I mentioned in passing in the Forbes post about science funding that I’m thoroughly sick of hearing about how the World Wide Web was invented at CERN. I got into an argument about this a while back on Twitter, too, but had to go do something else and couldn’t go into much detail. It’s probably worth explaining at greater-than-Twitter length, though, and a little too inside-baseball for Forbes, so I’ll write something about it here.

At its core, the “CERN invented WWW” argument is a “Basic research pays off in unexpected ways” argument, and in that sense, it’s fine. The problem is, it’s not anything more than that– its fine as an argument for funding basic research as a general matter, but it’s not an argument *for* anything in particular.

What bugs me is now when it’s used as a general “Basic research is good” argument, but that it’s used as a catch-all argument for giving particle physicists whatever they want for whatever they decide they want to do next. It’s used to steamroll past a number of other, perfectly valid, arguments about funding priorities within the general area of basic physics research, and that gets really tiresome.

Inventing WWW is great, but it’s not an argument for particle physics in particular, precisely because it was a weird spin-off that nobody expected, or knew what to do with. In fact, you can argue that much of the impact of the Web was enabled precisely because CERN didn’t really understand it, and Time Berners-Lee just went and did it, and gave the whole thing away. You can easily imagine a different arrangement where Web-like network technologies were developed by people who better understood the implications, and operated in a more proprietary way from the start.

As an argument for funding particle physics in particular, though, the argument undermines itself precisely due to the chance nature of the discovery. Past performance does not guarantee future results, and the fact that CERN stumbled into a transformative discovery once doesn’t mean you can expect anything remotely similar to happen again.

The success of the Web is all too often invoked as a way around a very different funding argument, though, where it doesn’t really apply, which is an argument about the relative importance of Big Science. That is, a side spin-off like the Web is a great argument for funding basic science in general, but it doesn’t say anything about the relative merits of spending a billion dollars on building a next-generation particle collider, as opposed to funding a thousand million-dollar grants for smaller projects in less abstract areas of physics.

There are arguments that go both ways on that, and none of them have anything to do with the Web. On the Big Science side, you can argue that working at an extremely large scale necessarily involves pushing the limits of engineering and networking and working in those big limits might offer greater opportunities for discovery. On the small-science side, you can argue that a greater diversity of projects and researchers offers more chances for the unexpected to happen compared to the same investment in a single enormous project.

I’m not sure what the right answer to that question is– given my background, I’m naturally inclined toward the “lots of small projects (in subfields like the one I work in)” model, but I can see some merit to the arguments about working at scale. I think it *is* a legitimate question, though, one that needs to be considered seriously, and not one that can be headed off by using WWW as a Get Funding Forever trump card for particle physics.

A bunch of people in my social-media feeds are sharing this post by Alana Cattapan titled Time-sucking academic job applications don’t know enormity of what they ask. It describes an ad asking for two sample course syllabi “not merely syllabi for courses previously taught — but rather syllabi for specific courses in the hiring department,” and expresses outrage at the imposition on the time of people applying for the job. She argues that the burden falls particularly heavily on groups that are already disadvantaged, such as people currently in contingent faculty positions.

It’s a good argument, as far as it goes, and as someone who has been on the hiring side of more faculty searches than I care to think about, the thought of having to review sample syllabi for every applicant in a pool is… not exactly an appealing prospect. At the same time, though, I can see how a hiring committee would end up implementing this for the best of reasons.

Many of the standard materials used in academic hiring are famously rife with biases– letters of reference being the most obviously problematic, but even the use of CV’s can create issues, as it lends itself to paper-counting and lazy credentialism (“They’re from Bigname University, they must be good…”). Given these well-known problems, I can see a chain of reasoning leading to the sample-syllabus request as a measure to help avoid biases in the hiring process. A sample syllabus is much more concrete than the usual “teaching philosophy” (which tends to be met with boilerplate piffle), particularly if it’s for a specific course familiar to the members of the hiring committee. It offers a relatively objective way to sort out who *really* understands what’s involved in teaching, that doesn’t rely on name recognition or personal networking. I can even imagine some faculty earnestly arguing that this would give an *advantage* to people in contingent-faculty jobs, who have lots of teaching experience and would thus be better able to craft a good syllabus than some wet-behind-the-ears grad student from a prestigious university.

And yet, Cattapan’s “too much burden on the applicant” argument is a good one. Which is just another reminder that academic hiring is a lot like Churchill’s famous quip about democracy: whatever system you’re using is the worst possible one, except for all the others.

And, like most discussions of academic hiring, this is frustrating because it dances around what’s really *the* central problem with academic hiring, namely that the job market for faculty positions absolutely sucks, and has for decades. A single tenure-track opening will generally draw triple-digit numbers of applications, and maybe 40% of those will be obviously unqualified. Which leaves the people doing the hiring with literally dozens of applications that they have to cut down *somehow*. It’s a process that will necessarily leave large numbers of perfectly well qualified people shut out of jobs through no particular fault of their own, just because there aren’t nearly enough jobs to go around.

Given that market situation, most arguments about why this or that method of winnowing the field of candidates is Bad feel frustratingly pointless. We can drop some measures as too burdensome for applicants, and others as too riddled with bias, but none of that changes the fact that somehow, 149 of 150 applicants need to be disappointed at the end of the process. And it’s never really clear what should replace those problematic methods that would do a substantially better job of weeding out 99.3% of the applicants without introducing new problems.

At some level the fairest thing to do would be to make the easy cut of removing the obviously unqualified and then using a random number generator to pick who gets invited to campus for interviews. I doubt that would make anybody any happier, though.

Don’t get me wrong, this isn’t a throw-up-your-hands anti-measurement argument. I’d love it if somebody could find a relatively objective and reasonably efficient means of picking job candidates out of a large pool, and I certainly think it’s worth exploring new and different ways of measuring academic “quality,” like the sort of thing Bee at Backreaction talks about. (I’d settle for more essays and blog posts saying “This is what you *should* do,” rather than “This is what you *shouldn’t* do”…) But it’s also important to note that all of these things are small perturbations to the real central problem of academic hiring, namely that there are too few jobs for too many applicants.

Yesterday Ryan Mandelbaum, at Gizmodo, posted a decidedly tongue-in-cheek piece about whether or not the universe is a computer simulation. (The piece was filed under the category “LOL.”)

The immediate impetus for Mandelbaum’s piece was a blog post by Sabine Hossenfelder, a physicist who will likely be familiar to regulars here in the nerdosphere. In her post, Sabine vents about the simulation speculations of philosophers like Nick Bostrom. She writes:

Proclaiming that “the programmer did it” doesn’t only not explain anything – it teleports us back to the age of mythology. The simulation hypothesis annoys me because it intrudes on the terrain of physicists. It’s a bold claim about the laws of nature that however doesn’t pay any attention to what we know about the laws of nature.

After hammering home that point, Sabine goes further, and says that the simulation hypothesis is almost *ruled out*, by (for example) the fact that our universe is Lorentz-invariant, and a simulation of our world by a discrete lattice of bits won’t reproduce Lorentz-invariance or other continuous symmetries.

In writing his post, Ryan Mandelbaum interviewed two people: Sabine and me.

I basically told Ryan that I agree with Sabine insofar as she argues that the simulation hypothesis is *lazy*—that it doesn’t pay its rent by doing real explanatory work, doesn’t even engage much with any of the deep things we’ve learned about the physical world—and disagree insofar as she argues that the simulation hypothesis faces some special difficulty because of Lorentz-invariance or other continuous phenomena in known physics. In short: blame it for being unfalsifiable rather than for being falsified!

Indeed, to whatever extent we believe the Bekenstein bound—and even more pointedly, to whatever extent we think the AdS/CFT correspondence says something about reality—we believe that in quantum gravity, any bounded physical system (with a short-wavelength cutoff, yada yada) lives in a Hilbert space of a finite number of qubits, perhaps ~10^{69} qubits per square meter of surface area. And as a corollary, if the cosmological constant is indeed constant (so that galaxies more than ~20 billion light years away are receding from us faster than light), then our entire observable universe can be described as a system of ~10^{122} qubits. The qubits would in some sense be the fundamental reality, from which Lorentz-invariant spacetime and all the rest would need to be recovered as low-energy effective descriptions. (I hasten to add: there’s of course nothing special about *qubits* here, any more than there is about bits in classical computation, compared to some other unit of information—nothing that says the Hilbert space dimension has to be a power of 2 or anything silly like that.) Anyway, this would mean that our observable universe could be simulated by a quantum computer—or even for that matter by a classical computer, to high precision, using a mere ~2^{10^122} time steps.

Sabine might respond that AdS/CFT and other quantum gravity ideas are mere theoretical speculations, not solid and established like special relativity. But crucially, if you believe that the observable universe couldn’t be simulated by a computer even in principle—that it has no mapping to any system of bits or qubits—then at some point the speculative shoe shifts to the other foot. The question becomes: do you reject the Church-Turing Thesis? Or, what amounts to the same thing: do you believe, like Roger Penrose, that it’s possible to build devices in nature that solve the halting problem or other uncomputable problems? If so, how? But if not, then how exactly does the universe *avoid* being computational, in the broad sense of the term?

I’d write more, but by coincidence, right now I’m at an It from Qubit meeting at Stanford, where everyone is talking about how to map quantum theories of gravity to quantum circuits acting on finite sets of qubits, and the questions in quantum circuit complexity that are thereby raised. It’s tremendously exciting—the mixture of attendees is among the most stimulating I’ve ever encountered, from Lenny Susskind and Don Page and Daniel Harlow to Umesh Vazirani and Dorit Aharonov and Mario Szegedy to Google’s Sergey Brin. But it should surprise no one that, amid all the discussion of computation and fundamental physics, the question of whether the universe “really” “is” a simulation has barely come up. Why would it, when there are so many more fruitful things to ask? All I can say with confidence is that, if our world *is* a simulation, then whoever is simulating it (God, or a bored teenager in the metaverse) seems to have a clear preference for the 2-norm over the 1-norm, and for the complex numbers over the reals.

I have tried hard to avoid political tracts on this blog, because I don't think that's why people necessarily want to read here. Political flamewars in the comments or loss of readers over differences of opinion are not outcomes I want. The recent proposed budget from the White House, however, inspires some observations. (I know the President's suggested budget is only the very beginning of the budgetary process, but it does tell you something about the administration priorities.)

The second law of thermodynamics tell us that some macroscopic processes tend to run only one direction. It's easier to disperse a drop of ink in a glass of water than to somehow reconstitute the drop of ink once the glass has been stirred.

In general, the response of a system to some input (say the response of a ferromagnet to an applied magnetic field, or the deformation of a blob of silly putty in response to an applied stress) can depend on the history of the material. Taking the input from A to B and back to A doesn't necessarily return the system to its original state. Cycling the input and ending up with a looping trajectory of the system in response because of that history dependence is called hysteresis. This happens because there is some inherent time scale for the system to respond to inputs, and if it can't keep up, there is lag.

The proposed budget would make sweeping changes to programs and efforts that, in some cases, took decades to put in place. Drastically reducing the size and scope of federal agencies is not something that can simply be undone by the next Congress or the next President. Cutting 20% of NIH or 17% of DOE Office of Science would have ripple effects for many years, and anyone who has worked in a large institution knows that big cuts are almost never restored. Expertise at EPA and NOAA can't just be rebuilt once eliminated.

People can have legitimate discussions and differences of opinion about the role of the government and what it should be funding. However, everyone should recognize that these are serious decisions, many of which are irreversible in practical terms. Acting otherwise is irresponsible and foolish.

Just a short post to note that Norwegian Academy of Science and Letters has just announced that the 2017 Abel prize has been awarded to Yves Meyer, “for his pivotal role in the development of the mathematical theory of wavelets”. The actual prize ceremony will be at Oslo in May. I am actually in Oslo […]

Just a short post to note that Norwegian Academy of Science and Letters has just announced that the 2017 Abel prize has been awarded to Yves Meyer, “for his pivotal role in the development of the mathematical theory of wavelets”. The actual prize ceremony will be at Oslo in May.

I am actually in Oslo myself currently, having just presented Meyer’s work at the announcement ceremony (and also having written a brief description of some of his work). The Abel prize has a somewhat unintuitive (and occasionally misunderstood) arrangement in which the presenter of the work of the prize is selected independently of the winner of the prize (I think in part so that the choice of presenter gives no clues as to the identity of the laureate). In particular, like other presenters before me (which in recent years have included Timothy Gowers, Jordan Ellenberg, and Alex Bellos), I agreed to present the laureate’s work before knowing who the laureate was! But in this case the task was very easy, because Meyer’s areas of (both pure and applied) harmonic analysis and PDE fell rather squarely within my own area of expertise. (I had previously written about some other work of Meyer in this blog post.) Indeed I had learned about Meyer’s wavelet constructions as a graduate student while taking a course from Ingrid Daubechies. Daubechies also made extremely important contributions to the theory of wavelets, but my understanding is that due to a conflict of interest arising from Daubechies’ presidency of the International Mathematical Union (which nominates members of the Abel prize committee) from 2011 to 2014, she was not eligible for the prize this year, and so I do not think this prize should be necessarily construed as a judgement on the relative contributions of Meyer and Daubechies to this field. (In any case I fully agree with the Abel prize committee’s citation of Meyer’s pivotal role in the development of the theory of wavelets.)

Filed under: math.CA, math.IT, non-technical Tagged: Abel prize, Yves Meyer

The $p$-norms have a nice multiplicativity property:

$\|(A x, A y, A z, B x, B y, B z)\|_p = \|(A, B)\|_p \, \|(x, y, z)\|_p$

for all $A, B, x, y, z \in \mathbb{R}$ — and similarly, of course, for any numbers of arguments.

Guillaume Aubrun and Ion Nechita showed that this condition completely characterizes the $p$-norms. In other words, *any* system of norms that’s multiplicative in this sense must be equal to $\|\cdot\|_p$ for some $p \in [1, \infty]$. And the amazing thing is, to prove this, they used some nontrivial probability theory.

All this is explained in this week’s functional equations notes, which start on page 26 here.

*Guest post by Simon Cho*

We continue the Kan Extension Seminar II with Max Kelly’s On the operads of J. P. May. As we will see, the main message of the paper is that (symmetric) operads enriched in a suitably nice category $\mathcal{V}$ arise naturally as monoids for a “substitution product” in the monoidal category $[\mathbf{P}, \mathcal{V}]$ (where $\mathbf{P}$ is a category that keeps track of the symmetry). Before we begin, I want to thank the organizers and participants of the Kan Extension Seminar (II) for the opportunity to read and discuss these nice papers with them.

Some time ago, in her excellent post about Hyland and Power’s paper, Evangelia described what Lawvere theories are about. We might think of Lawvere theories as a way to frame algebraic structure by stratifying the different components of an algebraic structure into roughly three ascending levels of specificity: the product structure, the specific algebraic operations (meaning, other than projections, etc.), and the models of that algebraic structure. These structures are manifested categorically through (respectively) the category $\aleph_0^{\text{op}}$ of finite sets and (the duals of) maps between them, a category $\mathcal{L}$ with finite products that has the same objects as $\aleph_0$, and some other category $\mathcal{C}$ with finite products. Then a Lawvere theory is just a strict product preserving functor $I: \aleph_0^{\text{op}} \rightarrow \mathcal{L}$, and a model or interpretation of a Lawvere theory is a (non-strict) product preserving functor $M: \mathcal{L} \rightarrow \mathcal{C}$.

Thus $\aleph_0^{\text{op}}$ specifies the bare product structure (with the attendant projections, etc.) which gives us a notion of what it means to be “$n$-ary” for some given $n$; $I$ then transfers this notion of arity to the category $\mathcal{L}$, whose shape describes the specific algebraic structure in question (think of the diagrams one uses to categorically define the group axioms, for example); $M$ then gives a particular manifestation of the algebraic structure $\mathcal{L}$ on an object $M \circ I (1) \in \mathcal{C}$.

The reason I bring this up is that I like to think of operads as what results when we make the following change of perspective on Lawvere theories: whereas models of Lawvere theories are essentially given by specifying a “ground set of elements” $A \in \mathcal{C}$ and taking as the $n$-ary operations morphisms $A^n \rightarrow A$, we now consider a hypothetical category whose ($n$-indexed) objects themselves are the homsets $\mathcal{C}(A^n, A)$, along with some machinery that keeps track of what happens when we permute the argument slots.

More precisely, consider the category $\mathbf{P}$ with objects the natural numbers, and morphisms $\mathbf{P}(m,n)$ given by $\mathbf{P}(n,n) = \Sigma_n$ (the symmetric group on $n$ letters) and $\mathbf{P}(m,n) = \emptyset$ for $m \neq n$.

Let $\mathcal{V}$ be a cosmos, that is, a complete and cocomplete symmetric monoidal closed category with identity $I$ and internal hom $[-,-]$.

Fix $A \in \mathcal{V}$. The assignment $n \mapsto [A^{\otimes n}, A]$ defines a functor $\mathbf{P} \rightarrow \mathcal{V}$ (where functoriality in $\mathbf{P}$ comes from the symmetry of the tensor product in $\mathcal{V}$). This turns out to be a typical example of a $\mathcal{V}$-operad, which we call the “endomorphism operad” on $A$. In order to actually define what an operad is, we need to lay some groundwork.

(A point of notation: we will henceforth denote $A^{\otimes n}$ by $A^n$.)

We’ll need the fact that the functor $\mathcal{V}(I, -): \mathcal{V} \rightarrow \textbf{Sets}$ has a left adjoint $F$ given by $FX = \coprod_X I$. $F$ takes the product to the tensor product (since it’s a left adjoint and tensor products in $\mathcal{V}$ distributes over coproducts), and in fact we can assume that it does so strictly. Henceforth for $X \in \textbf{Sets}$ and $A \in \mathcal{V}$ we write $X \otimes A$ to actually mean $FX \otimes A$.

We then get a cosmos structure on $\mathcal{F}$ which is given by Day convolution: for $T,S \in \mathcal{F}$ we have $T \otimes S = \int^{m,n} \mathbf{P}(m+n, - ) \otimes Tm \otimes Sn$ Since we are thinking of a given $T \in \mathcal{F}$ as a collection of operations (indexed by arity) on which we can act by permuting the argument slots, we can think of $(T \otimes S) k$ as a collection of the $k$-ary operations that we obtain by freely permuting $m$ argument slots of type $T$ and $n$ argument slots of type $S$ (where $m,n$ range over all pairs such that $m+n = k$), modulo respecting the previously given actions of $\Sigma_m$ (resp. $\Sigma_n$) on $Tm$ (resp. $Sn$).

The identity is then given by $\mathbf{P}(0,-) \otimes I$.

**Associativity and symmetry of the cosmos structure.** Now let $T,S, R \in \mathcal{F}$. If we unpack the definition, draw out some diagrams, and apply some abstract nonsense, we find that
$T \otimes (S \otimes R) \simeq (T \otimes S) \otimes R \simeq \int^{m+n+k} \mathbf{P}(m+n+k, - ) \otimes Tm \otimes Sn \otimes Rk$
which we can again assume are actually equalities.

Before we address the symmetry of this monoidal structure, we make a technical point. $\mathbf{P}$ itself has a symmetric monoidal structure, given by addition. Thus for $n_1, \dots, n_m \in \mathbf{P}$ we have $n_1 + \cdots + n_m \in \mathbf{P}$. There is evidently an action of $\Sigma_m$ on this term, which we require to be in the “wrong” direction, so that $\xi \in \Sigma_m$ induces $\langle \xi \rangle: n_{\xi 1} + \cdots + n_{\xi m} \rightarrow n_1 + \cdots + n_m$ rather than the other way around.

(However, for the symmetry of the monoidal structure on $\mathcal{V}$, given a product $A_1 \otimes \cdots \otimes A_m$ we require that the action of $\Sigma_m$ on this term is in the “correct” direction, i.e. $\xi \in \Sigma_m$ induces $\langle \xi \rangle: A_1 \otimes \cdots \otimes A_m \rightarrow A_{\xi 1} \otimes \cdots \otimes A_{\xi m}$.)

We thus have:

$\begin{matrix} T_1 \otimes \cdots \otimes T_m &=& \int^{n_1, \dots, n_m} \mathbf{P}(n_1 + \cdots n_m, - ) \otimes T_1 n_1 \otimes \cdots \otimes T_{m} n_m\\ &&\\ {\langle \xi \rangle} \Big \downarrow && \Big \downarrow {\mathbf{P}(\langle \xi \rangle, -) \otimes \langle \xi \rangle}\\ &&\\ T_{\xi 1} \otimes \cdots \otimes T_{\xi m} &=& \int^{n_1, \dots, n_m} \mathbf{P}(n_{\xi 1} + \cdots n_{\xi m}, - ) \otimes T_{\xi 1} n_{\xi 1} \otimes \cdots \otimes T_{\xi m} n_{\xi m}\\ \end{matrix}$

Now $\langle \xi \rangle: n_{\xi 1} + \cdots + n_{\xi m} \rightarrow n_1 + \cdots + n_m$ extends to an action $\langle \xi \rangle: T_1 \otimes \cdots \otimes T_m \rightarrow T_{\xi 1} \otimes \cdots \otimes T_{\xi m}$ as we saw previously. Therefore we now have a functor $\mathbf{P}^{\text{op}} \times \mathcal{F} \rightarrow \mathcal{F}$ given by $(m, T) \mapsto T^m$, a fact which we will later use.

**$\mathcal{F}$ as a $\mathcal{V}$-category.** There is a way in which we can regard $\mathcal{V}$ as a full coreflective subcategory of $\mathcal{F}$: consider the functor $\phi: \mathcal{F} \rightarrow \mathcal{V}$ given by $\phi T = T0$. This has a right adjoint $\psi: \mathcal{V} \rightarrow \mathcal{F}$ given by $\psi A = \mathbf{P}(0, -) \otimes A$.

The inclusion $\psi$ preserves all of the relevant monoidal structure, so we are justified in considering $A \in \mathcal{V}$ as either an object of $\mathcal{V}$ or of $\mathcal{F}$ (via the inclusion $\psi$). With this notation we can write, for $A \in \mathcal{V}$ and $T,S \in \mathcal{F}$: $\mathcal{F}(A \otimes T, S) \simeq \mathcal{V}(A, [T,S])$ If $T, S \in \mathcal{F}$ then their $\mathcal{F}$-valued hom is given by $[[T,S]]$, where for $k \in \mathbf{P}$ we have $[[T,S]]k = \int_n [Tn, S(n+k)]$ and their $\mathcal{V}$-valued hom, which makes $\mathcal{F}$ into a $\mathcal{V}$-category, is given by $[T,S] = \phi [[T,S]] = \int_n [Tn, Sn]$

Let us return to our motivating example of the endomorphism operad (which we denote by $\{A,A\}$) on $A$, for a fixed $A \in \mathcal{V}$. For now it’s just an object $\{A, A\} \in \mathcal{F}$; but it contains more structure than we’re currently using. Namely, for each $m, n_1, \dots, n_m \in \mathbf{P}$ we can give a morphism $[A^m, A] \otimes \left ( [A^{n_1}, A] \otimes \cdots \otimes [A^{n_m}, A] \right ) \rightarrow [A^{n_1 + \cdots + n_m}, A]$ coming from evaluation (see the section below about the little $n$-disks operad for details). We would like a general framework for expressing such a notion of composing operations.

**Definition of an operad.** Recall from the previous section that, for given $T \in \mathcal{F}$, we can consider $n \mapsto T^n$ as a functor $\mathbf{P}^{\text{op}} \rightarrow \mathcal{F}$. We can thus define a (non-symmetric!) product $T \circ S = \int^n Tn \otimes S^n$. It is easy to check that if $S \in \mathcal{V}$ then in fact $T \circ S \in \mathcal{V}$, so that $\circ$ can be considered as a functor either of type $\mathcal{F} \times \mathcal{F} \rightarrow \mathcal{F}$ or of type $\mathcal{F} \times \mathcal{V} \rightarrow \mathcal{V}$.

The clarity with which Kelly’s paper demonstrates the various important properties of this substitution product would be difficult for me to improve upon, so I simply list here the punchlines, and refer the reader to the original paper for their proofs:

For $T,S \in \mathcal{F}$ and $n \in \mathbf{P}$, we have $(T \circ S)^n \simeq T^n \circ S$ which is natural in $T, S, n$. Using this and a Fubini style argument we get associativity of $\circ$.

$J = \mathbf{P}(1, - )\otimes I$ is the identity for $\circ$.

For $S \in \mathcal{F}$, $- \circ S: \mathcal{F} \rightarrow \mathcal{F}$ has the right adjoint $\{S, -\}$ given by $\{S, R\}m = [S^m, R]$. Moreover if $A \in \mathcal{V}$ then we in fact have $\mathcal{V}(T \circ A, B) \simeq \mathcal{F} (T, \{A, B\})$.

We can now define an *operad* as a monoid for $\circ$, i.e. some $T \in \mathcal{F}$ equipped with $\mu: T \circ T \rightarrow T$ and $\eta: J \rightarrow T$ satisfying the monoid axioms. Operad morphisms are morphisms $T \rightarrow T^\prime$ that respect $\mu$ and $\eta$.

**$\{A, A\}$ as an operad.** Once again we turn back to the example of $\{A, A\} \in \mathcal{F}$. Note that our choice to denote the endomorphism operad $(n \mapsto [A^n, A])$ by $\{A, A\}$ agrees with the construction of $\{A, -\}$ as the right adjoint to $- \circ A$.

There is an evident evaluation map $\{A, A\} \circ A \xrightarrow{e} A$, so that we have the composition $\{A, A\} \circ \{A, A\} \circ A \xrightarrow{1 \circ e} \{A,A\} \circ A \xrightarrow{e} A$ which by adjunction gives us $\mu:\{A,A\} \circ \{A,A\} \rightarrow \{A,A\}$ which we take as our monoid multiplication. Similarly $J \circ A \simeq A$ corresponds by adjunction to $\eta: J \rightarrow \{A, A\}$. We thus have that $\{A,A\}$ is an operad. In fact it is the “universal” operad, in the following sense:

Every operad $T \in \mathcal{F}$ gives a monad $T \circ -$ on $\mathcal{F}$, or on $\mathcal{V}$ via restriction. Given $A \in \mathcal{F}$, algebra structures $h^{\prime}: T \circ A \rightarrow A$ for the monad $T \circ -$ on $A$ correspond precisely to operad morphisms $h: T \rightarrow \{A,A\}$. In this case we say that $h$ gives an algebra structure on $A$ for the operad $T$.

There are some other aspects of operads that the paper looks at, but for this post I will abuse artistic license to talk about something else that isn’t exactly in the paper (although it is indirectly referenced): May’s little $n$-disks operad. For a great introduction to the following material I recommend Emily Riehl’s notes on Kathryn Hess’s two-part (I,II) talk on operads in algebraic topology.

Let $\mathcal{V} = (\mathbf{Top}_{\text{nice}}, \times, \{*\})$ where $\mathbf{Top}_{\text{nice}}$ is one’s favorite cartesian closed category of topological spaces, with $\times$ the appropriate product in this category.

Fix some $n \in \mathbb{N}$. For $k \in \mathbf{P}$, we let $d_n(k) = \text{sEmb}(\coprod_{k} D^n, D^n)$, the space of standard embeddings of $k$ copies of the closed unit $n$-disk in $\mathbb{R}^n$ into the closed unit $n$-disk in $\mathbb{R}^n$. By the space of standard embeddings we mean the subspace of the mapping space consisting of the maps which restrict on each summand to affine maps $x \mapsto \lambda x + c$ with $0 \leq \lambda \leq 1$.

Given $\xi \in \mathbf{P}(k, k)$ we have the evident action $\langle \xi \rangle: \text{sEmb}(\coprod_{k} D^n, D^n) \rightarrow \text{sEmb}(\coprod_{\xi k} D^n, D^n)$, which gives us a functor $d_n: \mathbf{P} \rightarrow \mathbf{Top}_{\text{nice}}$, so $d_n \in \mathcal{F}$.

Fix some $k,l \in \mathbf{P}$; then $d_n^k(l) = \int^{m_1, \dots, m_k} \mathbf{P}(m_1 + \cdots + m_k, l) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k)$, which we can roughly think of as all the different ways we can partition a total of $l$ disks into $k$ blocks, with the $i^{\text{th}}$ block having $m_i$ disks, and then map each block of $m_i$ disks into a single disk, all the while being able to permute the $l$ disks amongst themselves (without necessarily having to respect the partitions).

We then get $\mu: d_n \circ d_n \rightarrow d_n$ by composing the disk embeddings. More precisely, for each $l$ we get a morphism $\mu_l: (d_n(k) \otimes d_n^k)l \simeq d_n(k) \otimes (d_n^k(l)) \rightarrow d_n(l)$ from the following considerations:

First we note that $\begin{aligned} d_n(k) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k) &= \text{sEmb}(\coprod_k D^n, D^n) \times (\prod_{1 \leq i \leq k} \text{sEmb}(\coprod_{m_i} D^n, D^n))\\ &\simeq \text{sEmb}(D^n, D^n)^k \times (\prod_{1 \leq i \leq k} \text{sEmb}(\coprod_{m_i} D^n, D^n))\\ &\simeq \prod_{1 \leq i \leq k} (\text{sEmb}(\coprod_{m_i} D^n, D^n) \times \text{sEmb}(D^n, D^n)). \end{aligned}$ Now for each $i$ there is a map $\text{sEmb}(\coprod_{m_i} D^n, D^n) \times \text{sEmb}(D^n, D^n) \rightarrow \text{sEmb}(\coprod_{m_i}D^n, D^n)$ induced from iterated evaluation by adjunction. Then by the above, this gives a morphism $\begin{aligned} d_n(k) \otimes d_n(m_1) \otimes \cdots \otimes d_n(m_k) &\rightarrow \prod_{1 \leq i \leq k} \text{sEmb} (\coprod_{m_i} D^n, D^n)\\ &\simeq \text{sEmb}(\coprod_{m_1 + \cdots + m_k} D^n, D^n)\\ &= d_n(m_1 + \cdots + m_k). \end{aligned}$

A big reason that the little $n$-disks operad is relevant to algebraic topology is that there is a big theorem stating that a space is weakly equivalent to an $n$-fold loop space if and only if it’s an algebra for $d_n$.

One direction is straightforward: consider a space $A$ and its $n$-fold loop space $\Omega^n A$. Given an element of $d_n (k)$ and $k$ choices of “little maps” $(D^n, \partial D^n) \rightarrow (A, \ast)$, we can stitch together these little maps into one large map $(D^n, \partial D^n) \rightarrow (A,\ast)$ according to the instructions specified by the chosen element of $d_n(k)$ (where we map everything in the complement of the $k$ little disks to the basepoint in $A$). Doing this for each $k$, we get an operad morphism $d_n \rightarrow \{\Omega^n A, \Omega^n A\}$.

The other direction is much harder, and Maru gave an absolutely fantastic sketch of the basic story in our group discussions, which I hope she will post in the comments; I refrain from including it in the body of this post, partially for reasons of length and partially because I would just end up repeating verbatim what she said in the discussion.

Today my research time was spent writing in the paper by Lauren Anderson (Flatiron) about the *TGAS* color–magnitude diagram. I think of it as being a probabilistic inference in which we put a prior on stellar distances and then infer the distance. But that isn't correct! It is an inference in which we put a prior on the color–magnitude diagram, and then, given noisy color and (apparent) magnitude information, this turns into an (effective, implicit) prior on distance. This *Duh!* moment led to some changes to the method section!

The stars group meeting today wandered into dangerous territory, because it got me on my soap box! The points of discussion were: Are there biases in the *Gaia TGAS* parallaxes? and How could we use proper motions responsibly to constrain stellar parallaxes? Keith Hawkins (Columbia) is working a bit on the former, and I am thinking of writing something short with Boris Leistedt (NYU) on the latter.

The reason it got me on my soap-box is a huge set of issues about whether catalogs should deliver likelihood or posterior information. My view—and (I think) the view of the *Gaia DPAC*—is that the *TGAS* measurements and uncertainties are *parameters of a parameterized model of the likelihood function*. They are not parameters of a posterior, nor the output of any Bayesian inference. If they *were* outputs of a Bayesian inference, they could not be used in hierarchical models or other kinds of subsequent inferences without a factoring out of the *Gaia*-team prior.

This view (and this issue) has implications for what we are doing with our (Liestedt, Hawkins, Anderson) models of the color–magnitude diagram. If we output posterior information, we have to also output prior information for our stuff to be used by normals, down-stream. Even *with* such output, the results are hard to use correctly. We have various papers, but they are hard to read!

One comment is that, if the *Gaia TGAS* contains likelihood information, then the right way to consider its possible biases or systematic errors is to build a better model of the likelihood function, given their outputs. That is, the systematics should be created to be adjustments to the likelihood function, not posterior outputs, if at all possible.

Another comment is that negative parallaxes make sense for a likelihood function, but not (really) for a posterior pdf. Usually a sensible prior will rule out negative parallaxes! But a sensible likelihood function will permit them. The fact that the *Gaia* catalogs will have negative parallaxes is related to the fact that it is better to give likelihood information. This all has huge implications for people (like me, like Portillo at Harvard, like Lang at Toronto) who are thinking about making probabilistic catalogs. It's a big, subtle, and complex deal.