, SteelyKid was doing something violent with toys (she’s very tough, as you can see from the featured image above), and in the process made the canonical falling-bomb whistle noise. And it occurred to me to wonder, why that sound?

I mean, I’ve seen footage of falling bombs and the canonical sound seems like an accurate enough. But I’m not sure I understand it from a physics perspective. Specifically, when you do the falling-bomb whistle, you do a whistle that decreases in pitch. But the bomb is falling faster as it approaches the ground, so if anything, I would expect the pitch to go up– if the sound is made by some sort of turbulence effect, you would guess that the frequency should increase as the speed goes up. (If it’s a deliberate effect from some sort of whistling thing, on the other hand, the pitch might be fairly constant.)

So, why is that? Is it a Doppler effect thing? (As you know, even if your name’s not Bob, the Doppler effect is a phenomenon involving waves from a moving source, that shifts the frequency an observer detects up when the source and observer are moving together, and down when they’re moving apart. This is why a police siren sounds higher in pitch as it comes toward you, and lower as it goes away.)

Well, if you dropped a bomb from a height of 1000m (just to have round numbers), the speed when it hit the ground in the absence of air resistance would be a bit less than half the speed of sound, which would give a comparable decrease in the frequency heard by the pilot of the plane that dropped the bomb (the Doppler shift of a sound at a given frequency is equal to the frequency at the source multiplied by the speed of the source divided by the speed of sound). Obviously, a real bomb would experience air resistance (it couldn’t make sound otherwise), so the actual shift would be less, and would eventually reach a constant pitch as the bomb reached terminal velocity. This gives a back-of-the-envelope (well, margin-of-the-class-notebook, because that’s where I actually did it) sense of the scale of the Doppler effect you could expect, and is basically consistent with the canonical whistle sound.

But if it’s a Doppler thing, the change in pitch should depend on the frame of reference. That is, the people in the plane would hear it go down, but people on the ground would hear it go up. But nobody ever does the falling-bomb noise in the reverse direction. And I have no personal war-zone experience to draw on, here– maybe I should ask my Senior Middle East Correspondent, who came under mortar fire in Libya a while back…

I suppose the always-decrease-in-pitch thing could just be psychology– if you’re playing war, it’s much more comfortable to put yourself in the position of the person dropping the bombs than the person being bombed.

On thinking about it a bit, though, you might be able to get a decrease in pitch from a falling bomb by being off to the side, because the velocity that matters is the velocity along the line of sight. So, if you were standing 1km away from a bomb falling from a 1km height, the initial downward velocity could be broken down into components that are toward you, and perpendicular to you, as shown in the figure– the red arrow is along the line-of sight, the orange perpendicular to it. Each component would be 70% of the initial speed, so that would give an upward Doppler shift of 70% of the maximum, as the Doppler shift is due only to the line-of-sight component.

Diagram of a falling bomb, showing the components of velocity.

Just before the bomb hits, though, the velocity would be nearly perpendicular to the line of sight, so the Doppler shift would be zero (to first order). Which would be a decrease in pitch from the starting condition. So maybe somebody on the ground would hear a decreasing pitch after all, at least provided they’re a fair distance from the impact point.

Of course, if the bomb was released from rest, it would be speeding up, and the increasing shift from that might be bigger than the angle effect. But once the bomb reached its terminal velocity, the angle effect would give you a reduction in frequency, in the same way that you get the characteristic “eeeeeeee-owwwwwww” effect as a fast car goes by– the Doppler shift moves the frequency up as it comes toward you, is zero just as it passes, and then shifts the frequency down as it recedes.

Which of these effects would win out is difficult to say. This, of course, demands some sort of VPython simulation, but I have too much other stuff to do this morning– I’ve already spent an order of magnitude more time on this post than I intended to…

And, of course (to head off an inevitable comment) I’m sure this is Google-able, but I offer it here as an example of the thought processes of a physicist confronted with an anomalous result. And also an example of the weird stuff that distracts me when I’m playing with the kids…

In my last post, I expressed the view that a particle accelerator with proton-proton collisions of (roughly) 100 TeV of energy, significantly more powerful than the currently operational Large Hadron Collider [LHC] that helped scientists discover the Higgs particle, is an obvious and important next steps in our process of learning about the elementary workings of nature. And I described how we don’t yet know whether it will be an exploratory machine or a machine with a clear scientific target; it will depend on what the LHC does or does not discover over the coming few years.

What will it mean, for the 100 TeV collider project and more generally, if the LHC, having made possible the discovery of the Higgs particle, provides us with no more clues?  Specifically, over the next few years, hundreds of tests of the Standard Model (the equations that govern the known particles and forces) will be carried out in measurements made by the ATLAS, CMS and LHCb experiments at the LHC. Suppose that, as it has so far, the Standard Model passes every test that the experiments carry out? In particular, suppose the Higgs particle discovered in 2012 appears, after a few more years of intensive study, to be, as far the LHC can reveal, a Standard Model Higgs — the simplest possible type of Higgs particle?

Before we go any further, let’s keep in mind that we already know that the Standard Model isn’t all there is to nature. The Standard Model does not provide a consistent theory of gravity, nor does it explain neutrino masses, dark matter or “dark energy” (also known as the cosmological constant). Moreover, many of its features are just things we have to accept without explanation, such as the strengths of the forces, the existence of “three generations” (i.e., that there are two heavier cousins of the electron, two for the up quark and two for the down quark), the values of the masses of the various particles, etc. However, even though the Standard Model has its limitations, it is possible that everything that can actually be measured at the LHC — which cannot measure neutrino masses or directly observe dark matter or dark energy — will be well-described by the Standard Model. What if this is the case?

Michelson and Morley, and What They Discovered

In science, giving strong evidence that something isn’t there can be as important as discovering something that is there — and it’s often harder to do, because you have to thoroughly exclude all possibilities. [It's very hard to show that your lost keys are nowhere in the house --- you have to convince yourself that you looked everywhere.] A famous example is the case of Albert Michelson, in his two experiments (one in 1881, a second with Edward Morley in 1887) trying to detect the “ether wind”.

Light had been shown to be a wave in the 1800s; and like all waves known at the time, it was assumed to be a wave in something material, just as sound waves are waves in air, and ocean waves are waves in water. This material was termed the “luminiferous ether”. As we can detect our motion through air or through water in various ways, it seemed that it should be possible to detect our motion through the ether, specifically by looking for the possibility that light traveling in different directions travels at slightly different speeds.  This is what Michelson and Morley were trying to do: detect the movement of the Earth through the luminiferous ether.

Both of Michelson’s measurements failed to detect any ether wind, and did so expertly and convincingly. And for the convincing method that he invented — an experimental device called an interferometer, which had many other uses too — Michelson won the Nobel Prize in 1907. Meanwhile the failure to detect the ether drove both FitzGerald and Lorentz to consider radical new ideas about how matter might be deformed as it moves through the ether. Although these ideas weren’t right, they were important steps that Einstein was able to re-purpose, even more radically, in his 1905 equations of special relativity.

In Michelson’s case, the failure to discover the ether was itself a discovery, recognized only in retrospect: a discovery that the ether did not exist. (Or, if you’d like to say that it does exist, which some people do, then what was discovered is that the ether is utterly unlike any normal material substance in which waves are observed; no matter how fast or in what direction you are moving relative to me, both of us are at rest relative to the ether.) So one must not be too quick to assume that a lack of discovery is actually a step backwards; it may actually be a huge step forward.

Epicycles or a Revolution?

There were various attempts to make sense of Michelson and Morley’s experiment.   Some interpretations involved  tweaks of the notion of the ether.  Tweaks of this type, in which some original idea (here, the ether) is retained, but adjusted somehow to explain the data, are often referred to as “epicycles” by scientists.   (This is analogous to the way an epicycle was used by Ptolemy to explain the complex motions of the planets in the sky, in order to retain an earth-centered universe; the sun-centered solar system requires no such epicycles.) A tweak of this sort could have been the right direction to explain Michelson and Morley’s data, but as it turned out, it was not. Instead, the non-detection of the ether wind required something more dramatic — for it turned out that waves of light, though at first glance very similar to other types of waves, were in fact extraordinarily different. There simply was no ether wind for Michelson and Morley to detect.

If the LHC discovers nothing beyond the Standard Model, we will face what I see as a similar mystery.  As I explained here, the Standard Model, with no other particles added to it, is a consistent but extraordinarily “unnatural” (i.e. extremely non-generic) example of a quantum field theory.  This is a big deal. Just as nineteenth-century physicists deeply understood both the theory of waves and many specific examples of waves in nature  and had excellent reasons to expect a detectable ether, twenty-first century physicists understand quantum field theory and naturalness both from the theoretical point of view and from many examples in nature, and have very good reasons to expect particle physics to be described by a natural theory.  (Our examples come both from condensed matter physics [e.g. metals, magnets, fluids, etc.] and from particle physics [e.g. the physics of hadrons].) Extremely unnatural systems — that is, physical systems described by quantum field theories that are highly non-generic — simply have not previously turned up in nature… which is just as we would expect from our theoretical understanding.

[Experts: As I emphasized in my Santa Barbara talk last week, appealing to anthropic arguments about the hierarchy between gravity and the other forces does not allow you to escape from the naturalness problem.]

So what might it mean if an unnatural quantum field theory describes all of the measurements at the LHC? It may mean that our understanding of particle physics requires an epicyclic change — a tweak.  The implications of a tweak would potentially be minor. A tweak might only require us to keep doing what we’re doing, exploring in the same direction but a little further, working a little harder — i.e. to keep colliding protons together, but go up in collision energy a bit more, from the LHC to the 100 TeV collider. For instance, perhaps the Standard Model is supplemented by additional particles that, rather than having masses that put them within reach of the LHC, as would inevitably be the case in a natural extension of the Standard Model (here’s an example), are just a little bit heavier than expected. In this case the world would be somewhat unnatural, but not too much, perhaps through some relatively minor accident of nature; and a 100 TeV collider would have enough energy per collision to discover and reveal the nature of these particles.

Or perhaps a tweak is entirely the wrong idea, and instead our understanding is fundamentally amiss. Perhaps another Einstein will be needed to radically reshape the way we think about what we know.  A dramatic rethink is both more exciting and more disturbing. It was an intellectual challenge for 19th century physicists to imagine, from the result of the Michelson-Morley experiment, that key clues to its explanation would be found in seeking violations of Newton’s equations for how energy and momentum depend on velocity. (The first experiments on this issue were carried out in 1901, but definitive experiments took another 15 years.) It was an even greater challenge to envision that the already-known unexplained shift in the orbit of Mercury would also be related to the Michelson-Morley (non)-discovery, as Einstein, in trying to adjust Newton’s gravity to make it consistent with the theory of special relativity, showed in 1913.

My point is that the experiments that were needed to properly interpret Michelson-Morley’s result

• did not involve trying to detect motion through the ether,
• did not involve building even more powerful and accurate interferometers,
• and were not immediately obvious to the practitioners in 1888.

This should give us pause. We might, if we continue as we are, be heading in the wrong direction.

Difficult as it is to do, we have to take seriously the possibility that if (and remember this is still a very big “if”) the LHC finds only what is predicted by the Standard Model, the reason may involve a significant reorganization of our knowledge, perhaps even as great as relativity’s re-making of our concepts of space and time. Were that the case, it is possible that higher-energy colliders would tell us nothing, and give us no clues at all. An exploratory 100 TeV collider is not guaranteed to reveal secrets of nature, any more than a better version of Michelson-Morley’s interferometer would have been guaranteed to do so. It may be that a completely different direction of exploration, including directions that currently would seem silly or pointless, will be necessary.

This is not to say that a 100 TeV collider isn’t needed!  It might be that all we need is a tweak of our current understanding, and then such a machine is exactly what we need, and will be the only way to resolve the current mysteries.  Or it might be that the 100 TeV machine is just what we need to learn something revolutionary.  But we also need to be looking for other lines of investigation, perhaps ones that today would sound unrelated to particle physics, or even unrelated to any known fundamental question about nature.

Let me provide one example from recent history — one which did not lead to a discovery, but still illustrates that this is not all about 19th century history.

An Example

One of the great contributions to science of Nima Arkani-Hamed, Savas Dimopoulos and Gia Dvali was to observe (in a 1998 paper I’ll refer to as ADD, after the authors’ initials) that no one had ever excluded the possibility that we, and all the particles from which we’re made, can move around freely in three spatial dimensions, but are stuck (as it were) as though to the corner edge of a thin rod — a rod as much as one millimeter wide, into which only gravitational fields (but not, for example, electric fields or magnetic fields) may penetrate.  Moreover, they emphasized that the presence of these extra dimensions might explain why gravity is so much weaker than the other known forces.

Fig. 1: ADD’s paper pointed out that no experiment as of 1998 could yet rule out the possibility that our familiar three-dimensional world is a corner of a five-dimensional world, where the two extra dimensions are finite but perhaps as large as a millimeter.

Given the incredible number of experiments over the past two centuries that have probed distances vastly smaller than a millimeter, the claim that there could exist millimeter-sized unknown dimensions was amazing, and came as a tremendous shock — certainly to me. At first, I simply didn’t believe that the ADD paper could be right.  But it was.

One of the most important immediate effects of the ADD paper was to generate a strong motivation for a new class of experiments that could be done, rather inexpensively, on the top of a table. If the world were as they imagined it might be, then Newton’s (and Einstein’s) law for gravity, which states that the force between two stationary objects depends on the distance r between them as 1/r², would increase faster than this at distances shorter than the width of the rod in Figure 1.  This is illustrated in Figure 2.

Fig. 2: If the world were as sketched in Figure 1, then Newton/Einstein’s law of gravity would be violated at distances shorter than the width of the rod in Figure 1. The blue line shows Newton/Einstein’s prediction; the red line shows what a universe like that in Figure 1 would predict instead. Experiments done in the last few years agree with the blue curve down to a small fraction of a millimeter.

These experiments are not easy — gravity is very, very weak compared to electrical forces, and lots of electrical effects can show up at very short distances and have to be cleverly avoided. But some of the best experimentalists in the world figured out how to do it (see here and here). After the experiments were done, Newton/Einstein’s law was verified down to a few hundredths of a millimeter.  If we live on the corner of a rod, as in Figure 1, it’s much, much smaller than a millimeter in width.

But it could have been true. And if it had, it might not have been discovered by a huge particle accelerator. It might have been discovered in these small inexpensive experiments that could have been performed years earlier. The experiments weren’t carried out earlier mainly because no one had pointed out quite how important they could be.

Ok Fine; What Other Experiments Should We Do?

So what are the non-obvious experiments we should be doing now or in the near future?  Well, if I had a really good suggestion for a new class of experiments, I would tell you — or rather, I would write about it in a scientific paper. (Actually, I do know of an important class of measurements, and I have written a scientific paper about them; but these are measurements to be done at the LHC, and don’t involve a entirely new experiment.)  Although I’m thinking about these things, I do not yet have any good ideas.  Until I do, or someone else does, this is all just talk — and talk does not impress physicists.

Indeed, you might object that my remarks in this post have been almost without content, and possibly without merit.  I agree with that objection.

Still, I have some reasons for making these points. In part, I want to highlight, for a wide audience, the possible historic importance of what might now be happening in particle physics. And I especially want to draw the attention of young people. There have been experts in my field who have written that non-discoveries at the LHC constitute a “nightmare scenario” for particle physics… that there might be nothing for particle physicists to do for a long time. But I want to point out that on the contrary, not only may it not be a nightmare, it might actually represent an extraordinary opportunity. Not discovering the ether opened people’s minds, and eventually opened the door for Einstein to walk through. And if the LHC shows us that particle physics is not described by a natural quantum field theory, it may, similarly, open the door for a young person to show us that our understanding of quantum field theory and naturalness, while as intelligent and sensible and precise as the 19th century understanding of waves, does not apply unaltered to particle physics, and must be significantly revised.

Of course the LHC is still a young machine, and it may still permit additional major discoveries, rendering everything I’ve said here moot. But young people entering the field, or soon to enter it, should not assume that the experts necessarily understand where the field’s future lies. Like FitzGerald and Lorentz, even the most brilliant and creative among us might be suffering from our own hard-won and well-established assumptions, and we might soon need the vision of a brilliant young genius — perhaps a theorist with a clever set of equations, or perhaps an experimentalist with a clever new question and a clever measurement to answer it — to set us straight, and put us onto the right path.

As in the previous post, all computations here are at the formal level only.

In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with Lagrangian space and Eulerian space , we let be the space of volume-preserving, orientation-preserving maps from Lagrangian space to Eulerian space. Given a curve , we can define the Lagrangian velocity field as the time derivative of , and the Eulerian velocity field . The volume-preserving nature of ensures that is a divergence-free vector field:

as we shall see below the fold, critical points of this functional (with appropriate boundary conditions) obey the generalised Euler equations

where the stream function is given by the formula

If we take the curl of the generalised Euler equation (2), we obtain (after some computation) the surface quasi-geostrophic equation

This equation has strong analogies with the three-dimensional incompressible Euler equations, and can be viewed as a simplified model for that system; see this paper of Constantin, Majda, and Tabak for details.

Now we can specialise the general conservation laws derived previously to this setting. The conserved Hamiltonian is

(a law previously observed for this equation in the abovementioned paper of Constantin, Majda, and Tabak). As commutes with rigid motions, we also have (formally, at least) conservation of momentum

(which up to trivial transformations is also expressible in impulse form as , after integration by parts), and conservation of angular momentum

(which up to trivial transformations is ). Finally, diffeomorphism invariance gives pointwise conservation of Lagrangian vorticity , thus is transported by the flow (which is also evident from (3). In particular, all integrals of the form for a fixed function are conserved by the flow.

— 1. Euler-Lagrange calculations —

We now justify the claim that stationary points of the functional obey (2). We consider continuous deformations of the critical point , thus now depends on both and . We already have the Eulerian velocity field , which is related to the derivative of by the formula

similarly we may introduce a deformation field by

The vector field is divergence free and has to obey appropriate vanishing conditions at infinity, but is otherwise unconstrained. If we compute using the above two equations and the chain rule, we arrive at the “zero-curvature” condition

so in particular

On the other hand, as is a critical point, we have

Inserting (4) and integrating by parts (and using the divergence-free nature of ), this expression can be rewritten as

Since is essentially an arbitrary divergence-free vector field, the expression inside parentheses must vanish, and the equation (2) follows.

In the morning, Andrea Maccio (MPIA) gave a nice talk about making star-formation and AGN feedback in numerical simulations of galaxy formation much more realistic. He is very negative about the possibility that AGN can stop star formation: AGN emit jets, which punch through the central part of the ISM and don't really heat all of the necessary volume. He also showed that the the dark-energy equation of state can affect galaxy evolution, not because the dark energy has any direct effect on how structures form, but because it changes the timing of gravitational collapse and star-formation episodes. That got the audience all in a tizzy: Can we infer the dark-energy equation of state from the radial distribution of stars in a galaxy? The answer is no: It looks like this effect is strongly degenerate with feedback parameters, but it is super-intriguing.

Late in the day, Foreman-Mackey and I checked in with Dawson about the in-transit noise. She has some systems that show a very strong effect of higher noise in transit than out. We suggested improvements to the statistical tests, and Dawson will try to move to smaller and smaller planets tomorrow.

This morning was a "hack session" in the Josh-Peek-organized reading group on astrostatistics up at Columbia called #NYCastroML. I helped Kelle Cruz (CUNY) and others build a mixture-of-Gaussians model of the WISE point-source catalog. Well, we just worked on single-Gaussian models, but we are getting ready to do multiple Gaussians. We got a long way, although, as usual, most of the session was really "data munging" and not "data analysis". That's not uncommon: Getting the data into a consistent, useful, checked state is often the hardest part of the project. And, as far as I know, there is no "theory" of this part of data analysis. It just is.

Bekki Dawson (Berkeley) showed up for a few days, in which she will work with Foreman-Mackey and Angus and me and also give our astro seminar. We discussed her observation (not new) that the photometric noise in a star light curve is often higher in transit than out of transit. This is explained by there being strong surface features on the star that get selectively blocked by the planet but affect the out-of-transit lightcurve only in an integrated way. Dawson's concern is that this excess noise is not being incorporated in completeness and sensitivity tests with the Kepler data; that is, we might be being over-optimistic about our small-planet samples. We made a plan to test this, and, if necessary, make more realistic artificial planet injections for better completeness and sensitivity studies.

Been falling behind on my favorite thing to do on the blog: post summaries of my own research papers. Back in October I submitted a paper with two Caltech colleagues, postdoc Stefan Leichenauer and grad student Jason Pollack, on the intriguing intersection of effective field theory (EFT) and cosmological large-scale structure (LSS). Now’s a good time to bring it up, as there’s a great popular-level discussion of the idea by Natalie Wolchover in Quanta.

So what is the connection between EFT and LSS? An effective field theory, as loyal readers know, an “effective field theory” is a way to describe what happens at low energies (or, equivalently, long wavelengths) without having a complete picture of what’s going on at higher energies. In particle physics, we can calculate processes in the Standard Model perfectly well without having a complete picture of grand unification or quantum gravity. It’s not that higher energies are unimportant, it’s just that all of their effects on low-energy physics can be summed up in their contributions to just a handful of measurable parameters.

In cosmology, we consider the evolution of LSS from tiny perturbations at early times to the splendor of galaxies and clusters that we see today. It’s really a story of particles — photons, atoms, dark matter particles — more than a field theory (although of course there’s an even deeper description in which everything is a field theory, but that’s far removed from cosmology). So the right tool is the Boltzmann equation — not the entropy formula that appears on his tombstone, but the equation that tells us how a distribution of particles evolves in phase space. However, the number of particles in the universe is very large indeed, so it’s the most obvious thing in the world to make an approximation by “smoothing” the particle distribution into an effective fluid. That fluid has a density and a velocity, but also has parameters like an effective speed of sound and viscosity. As Leonardo Senatore, one of the pioneers of this approach, says in Quanta, the viscosity of the universe is approximately equal to that of chocolate syrup.

So the goal of the EFT of LSS program (which is still in its infancy, although there is an important prehistory) is to derive the correct theory of the effective cosmological fluid. That is, to determine how all of the complicated churning dynamics at the scales of galaxies and clusters feeds back onto what happens at larger distances where things are relatively smooth and well-behaved. It turns out that this is more than a fun thing for theorists to spend their time with; getting the EFT right lets us describe what happens even at some length scales that are formally “nonlinear,” and therefore would conventionally be thought of as inaccessible to anything but numerical simulations. I really think it’s the way forward for comparing theoretical predictions to the wave of precision data we are blessed with in cosmology.

Here is the abstract for the paper I wrote with Stefan and Jason:

A Consistent Effective Theory of Long-Wavelength Cosmological Perturbations
Sean M. Carroll, Stefan Leichenauer, Jason Pollack

Effective field theory provides a perturbative framework to study the evolution of cosmological large-scale structure. We investigate the underpinnings of this approach, and suggest new ways to compute correlation functions of cosmological observables. We find that, in contrast with quantum field theory, the appropriate effective theory of classical cosmological perturbations involves interactions that are nonlocal in time. We describe an alternative to the usual approach of smoothing the perturbations, based on a path-integral formulation of the renormalization group equations. This technique allows for improved handling of short-distance modes that are perturbatively generated by long-distance interactions.

As useful as the EFT of LSS approach is, our own contribution is mostly on the formalism side of things. (You will search in vain for any nice plots comparing predictions to data in our paper — but do check out the references.) We try to be especially careful in establishing the foundations of the approach, and along the way we show that it’s not really a “field” theory in the conventional sense, as there are interactions that are nonlocal in time (a result also found by Carrasco, Foreman, Green, and Senatore). This is a formal worry, but doesn’t necessarily mean that the theory is badly behaved; one just has to work a bit to understand the time-dependence of the effective coupling constants.

Here is a video from a physics colloquium I gave at NYU on our paper. A colloquium is intermediate in level between a public talk and a technical seminar, so there are some heavy equations at the end but the beginning is pretty motivational. Enjoy!

Given the academic circles I run in, it’s not surprising that one of the most repeated stories crossing my social media feeds yesterday had to do with the changes to the SAT. Starting in 2015, the essay section will no longer be mandatory, and they’re going to reconfigure the reading and math sections to emphasize different categories of questions.

My slightly cynical take on this is that changes seem to be driven more by marketing than education– stories about this all mention that the changes make the new SAT more like the ACT, which has been gaining in popularity in recent years. Which cuts into the income of ETS, so they’re re-arranging things to keep the cash flow coming. There’s a lot of buzzwordplay about this– “critical thinking” and so on– but most of it sounds like the sort of stuff you come up with after the fact, to justify something you’ve already decided to do for more concrete reasons. But then, I might be a little biased, because the SAT’s essay section was responsible for my first taste of Internet fame, back when Dave Munger and I did the Blogger SAT Challenge in 2006.

One interesting (to me, anyway, and it’s my blog…) spin-off of this was a Twitter exchange with a former student, who tweeted “Memorizing big words is not a useful skill.” I don’t disagree with that, but it illustrates the big problem with the SAT’s, and massive standardized tests in general. And also maybe a bit of a shift in the way we approach language.

What I mean by that is that I think the vocabulary section was never intended to be a memorization test, but a proxy test for wide reading. I say this mostly because while I did very well on the Verbal portion of the SAT back in the day (780, and this was in 1988 before they started “recentering” things), I never spent time memorizing anything in preparation for the test. I didn’t really need to, because I spent a lot of time reading. A lot. Mostly genre fiction by that point, but really, just about anything with words in a row.

And the thing is, if you do that, you pick up a bunch of vocabulary more or less incidentally, and also the ability to figure out approximate meanings from context and other clues. Which is more or less what the SAT vocabulary section was testing– “Here are some uncommon words, can you figure out what they mean?”

I think the original idea of the SAT vocabulary questions was to pick up those students who did a lot of reading, on the theory that doing a lot of reading (and thus acquiring a large vocabulary) was a good proxy for “scholastic aptitude.” The problem, of course, is that this can be gamed. If you don’t do a lot of reading, you can still do well on the test provided you’re willing to devote a bunch of time to memorizing the definitions of obscure words. Which is dumb and pointless, but probably takes less time than getting a large vocabulary the “right” way. Which makes it an attractive option for the indolent children of the idle rich, contributing to the birth of the modern test-prep industry.

So, to the extent that, over time, this has become more of a memorization test than a “reading widely” test, it’s probably good to dump the vocabulary section. But the whole business remains in a sort of Red Queen race condition– whatever they replace the vocabulary questions with will quickly be gamed in the same way that the vocabulary section was. There’s just too much money around for it not to– students will flounder with the new test for a year or two, but five years from now, high-schoolers will be doing something just as academically pointless as memorizing lists of big word, and ETS will start looking at new changes.

The fundamental problem, of course, is one of scale. The SAT is taken by millions of students at a time, and needs to be graded quickly and consistently. Which requires machine-gradable multiple-choice questions or the functional equivalent thereof– even the essays get looked at for only a couple of minutes apiece, because that’s the only way to power through that many tests.

This constraint opens the door for gaming the tests, and is probably part of why the scores are only weakly correlated with future college performance. Students can do very well on the SAT via a lot of strategies that exploit weaknesses introduced by the scale, but when they get to college and their work gets more individual scrutiny, the weaknesses hidden by the bubble-sheet tests get exposed.

So, what’s the “shift in the way we approach language” bit? Another thing crossing my social media feeds was a post on Facebook from a scientist who had written an article that was (in the author’s opinion) excessively edited, asking whether to argue over the changes. Responses were sort of split, with some people saying that it couldn’t hurt, while others said it was futile. A few of the latter camp suggested that the editing was probably necessary to remove jargon terms.

One of the words cited as possible jargon was “herbivory,” as in “behaving like a herbivore.” Which made me sort of scratch my head, because that doesn’t seem like an especially challenging word, particularly for an educated audience. And this was close on the heels of my “What Is Color?” post, which got a bunch of comments about my use of “disturbance” and “spectrometer.” Which, again, I was a little surprised by– “disturbance” doesn’t seem like that hard a word (I wouldn’t hesitate to use it with SteelyKid, and she’s half the age of the target audience), and “spectrometer” is defined in the video.

And this seems like something where the expectations have changed a lot. In the process of researching the book-in-progress, I’ve read a lot of stuff about the history of science. Most of this is of recent vintage, but some of the stuff I read was basically popular science writing from the 1940′s and 50′s. And it’s striking how much more the writers of that time expect from their audience, in an SAT vocabulary sort of sense– they throw out moderately obscure words, and expect readers to roll with them. A lot of the stuff I was reading seemed like it would be flagged for excessive jargon by the folks who regularly opine on such things in the science-communication social media I follow.

There’s a sense, of course, in which this is a good thing–the target audience these days is much broader and more diverse in a socioeconomic kind of sense. A lot of those authors also expect their readers to get allusions to Classical mythology, which probably wasn’t elementary-school stuff in the 40′s, either. I don’t think it’s a bad thing to move beyond aiming pop-science writing at folks who have already been to Oxford.

But at the same time, the collision of these two strands of my social-media universe makes me wonder if we haven’t capitulated to the SAT-prep vision of the world. That is, we’ve internalized the idea that acquisition of vocabulary is something that happens through pointless memorization of lists of words, rather than as an organic part of the reading process. We seem to be much less accepting of writing that expects something of the reader, instead driving everything toward the Up Goer Five common denominator. And that, I think, is not a positive development, from either a science literacy or a college prep standpoint.

I’ve put up a web page for How Not To Be Wrong; the reason it looks good is that I was smart enough to hire Will Amato instead of trying to do it myself.  It goes THWOCK!  But if you look at it on a phone you won’t get to hear it go thwock because thwock is Flash.

If you think static web pages are dead and believe in the glorious social future, go like my author page on Facebook, and you’ll get news about reviews, events, media, etc.

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection of one or more fields on a domain taking values in a space will solve the differential equation of interest if and only if is a critical point to the functional

involving the fields and their first derivatives , where the Lagrangian is a function on the vector bundle over consisting of triples with , , and a linear transformation; we also usually keep the boundary data of fixed in case has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on and , which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise as and as , in which case can be viewed locally as a function on .

Example 1 (Geodesic flow) Take and to be a Riemannian manifold, which we will write locally in coordinates as with metric for . A geodesic is then a critical point (keeping fixed) of the energy functional

or in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work really formally, as the manifold is now infinite dimensional).

More generally, if is itself a Riemannian manifold, which we write locally in coordinates as with metric for , then a harmonic map is a critical point of the energy functional

or in coordinates (again ignoring coordinate patch issues)

If we replace the Riemannian manifold by a Lorentzian manifold, such as Minkowski space , then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ).

Example 2 (-particle interactions) Take and ; then a function can be interpreted as a collection of trajectories in space, which we give a physical interpretation as the trajectories of particles. If we assign each particle a positive mass , and also introduce a potential energy function , then it turns out that Newton’s laws of motion in this context (with the force on the particle being given by the conservative force ) are equivalent to the trajectories being a critical point of the action functional

whenever is a (smooth) deformation with (and with respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

Write for the infinitesimal deformation of . By the chain rule, can be expressed in terms of . In coordinates, we have

where we parameterise by , and we use subscripts on to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of , and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves , where vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form for some class of functions that vanishes on the boundary, then there must be an associated differential identity that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand should be expressible as a spatial divergence. This is indeed the case:

Proposition 1 (Formal) Let be a critical point of the functional defined in (1). Then for any deformation with , we have

where is the vector field that is expressible in coordinates as

Proof: Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

The same computation, together with an integration by parts, shows that (2) may be rewritten as

Since is unconstrained on the interior of , the claim (6) follows (at a formal level, at least).

Many variational problems also enjoy one-parameter continuous symmetries: given any field (not necessarily a critical point), one can place that field in a one-parameter family with , such that

for all ; in particular,

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

whenever arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain is the Euclidean space , and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable , thus

then we obtain translation symmetries

for , where is the standard basis for . For a fixed , the left-hand side of (7) then becomes

where . Another common type of symmetry is a pointwise symmetry, in which

for all , in which case (7) clearly holds with .

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem) Suppose that is a critical point of the functional (1), and let be a one-parameter continuous symmetry with . Let be the vector field in (5), and let be the vector field in (7). Then we have the pointwise conservation law

In particular, for one-dimensional variational problems, in which , we have the conservation law for all (assuming of course that is connected and contains ).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if and the Lagrangian has no explicit time dependence, thus

then by using the time translation symmetry , we have

as discussed previously, whereas we have , and hence by (5)

and so Noether’s theorem gives conservation of the Hamiltonian

For instance, for geodesic flow, the Hamiltonian works out to be

For pointwise symmetries (9), vanishes, and so Noether’s theorem simplifies to ; in the one-dimensional case , we thus see from (5) that the quantity

is conserved in time. For instance, for the -particle system in Example 2, if we have the translation invariance

for all , then we have the pointwise translation symmetry

for all , and some , in which case , and the conserved quantity (11) becomes

as was arbitrary, this establishes conservation of the total momentum

Similarly, if we have the rotation invariance

for any and , then we have the pointwise rotation symmetry

for any skew-symmetric real matrix , in which case , and the conserved quantity (11) becomes

since is an arbitrary skew-symmetric matrix, this establishes conservation of the total angular momentum

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

— 1. Euler’s equations —

The geometric setup for the geodesic interpretation of Euler’s equations of fluid flow is as follows. We will need two copies and of Euclidean space , with two different structures. Firstly, we will have Lagrangian space

which is (viewed as a smooth manifold), together with the standard volume form . This space should be thought of as the space of “labels” of the particles of the fluid, and its coordinates are known as Lagrangian coordinates. The symmetry group of this space is the space of orientation-preserving and volume-preserving diffeomorphisms.

Secondly, we will need the Eulerian space

which is the smooth manifold together with the Euclidean metric and the standard volume form . This space is the physical space of “positions” of the particles of the fluid. The symmetry space of this space is the space of orientation-preserving rigid motions of Euclidean space.

Let be the space of diffeomorphisms from Lagrangian space to Eulerian space that preserve volume and orientation; this can be viewed as an infinite-dimensional manifold. A single element of describes the positions of an incompressible fluid at a snapshot in time; an incompressible fluid flow is then described by a curve . The time derivative of such a curve can be viewed in Eulerian coordinates as the velocity field

If one then defines the Lagrangian

then one can show that the critical points of this Lagrangian (formally) correspond to solutions to the Euler equations using the correspondence (12): see this previous blog post for details.

Applying a volume-preserving change of coordinates, the Lagrangian can also be expressed as

There are then three types of symmetries that are evident for this Lagrangian: time symmetry; symmetry on the Eulerian space; and symmetry on the Lagrangian space.

We begin with time symmetry, , which comes from the fact that the Lagrangian does not depend explicitly on the time variable . As discussed before, this gives conservation of the Hamiltonian (10), which (formally, at least) becomes

thus giving the familiar energy conservation law for the Euler equations.

Now we use the symmetry group that acts on the Eulerian space , and hence on fluid flows . This is a pointwise symmetry of the Lagrangian, and formally gives conservation of total momentum

and total angular momentum

in exact analogy with the situation with the -body system. (Indeed, one can formally view the Euler equations as an limit of a certain family of -body systems, which is indeed how these equations are physically derived.)

Finally, we consider symmetries on the Lagrangian space . Any divergence-free vector field on gives a one-parameter group of volume-preserving, orientation-preserving diffeomorphisms on , which then act on fluid flows by the formula

This is a pointwise symmetry of the Lagrangian, with infinitesimal derivative

Applying (11), we thus (formally) conclude that the quantity

is conserved. We can write this in coordinates as

We can specialise this conservation law by working with specific choices of divergence-free vector field . For instance, suppose we have a closed loop , which we parameterise by unit speed: . For an infinitesimal , we can then create a divergence-free vector field by setting when lies in the (transverse) -neighbourhood of , and zero otherwise. It is geometrically obvious that this field is divergence-free (up to errors of ). The conserved quantity (13) is then equal to

up to lower order terms, so that the quantity

is conserved. Writing for the curve , we see from the chain rule that this is equal to

giving Kelvin’s circulation theorem.

More generally, we can generate a divergence-free vector field from an alternating -vector by taking a further divergence:

Integrating by parts, we can then write (13) as

since is an arbitrary alternating -tensor, we conclude that for each and , the quantity

is conserved in time. Writing and using the chain rule, this becomes

which after interchange of the indices may be rewritten in terms of the vorticity as

giving the pointwise conservation of the pullback of the vorticity in Lagrangian coordinates. (Of course, this is just the differential form of Kelvin’s circulation theorem; it also implies conservation of the vortex stream lines in Lagrangian coordinates.)

Finally, as the vorticity is divergence-free (when viewed as a polar vector field), the pullback is also. If we then set to be the vector field associated to the conserved quantity , the quantity (13) can then be rewritten as the helicity

which is then also conserved; a similar argument gives conservation of the helicity on any set that is the union of stream lines.

Remark 1 The above Lagrangian mechanics calculations can also be recast into a Hamiltonian mechanics formalism; see for instance this paper of Olver for a Hamiltonian perspective on the conservation laws for the Euler equations.

Filed under: expository, math.AP, math.CA Tagged: calculus of variations, conservation laws, Euler-Arnold equation, incompressible Euler equations, Noether's theorem, symmetry

When people talk informally about topology, we always say that the basic idea of equivalence is that two spaces are equivalent if they can be bent, stretched, smushed, or twisted into each other, without tearing or gluing. A mug is the same shape as a donut, because you can make a donut out of clay, and then shape that donut into a mug without tearing, punching holes, or gluing pieces together. A sphere is the same shape as a cube, because if you’ve got a clay sphere, you can easily reshape it into a cube, and vice-versa.

Homeomorphism is the actual formal definition of that sense of equivalence. The intuition is fantastic – it’s one of the best informal description of a difficult formal concept that I know of in math! But it’s not ideal. WHen you take a formal idea and make it informal, you always lose some details.

What we’re going to do here is try to work our way gradually through the idea of transformability and topological equivalence, so that we can really understand it. Before we can do that, we need to be able to talk about what a continuous transformation is. To talk about continuous transformations, we need to be able to talk about some topological ideas called homotopy and isotopy. And to be able to define those, we need to be able to use topological products. (Whew! Nothing is ever easy, is it?) So today’s post is really about topological products!

The easiest way that I can think of to explain the product of two topological spaces is to say that it’s a way of combining the structures of the spaces by adding dimensions. For example, if you start with two spaces each of which is a line segment, the product of those two spaces is a square (or a circle, or an octagon, or …) You started with two one-dimensional spaces, and used them to create a new two-dimensional space. If you start with a circle and a line, the product is a cylinder.

In more formal terms, topological products are a direct extension of cartesian set products. As the mantra goes, topological spaces are just sets with structure, which means that the cartesian product of two topological sets is just the cartesian products of their point-sets, plus a combined structure that preserves attributes of the original structure of the spaces.

Let’s start with a reminder of what the cartesian product of two sets is. Given a set and a set , the cartestian product is defined as the set of all possible pairs , where and . If and , then .

In category theory, we take the basic idea of the cartesian product, and extend it to a general product of different mathematical objects. It does this by using the idea of projections. In this model, instead of saying that the product of sets and is a set of pairs , we can instead say that the product is a set of objects, and two functions and . (To be complete, we’d need to add some conditions, but the idea should be clear from this much.) Given any object in the the product set , will give us the projection of that object onto . This becomes more interesting when we consider sets of objects. The A-projection of a collection of points from the product set is the shadow that those points cast onto the set A.

A topological product is easiest to understand with that categorical approach. The set of points in a product category is the cartesian product of the sets of points in and the sets of points in . The trick, with topologies, is that you need to describe the topological structure of the product set: you need to be able to say what the neighorhoods are. There are lots of ways that you could define the neighborhoods of the product, but we define it as the topological space with the smallest collection of open-sets. To understand how we get that, the projections of the category theoretical approach make it much easier.

Informally, the neighborhoods in the product are things that cast shadows into the topological spaces and which are neighborhoods in and .

Suppose we have topological spaces A and B. If is the product topology , then it has projection functions and .

The projection functions from the product need to maintain the topological structure of the original topologies. That means that the projection function must be continuous. And that, in turn, means that the inverse image of the projection function is an open set. So: for each open set in , is an open-set in .

Let’s look at an example. We’ll start with two simple topological spaces – a cartesian plane (2d), and a line (1d). In the plane, the neighborhoods are open circles; in the line, the neighborhoods are open intervals. I’ve illustrated those below.

The product of those two is a three dimensional space. The neighborhoods in this space are cylinders. If you use the projection from the product to the plane, you get open circles – the neighborhood structure of the plane. If you use the projection from the product to the line, you get open intervals – the neighborhood structure of the line.

One interesting side-point here. One thing that we come across constantly in this kind of formal math is the axiom of choice. The AoC is an annoying bugger, because it varies from being obviously true to being obviously ridiculously false. Topological products is one of the places where it’s obviously true. The axiom choice is equivalent to the statement that given a collection of non-empty topological spaces, the product space is not empty. Obvious, right? But then look at the Banach-Tarski paradox.

Guest post by Nick Gurski

I have been thinking about various sorts of operads with my PhD student Alex Corner, and have become interested in the following very concrete question: what are examples of operads in the category of finite groups under the cartesian product? I don’t know any really interesting examples, but maybe you do! After the break I will explain why I got interested in this question, and tell you about some examples that I do know.

Alex and I started off thinking about various sorts of things you might do with operads in $\mathbf{Cat}$, and were eventually forced into what we currently call an action operad. This is an operad $G$ whose job it is to act on the objects of other operads. The key examples to keep in mind are the terminal operad (each set is just a singleton), the symmetric operad (the $n$th set is the $n$th symmetric group), and the braid operad (the $n$th set is the $n$th braid group). The technical definition involves an operad $G$, a group structure on each set $G(n)$, a map of operads $\pi:G \rightarrow \Sigma$ to the symmetric operad which is levelwise a group homomorphism, and a final condition (when it makes sense) relating operadic composition, $\mu$, with group multiplication:

$$\mu(g; f_1, \ldots, f_n) \cdot \mu(g'; f_1', \ldots, f_n') = \mu (g g'; f_{\pi(g')(1)}f_{1}', \ldots, f_{\pi(g')(n)}f_{n}').$$

These ideas have cropped up before, for example in Nathalie Wahl’s thesis or this preprint of Wenbin Zhang. Once you have this definition, you can define operads which are equivariant with respect to $G$: you have an operad $P$, a group action of $G(n)$ on $P(n)$ for each $n$, and some equivariance conditions that generalize the equivariance conditions for a symmetric operad.

This isn’t the only thing you can do with an action operad, you can also think about the 2-monad on $\mathbf{Cat}$ whose algebras are strict monoidal categories where $G(n)$ acts naturally on $n$-fold tensor products. If you do this with the symmetric operad, you get symmetric strict monoidal categories (or permutative categories, if you are a topologist); if you do this with the (ribbon) braid operad, you get (ribbon) braided strict monoidal categories; and if you do this with the action operad of all terminal groups, you get back plain old strict monoidal categories. The only other “naturally-occurring” example of an action operad that I know of is the operad of $n$-fruit cactus groups, $J_{n}$. These groups come up in the representation theory of quantum groups, particularly the theory of crystals, and the monoidal structure you get out here is something Drinfeld called a coboundary category. I can give you a generators-and-relations definition of these groups, at which point I would have completely exhausted my understanding of this operad. The best reference that I know of is the paper Crystals and coboundary categories by Henriques and Kamnitzer.

What does this have to do with my question about operads of finite groups? Well, as it turns out, the structure map for an action operad $\pi:G \rightarrow \Sigma$ only has two options: it can be surjective, or it can be the zero map (i.e., everything maps to the identity permutation). Furthermore, that condition I wrote down relating group multiplication and operadic composition says that giving an action operad with $\pi$ the zero map is equivalent to giving an operad in which the operadic composition maps all preserve group multiplication. Alex and I already showed that the operadic composition of all identity elements is the identity element in the target, in other words an action operad with $\pi$ the zero map is just an operad in the category of groups using the cartesian product.

You can take kernels of maps between action operads, so in particular given any action operad $G$ you can take the kernel of $\pi:G \rightarrow \Sigma$; this gives you an operad in groups. For the examples above, you get finite groups when $G$ is the terminal operad or $G = \Sigma$ (as the terminal operad is obviously the kernel in the case of $G = \Sigma$), but for the rest of the examples you get an operad in the category of groups, but most of those groups are infinite. The groups involved are the so-called pure versions: pure braids, pure ribbon braids, and pure $n$-fruit cacti. One can then think of an action operad with surjective map $\pi$ as being an extension of the symmetric operad by an operad in the category of groups (the “pure” version), and that action operad is finite if and only if the operad in the category of groups is one containing only finite groups. We can translate this back into thinking about monoidal categories by then noting if we have some notion of strict monoidal category in which $n$th tensor powers come equipped with a natural action of a finite group $G(n)$ for all $n$, then we must be able to dig up an operad in the category of finite groups.

Now let’s talk concrete examples: what operads do I know in the category of finite groups? Well, there is obviously the terminal operad, but I can go very slightly further in that I can tell you how to construct some new ones. Here are two methods you can use to construct operads of finite groups.

• Let $A$ be a finite abelian group. Then there is an operad $\underline{A}$ where $\underline{A}(n) = A^{n}$ (the power here is a cartesian power). The operadic composition map $$A^{n} \times A^{k_{1}} \times \cdots \times A^{k_{n}} \rightarrow A^{\Sigma k_{i}}$$ takes the vector $(a_{1}, \ldots, a_{n})$ in the first coordinate and duplicates the $i$th coordinate $k_{i}$ times, then adds the result to the vector you get by just concatenating the $n$ vectors in the $A^{k_{i}}$. Here $A$ must be abelian as it appears as $\underline{A}(1)$, and $G(1)$ must be abelian for any action operad by an Eckmann-Hilton argument.
• Now let $G$ be any finite group, but in fact you will see that we don’t use anything interesting about the group structure here. There is an operad $G^{c2}$ with $$G^{c2}(n) = G^{\binom{n}{2}}$$ given the pointwise group structure. You should think of this as the set of functions from $\{ (i,j): 1 \leq i \lt j \leq n \}$ to $G$. Operad composition is a map $$G^{\binom{n}{2}} \times G^{\binom{k_{1}}{2}} \times \cdots \times G^{\binom{k_{n}}{2}} \rightarrow G^{\binom{\Sigma k_{i}}{2}}.$$ If we are given $1 \leq a \lt b \leq \sum k_{i}$, we must give back an element of $G$. If there is some $r$ such that $$\sum_{i=1}^{r-1} k_{i} \leq a \lt b \lt \sum_{i=1}^{r} k_{i},$$ then we use the function coming from $G^{c2}(r)$ evaluated on $$1 \leq a +1 - \sum_{i=1}^{r-1} k_{i} \lt b+1- \sum_{i=1}^{r-1} k_{i} \leq k_{r}.$$ If not, then there exist $r \lt s$ such that $a$ lies in the $r$th “interval” as we had before and $b$ lies in the $s$th interval, so then you use the function coming from $G^{c2}(n)$ evaluated on $r \lt s$.

I am reasonably content with the first of these constructions, I understand how to do the second but don’t really know where it comes from, and I have some ideas about how one could try to insert the finite group of their choice as $G(0)$ but haven’t checked the details, and know basically nothing else. Furthermore, I don’t know how these interact with each other, or how you can form extensions of $\Sigma$ with them outside of some obvious constructions.

Those are just the two straightforward approaches that I know of to construct operads in the category of finite groups. You can also try to construct these operads from operads of topological spaces or simplicial sets, but once again I don’t know of an example that produces finite things (apart from ones giving the groups above). Do you know any others?

• I saw a very nice talk by Jelena Vuckovic about doing nonlinear and cavity optics, with (self-assembled InAs) quantum dots as the emitters, and the cavity being formed in 2d photonic band gap systems.  The latest work looks at nonlinear effects like photon blockade, and makes contact to some work involving "circuit" quantum electrodynamics (see here).
• I went to a talk by Ken Golden, who taught me sophomore differential equations, and he gave a fascinating presentation about applying rigorous math (percolation theory, treating microstructured composites like effective media) to the challenging problem of understanding melting polar sea ice.  As a side note, he showed a great picture that is an example of the "quasistatic limit" - long wavelength surface ocean waves don't "see" individual ice floes, but instead propagate in an effective medium.
• There was a great invited session about oxide heterostructures.  Mobilities are improving (under the right conditions) to the point where some ways of learning about the band structure through electronic transport are now becoming possible.  Very impressive was a talk by Shahal Ilani, where he presented a very compelling view of the importance of structural domains ("ferroelasticity") in the underlying strontium titanate - when those domains are under control, transport becomes much more clean, revealing the apparent existence of a magnetically interesting phase at high carrier density and high in-plane magnetic field.

guest post by Vanessa Schweizer

This is my first post to Azimuth. It’s a companion to the one by Alaistair Jamieson-Lane. I’m an assistant professor at the University of Waterloo in Canada with the Centre for Knowledge Integration, or CKI. Through our teaching and research, the CKI focuses on integrating what appears, at first blush, to be drastically different fields in order to make the world a better place. The very topics I would like to cover today, which are mathematics and policy design, are an example of our flavour of knowledge integration. However, before getting into that, perhaps some background on how I got here would be helpful.

The conundrum of complex systems

For about eight years, I have focused on various problems related to long-term forecasting of social and technological change (long-term meaning in excess of 10 years). I became interested in these problems because they are particularly relevant to how we understand and respond to global environmental changes such as climate change.

In case you don’t know much about global warming or what the fuss is about, part of what makes the problem particularly difficult is that the feedback from the physical climate system to human political and economic systems is exceedingly slow. It is so slow, that under traditional economic and political analyses, an optimal policy strategy may appear to be to wait before making any major decisions – that is, wait for scientific knowledge and technologies to improve, or at least wait until the next election [1]. Let somebody else make the tough (and potentially politically unpopular) decisions!

The problem with waiting is that the greenhouse gases that scientists are most concerned about stay in the atmosphere for decades or centuries. They are also churned out by the gigatonne each year. Thus the warming trends that we have experienced for the past 30 years, for instance, are the cumulative result of emissions that happened not only recently but also long ago—in the case of carbon dioxide, as far back as the turn of the 20th century. The world in the 1910s was quainter than it is now, and as more economies around the globe industrialize and modernize, it is natural to wonder: how will we manage to power it all? Will we still rely so heavily on fossil fuels, which are the primary source of our carbon dioxide emissions?

Such questions are part of what makes climate change a controversial topic. Present-day policy decisions about energy use will influence the climatic conditions of the future, so what kind of future (both near-term and long-term) do we want?

Futures studies and trying to learn from the past

Many approaches can be taken to answer the question of what kind of future we want. An approach familiar to the political world is for a leader to espouse his or her particular hopes and concerns for the future, then work to convince others that those ideas are more relevant than someone else’s. Alternatively, economists do better by developing and investigating different simulations of economic developments over time; however, the predictive power of even these tools drops off precipitously beyond the 10-year time horizon.

The limitations of these approaches should not be too surprising, since any stockbroker will say that when making financial investments, past performance is not necessarily indicative of future results. We can expect the same problem with rhetorical appeals, or economic models, that are based on past performances or empirical (which also implies historical) relationships.

A different take on foresight

A different approach avoids the frustration of proving history to be a fickle tutor for the future. By setting aside the supposition that we must be able to explain why the future might play out a particular way (that is, to know the ‘history’ of a possible future outcome), alternative futures 20, 50, or 100 years hence can be conceptualized as different sets of conditions that may substantially diverge from what we see today and have seen before. This perspective is employed in cross-impact balance analysis, an algorithm that searches for conditions that can be demonstrated to be self-consistent [3].

Findings from cross-impact balance analyses have been informative for scientific assessments produced by the Intergovernmental Panel on Climate Change Research, or IPCC. To present a coherent picture of the climate change problem, the IPCC has coordinated scenario studies across economic and policy analysts as well as climate scientists since the 1990s. Prior to the development of the cross-impact balance method, these researchers had to do their best to identify appropriate ranges for rates of population growth, economic growth, energy efficiency improvements, etc. through their best judgment.

A retrospective using cross-impact balances on the first Special Report on Emissions Scenarios found that the researchers did a good job in many respects. However, they underrepresented the large number of alternative futures that would result in high greenhouse gas emissions in the absence of climate policy [4].

As part of the latest update to these coordinated scenarios, climate change researchers decided it would be useful to organize alternative futures according socio-economic conditions that pose greater or fewer challenges to mitigation and adaptation. Mitigation refers to policy actions that decrease greenhouse gas emissions, while adaptation refers to reducing harms due to climate change or to taking advantage of benefits. Some climate change researchers argued that it would be sufficient to consider alternative futures where challenges to mitigation and adaptation co-varied, e.g. three families of futures where mitigation and adaptation challenges would be low, medium, or high.

Instead, cross-impact balances revealed that mixed-outcome futures—such as socio-economic conditions simultaneously producing fewer challenges to mitigation but greater challenges to adaptation—could not be completely ignored. This counter-intuitive finding, among others, brought the importance of quality of governance to the fore [5].

Although it is generally recognized that quality of governance—e.g. control of corruption and the rule of law—affects quality of life [6], many in the climate change research community have focused on technological improvements, such as drought-resistant crops, or economic incentives, such as carbon prices, for mitigation and adaptation. The cross-impact balance results underscored that should global patterns of quality of governance across nations take a turn for the worse, poor governance could stymie these efforts. This is because the influence of quality of governance is pervasive; where corruption is permitted at the highest levels of power, it may be permitted at other levels as well—including levels that are responsible for building schools, teaching literacy, maintaining roads, enforcing public order, and so forth.

The cross-impact balance study revealed this in the abstract, as summarized in the example matrices below. Alastair included a matrix like these in his post, where he explained that numerical judgments in such a matrix can be used to calculate the net impact of simultaneous influences on system factors. My purpose in presenting these matrices is a bit different, as the matrix structure can also explain why particular outcomes behave as system attractors.

In this example, a solid light gray square means that the row factor directly influences the column factor some amount, while white space means that there is no direct influence:

Dark gray squares along the diagonal have no meaning, since everything is perfectly correlated to itself. The pink squares highlight the rows for the factors “quality of governance” and “economy.” The importance of these rows is more apparent here; the matrix above is a truncated version of this more detailed one:

(Click to enlarge.)

The pink rows are highlighted because of a striking property of these factors. They are the two most influential factors of the system, as you can see from how many solid squares appear in their rows. The direct influence of quality of governance is second only to the economy. (Careful observers will note that the economy directly influences quality of governance, while quality of governance directly influences the economy). Other scholars have meticulously documented similar findings through observations [7].

As a method for climate policy analysis, cross-impact balances fill an important gap between genius forecasting (i.e., ideas about the far-off future espoused by one person) and scientific judgments that, in the face of deep uncertainty, are overconfident (i.e. neglecting the ‘fat’ or ‘long’ tails of a distribution).

Wanted: intrepid explorers of future possibilities

However, alternative visions of the future are only part of the information that’s needed to create the future that is desired. Descriptions of courses of action that are likely to get us there are also helpful. In this regard, the post by Jamieson-Lane describes early work on modifying cross-impact balances for studying transition scenarios rather than searching primarily for system attractors.

This is where you, as the mathematician or physicist, come in! I have been working with cross-impact balances as a policy analyst, and I can see the potential of this method to revolutionize policy discussions—not only for climate change but also for policy design in general. However, as pointed out by entrepreneurship professor Karl T. Ulrich, design problems are NP-complete. Those of us with lesser math skills can be easily intimidated by the scope of such search problems. For this reason, many analysts have resigned themselves to ad hoc explorations of the vast space of future possibilities. However, some analysts like me think it is important to develop methods that do better. I hope that some of you Azimuth readers may be up for collaborating with like-minded individuals on the challenge!

References

The graph of carbon emissions is from reference [2]; the pictures of the matrices are adapted from reference [5]:

[1] M. Granger Morgan, Milind Kandlikar, James Risbey and Hadi Dowlatabadi, Why conventional tools for policy analysis are often inadequate for problems of global change, Climatic Change 41 (1999), 271–281.

[2] T.F. Stocker et al., Technical Summary, in Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (2013), T.F. Stocker, D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex, and P.M. Midgley (eds.) Cambridge University Press, New York.

[3] Wolfgang Weimer-Jehle, Cross-impact balances: a system-theoretical approach to cross-impact analysis, Technological Forecasting & Social Change 73 (2006), 334–361.

[4] Vanessa J. Schweizer and Elmar Kriegler, Improving environmental change research with systematic techniques for qualitative scenarios, Environmental Research Letters 7 (2012), 044011.

[5] Vanessa J. Schweizer and Brian C. O’Neill, Systematic construction of global socioeconomic pathways using internally consistent element combinations, Climatic Change 122 (2014), 431–445.

[6] Daniel Kaufman, Aart Kray and Massimo Mastruzzi, Worldwide Governance Indicators (2013), The World Bank Group.

[7] Daron Acemoglu and James Robinson, The Origins of Power, Prosperity, and Poverty: Why Nations Fail. Website.

The following post is a collaboration between visiting undergraduates Evan Giarta from Stanford University and Joy Hui from Harvard University. As mentors for the 2013 Caltech InnoWorks Academy, Evan and Joy agreed to share their experience with the audience of this blog.

All throughout modern history, science and mathematics have been the foundation for engineering new, world-advancing technologies. From the wheel and sailboat to the automobile and jumbo jet, the fields of science, technology, engineering and math (STEM) have helped our world and its people move faster, farther, and forward. Now, more than ever, products that were unimaginable a century ago are used every day in households and businesses all over the earth, products made possible by generations of scientists, mathematicians, engineers and technologists.

There is, however, some troublesome news regarding the state of our nation’s math and science education. For the past few years, education and news reports have ranked the United States behind other developed countries in science and mathematics. In fact, the proportion of students that score in the most advanced levels of math and science in the United States is significantly lower than that of several other nations. This reality brings to light a stark concern: If proficiency in science and math is necessary for the engineering of novel technologies and breakthrough discoveries, but is something the next generation will lack, what will become of the production industry, our economy, and global security? While the answer to this question might range from complete and utter chaos to little or no effect, it seems reasonable to try and avoid finding out. Rather, we–the community of collective scientists, technologists, engineers, mathematicians–should seek to solve this problem: What can we do to restore the integrity and substance of the educational system, especially in science and math?

Although there is no easy answer to this question, there are ongoing efforts to address this important issue, one of which is the InnoWorks Academy. Founded in 2003 by a group of Duke undergraduates, the United InnoWorks Academy has developed summer programs that encourage underprivileged middle school students to explore the fields of science, technology, engineering, math, and medicine, (STEM^2) free of charge. The academy is sponsored by a variety of businesses and organizations, including GlaxoSmithKline, Cisco, Project Performance Corporation, Do Something, St. Andrew’s School, University of Pennsylvania, University of California, Los Angeles, and many others. InnoWorks Academy was a winner of the 2007 BRICK Awards and received both the MIT Global Community Choice Award and the MIT IDEAS Challenge Grand Prize in 2011. In ten short years, the United InnoWorks Academy has successfully conducted over 50 summer programs for more than 2,000 students through the contributions of over 1,000 volunteers. Currently, InnoWorks has chapters at about a dozen universities and is scheduled to add three more chapters this coming summer.

In early August of 2013, the Caltech InnoWorks chapter hosted its second annual summer camp, and Joy and I (Evan) were privileged to be invited as program mentors. As people on the inside, we got a first hand look at how the organization prepares and runs the week-long event and what the students themselves experience in the hands-on, interactive and collaborative one-of-a-kind opportunity that is InnoWorks.

Since Joy and I kept a diary of each day’s activities, we thought you may like to see what our middle-schoolers experienced during that week. Here is a play-by-play of the first few days from Joy’s perspective.

Monday, August 5, 2013 was the first day of our camp! Before I go on with talking about the cool things we did that day, I’d like to introduce my team. The self-titled YOLOSWAG consisted of Michael, Chase, Evan, Phaelan and myself. Michael was the oldest, and a little shy at first, but he definitely started talking when he got comfortable. Chase was respectful and polite, and we hit it off immediately. Evan loved science and had lots of questions and knew a TON. Phaelan was the only other girl on the team, but she was very nice and friendly to the other students, and eager to help with anything she could. We all seem different, right? But wait, here’s the best part: we were all die-hard Percy Jackson (property of Rick Riordan) fans! We were definitely the best team.

Anyway, back to the actual stuff we did. One of the first things we saw was a cloud demo, essentially the creation of a cloud in a fish tank, with lots of dry ice and water. The demonstrator, Rob Usiskin, stuck a lot of dry ice in the (empty) fish tank, and poured some water into the tank, which caused the water to turn into a fog, which turns out to be the exact form of a cloud! Add a bubble maker, a question of whether bubbles will float or sink on the cloud, and a room full of InnoWorks campers (about 40 of them), and you will get an hour of general excitement. See the pictures for yourself! (The bubbles, float, by the way, even when the cloud is invisible!)

Following the demonstration, we made Soap Boats. We were apparently supposed to cut index cards into the shape of boats, and dab a little bit of soap on the bottom of the boat, and set the boat in the water. These “Soap Boats” were supposed to be propelled forward by the soap’s ability to decrease the surface tension of the water it touched. Ours, appropriately named “Titanic,” however, simply sat in the water until the water soaked through, and the Titanic sank for the second time in history. Many other teams’ boats fared about the same, but we certainly had a blast designing and naming our Soap Boat!

The last activity of the day was a secret message decoded with bio-luminescence. Each team was given a vial of dried up ostracods, which are sea creatures found glowing in the darkness of the deep sea. Then, each team crushed the ostracods and mixed the resulting powder with water to catalyze the bio-luminescence. Every mentor had written a secret message on a slip of paper, folded it, and handed it to their teams to decipher in the dark (Mine said: YOLOSWAG for the win). The convenient cleaning closet provided said darkness–Spiros, our faculty mentor, suggested that it might also provide passage to Narnia. I don’t think we lost any of our students that day, so no Narnia-traveling was done; by students. Nevertheless, it was a fun-filled and action-packed day, and a great start to an eventful week.

Tuesday was full of fun, hands-on activities. After a short lesson on the effects of air resistance and gravity on free-falling objects, we demonstrated the concept with a thin sheet of paper and a large textbook. As expected, when placed side by side and dropped, air resistance caused the sheet of paper to hit the ground much later than the textbook. But when the sheet of paper was placed directly above the textbook, to the shock of students, both items fell at the exact same rate. Though thorough in their knowledge of physical laws, the connection between their conceptual understanding and real life application had yet to be established. And as a result, when asked to explain what happened and why, the best answer one could muster was simply, “SCIENCE!”.

Following an exercise consisting of blow dryers and floating ping pong balls, the kids received a brief tutorial on how tornadoes are formed by air moving through high and low pressure regions and gusts of vertically rising winds. Due to the forces it is producing and acting upon, the tornado would then be able to more or less sustain itself. To explore this concept further, students constructed water tornado machines by taping two soda bottles together at their openings. Laughter and wetness ensued. Some groups added small trinkets in their tornado machine to observed the water tornado’s effect on “debris”. One team in particular inserted duct tape sharks, and aptly renamed themselves as Sharknado.

In the afternoon, the campers were presented a lesson on the transportation of sailboats and aircraft. Contrary to what most people intuit, the fastest way to control a boat is not to flow in the direction of the wind, but to place the sail at the heading which produces a net force, which can be explained by Bernoulli’s Principle. It states that faster moving fluid has less pressure than slower moving fluid, therefore producing a force from the slower moving side to the faster moving side. Though in sailing this force is initially barely noticeable, over time it creates a large impulse to move the craft at considerable speeds. The same principle can also explain the way a plane takes flight.

With the importance of good design in mind, students were tasked with prototyping the fastest water-bottle boat. Given a solar panel, electric motor, various propellers, empty bottle, tape, and other construction essentials, kids started with basic designs, then diversified in order to gain an edge against other teams. Some teams boasted two-bottle designs, and others used one, each type having trade-offs in speed and stability. Few implemented style upgrades with graphics and colors, and even fewer leveraged performance modifications with ballast and crude control systems. But with a tough deadline to meet, not all boats met their intended specifications. Nonetheless, the races commenced and each team’s innovation was tested in the torrents of Caltech’s Beckman Fountain. Some failed, but those that survived were rewarded accordingly.

By the end of the second day, many of the campers’ initial shyness had been replaced with conversation and budding new friendships. Lunch hour and break times allowed time for kids and mentors alike to hang out and enjoy themselves in California’s summer sun in-between discovering the applications of science and math to engineering, medicine, and technology. These moments of discovery, no matter how rare, are the reasons why we do what we do as we continue our research, studies, and work to improve the world we live in.

In which we move out of the original trilogy, and into J.J. Abrams territory. Cue the lens flare!

This week’s random assortment of topics includes travel, airports, physics models of loading and unloading planes, uses and abuses of curve fitting, odd stuff we get sent to review, and high-speed video cameras.

Miscellaneous links:
– Rhett’s Atlanta airport post.

– The airplane loading study I was thinking of is Optimal boarding method for airline passengers, by Jason Steffen, from 2008.

–The Slo Mo Guys, Smarter Every Day, Veritasium.

–Trespassing on Einstein’s Lawn by Amanda Gefter.

That’s it for this week…

In algebraic geometry, we like to make statements like: “two conics meet at points”, “a degree four plane curve has bitangents”, “given four lines in three space, there are lines that meet all of them”. In each of these, we are saying that, as some parameter (the conics, the degree four curve, the lines) changes, the number of solutions to some equation stays constant. The “principle of conservation of number” refers to various theorems which make this precise.

In my experience, students in algebraic geometry tend to pick up the rough idea but remain hazy on the details, most likely because there are many different ways to make these details precise. I decided to try and write down all the basic results I could think of along these lines.

Let be some parameter space such as the space of pairs of two conics. Let be some space of solutions, such as the space of triples where is a point on . Let be a map, such as projection onto the components. We want theorems which will discuss the size of the fibers of , in terms of some global degree of the map .

We work over some field . For simplicity of presentation, we’ll assume that is affine, meaning that it is a subset of defined by polynomial equations

We’ll write for the ring .

It would be silly to ask for any such results if were disconnected. A very basic observation of algebraic geometry is that is connected if and only if has no nontrivial idempotents. In fact, we will ask for something stronger: That is an integral domain. The terminology for this is that is irreducible. From now on, we will make:

Assumption is irreducible. ( is an integral domain.)

If is also affine, with corresponding ring , then is an module. We define the degree of in this case to be the dimension of as a vector space. Degree can be defined in much greater generality; we will feel free to refer to it in greater generality without giving the definition. We will denote the degree of by . Roughly, we want theorems which say that the fibers of have size .

Here is our first result.

Theorem (Shafarevich, II.6.3, Theorem 4) If has characteristic zero and is algebraically closed then for almost all in . More precisely, there is some polynomial , not identically zero on , so that implies .

Warning This isn’t true if is not algebraically closed: Consider the map from .

Warning This isn’t true in characteristic : Consider .

We now want results which let us say something, not just about almost all , but about all .

Naive size

We will at first focus on counting the size of in a naive sense: We think of as sitting in (or in ) and we literally count points of the fiber. We can’t hope for the fibers to always be of full size because even the nicest map, , has fiber of size , not , over the point . So, using the naive size, we can only hope for upper bounds.

There are two additional problems. The first one is if we have something like projecting onto the coordinate. In this case, the degree is but the fiber over has size . When is affine, with corresponding ring , we can fix this by requiring that is torsion free as an -module. In general, the right condition is that no irreducible component of maps to a proper subvariety of .

More subtly, suppose that is a nodal curve, such as , and is its desingularization. (In this case, the line with as the map .) Then the degree of the map is , but the fiber over is , of size . The hypothesis to rule this out is that is integrally closed in its fraction field. By definition, this is the same as saying that is normal.

Once we rule out these possibilities, we have

Theorem (Shafarevich, II.6.3, Theorem 3) If is normal, and no irreducible component of maps to a proper subvariety of , then every fiber of has naive size .

I can’t resist mentioning a result which far harder than these:

Theorem (A consequence of Zariski’s Main Theorem) Let be normal and let have degree . Assume that no irreducible component of maps to a proper subvariety of . For any in , the number of connected components of is at most $d$.

Scheme theoretic size

We now consider counting size in a less naive way. Again, for simplicity, suppose that is affine, with corresponding ring . Let be a point of , so there is a map of rings by . Consider the ring , where acts on by the above map. The maps from this ring to are the point in . Thus, is an upper bound for the number of points of above . We will call this dimension the scheme theoretic size of the fiber. Once again, it can be defined when is not affine as well.

We have the following cautionary example: Let mapping onto the coordinate. Then the degree is , but the fiber above has size , either scheme theoretically or naively. To rule this out, we impose that is finite over . By definition, this means that is affine, and is a finitely generated module.

You might worry about how we could ever prove that is affine if it is not given to us as a closed subset of . Fortunately, we have:

Theorem (Hartshorne, Exercise III.11.2) If is projective with finite fibers, then it is a finite map. Here projective means that is a closed subset of , projecting onto . (This is not the morally right definition of a projective map, but if you are ready for the right definition, then you should be working with “proper” rather than “projective” anyway.)

We then have

Theorem (Hartshorne, Exercise II.5.8) If is finite over , and no irreducible component of maps to a proper subvariety of , then every fiber of has scheme theoretic size .

Flatness

Theorem Let be a finite map. Then all fibers have scheme theoretic size if and only if is flat over .

Unfortunately, flat is a rather technical condition. The first thing to understand is that some nice looking maps can fail to be flat:

Warning Let be , let and let the map be . This is a finite map. (We can alternately describe as .) This map is degree , but the fiber over has scheme theoretic size (and naive size ).

If your eye is well enough trained that this doesn’t look nice to you, try the examples here.

There are two good conditions that imply flatness:

Theorem (Hartshorne III.9.7) If is normal and one dimensional, and no irreducible component of maps to a proper subvariety of , then is flat over .

Theorem (The miracle flatness theorem) If is Cohen-Macaulay, is smooth of the same dimension as , and is finite, then is flat.

Advances in accelerators built for fundamental physics research have inspired improved cancer treatment facilities. But will one of the most promising—a carbon ion treatment facility—be built in the U.S.? Participants at a symposium organized by Brookhaven Lab for the 2014 AAAS meeting explored the science and surrounding issues.

by Karen McNulty Walsh

Accelerator physicists are natural-born problem solvers, finding ever more powerful ways to generate and steer particle beams for research into the mysteries of physics, materials, and matter. And from the very beginning, this field born at the dawn of the atomic age has actively sought ways to apply advanced technologies to tackle more practical problems. At the top of the list—even in those early days— was taking aim at cancer, the second leading cause of death in the U.S. today, affecting one in two men and one in three women.

Using beams of accelerated protons or heavier ions such as carbon, oncologists can deliver cell-killing energy to precisely targeted tumors—and do so without causing extensive damage to surrounding healthy tissue, eliminating the major drawback of conventional radiation therapy using x-rays.

“This is cancer care aimed at curing cancer, not just treating it,” said Ken Peach, a physicist and professor at the Particle Therapy Cancer Research Institute at Oxford University.

Peach was one of six participants in a symposium exploring the latest advances and challenges in this field—and a related press briefing attended by more than 30 science journalists—at the 2014 meeting of the American Association for the Advancement of Science in Chicago on February 16. The session, “Targeting Tumors: Ion Beam Accelerators Take Aim at Cancer,” was organized by the U.S. Department of Energy’s (DOE’s) Brookhaven National Laboratory, an active partner in an effort to build a prototype carbon-ion accelerator for medical research and therapy. Brookhaven Lab is also currently the only place in the U.S. where scientists can conduct fundamental radiobiological studies of how beams of ions heavier than protons, such as carbon ions, affect cells and DNA.

Participants in a symposium and press briefing exploring the latest advances and challenges in particle therapy for cancer at the 2014 AAAS meeting: Eric Colby (U.S. Department of Energy), Jim Deye (National Cancer Institute), Hak Choy (University of Texas Southwestern Medical Center), Kathryn Held (Harvard Medical School and Massachusetts General Hospital), Stephen Peggs (Brookhaven National Laboratory and Stony Brook University), and Ken Peach (Oxford University). (Credit: AAAS)

“We could cure a very high percentage of tumors if we could give sufficiently high doses of radiation, but we can’t because of the damage to healthy tissue,” said radiation biologist Kathryn Held of Harvard Medical School and Massachusetts General Hospital during her presentation. “That’s the advantage of particles. We can tailor the dose to the tumor and limit the amount of damage in the critical surrounding normal tissues.”

Yet despite the promise of this approach and the emergence of encouraging clinical results from carbon treatment facilities in Asia and Europe, there are currently no carbon therapy centers operating in the U.S.

Participants in the Brookhaven-organized session agreed: That situation has to change—especially since the very idea of particle therapy was born in the U.S.

Physicists as pioneers

“When Harvard physicist Robert Wilson, who later became the first director of Fermilab, was asked to explore the potential dangers of proton particle radiation [just after World War II], he flipped the problem on its head and described how proton beams might be extremely useful—as effective killers of cancer cells,” said Stephen Peggs, an accelerator physicist at Brookhaven Lab and adjunct professor at Stony Brook University.

As Peggs explained, the reason is simple: Unlike conventional x-rays, which deposit energy—and cause damage—all along their path as they travel through healthy tissue en route to a tumor (and beyond it), protons and other ions deposit most of their energy where the beam stops. Using magnets, accelerators can steer these charged particles left, right, up, and down and vary the energy of the beam to precisely place the cell-killing energy right where it’s needed: in the tumor.

The first implementation of particle therapy used helium and other ions generated by the Bevatron at Berkeley Lab. Those spin-off studies “established a foundation for all subsequent ion therapy,” Peggs said. And as accelerators for physics research grew in size, pioneering experiments in particle therapy continued, operating “parasitically” until the very first accelerator built for hospital-based proton therapy was completed with the help of DOE scientists at Fermilab in 1990.

But even before that machine left Illinois for Loma Linda University Medical Center in California, physicists were thinking about how it could be made better. The mantra of making machines smaller, faster, cheaper—and capable of accelerating more kinds of ions—has driven the field since then.

Advances in magnet technology, including compact superconducting magnets and beam-delivery systems developed at Brookhaven Lab, hold great promise for new machines. Peggs is working to incorporate these technologies in a prototype ‘ion Rapid Cycling Medical Synchrotron’ (iRCMS) capable of delivering protons and/or carbon ions for radiobiology research and for treating patients.

Brookhaven Lab accelerator physicist Stephen Peggs with magnet technology that could reduce the size of particle accelerators needed to steer heavy ion beams and deliver cell-killing energy to precisely targeted tumors while sparing surrounding healthy tissue.

Small machine, big particle impact

The benefits of using charged particles heavier than protons (e.g., carbon ions) stem not only from their physical properties—they stop and deposit their energy over an even smaller and better targeted tumor volume than protons—but also a range of biological advantages they have over x-rays.

As Kathryn Held elaborated in her talk, compared with x-ray photons, “carbon ions are much more effective at killing tumor cells. They put a huge hole through DNA compared to the small pinprick caused by x-rays, which causes clustered or complex DNA damage that is less accurately repaired between treatments—less repaired, period—and thus more lethal [to the tumor].” Carbon ions also appear to be more effective than x-rays at killing oxygen-deprived tumor cells, and might be most effective in fewer higher doses, “but we need more basic biological studies to really understand these effects,” Held said.

Different types of radiation treatment cause different kinds of damage to the DNA in a tumor cell. X-ray photons (top arrow) cause fairly simple damage (purple area) that cancer cells can sometimes repair between treatments. Charged particles—particularly ions heavier than protons (bottom arrow)—cause more and more complex forms of damage, resulting in less repair and a more lethal effect on the tumor. (Credit: NASA)

Held conducts research at the NASA Space Radiation Laboratory (NSRL) at Brookhaven Lab, an accelerator-based facility designed to fully understand risks and design protections for future astronauts exposed to radiation. But much of that research is relevant to understanding the mechanisms and basic radiobiological responses that can apply to the treatment of cancer. But additional facilities and funding are needed for research specifically aimed at understanding the radiobiological effects of heavier ions for potential cancer therapies, Held emphasized.

Hak Choy, a radiation oncologist and chair in the Department of Radiation Oncology at the University of Texas Southwestern Medical Center, presented compelling clinical data on the benefits of proton particle therapy, including improved outcomes and reduced side effects when compared with conventional radiation, particularly for treating tumors in sensitive areas such as the brain and spine and in children. “When you can target the tumor and spare critical tissue you get fewer side effects,” he said.

Data from Japan and Europe suggest that carbon ions could be three or four times more biologically potent than protons, Choy said, backing that claim with impressive survival statistics for certain types of cancers where carbon therapy surpassed protons, and was even better than surgery for one type of salivary gland cancer. “And carbon therapy is noninvasive,” he emphasized.

To learn more about this promising technology and the challenges of building a carbon ion treatment/research facility in the U.S., including perspectives from the National Cancer Institute, DOE and a discussion about economics, read the full summary of the AAAS symposium here: http://www.bnl.gov/newsroom/news.php?a=24672.

Karen McNulty Walsh is a science writer in the Media & Communications Office at Brookhaven National Laboratory.