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January 10, 2018

On the Magnitude Function of Domains in Euclidean Space, I

Posted by Simon Willerton

Guest post by Heiko Gimperlein and Magnus Goffeng.

The magnitude of a metric space was born, nearly ten years ago, on this blog, although it went by the name of cardinality back then. There has been much development since (for instance, see Tom Leinster and Mark Meckes’ survey of what was known in 2016). Basic geometric questions about magnitude, however, remain open even for compact subsets of n\mathbb{R}^n: Tom Leinster and Simon Willerton suggested that magnitude could be computed from intrinsic volumes, and the algebraic origin of magnitude created hopes for an inclusion-exclusion principle.

In this sequence of three posts we would like to discuss our recent article, which is about asymptotic geometric content in the magnitude function and also how it relates to scattering theory.

For “nice” compact domains in n\mathbb{R}^n we prove an asymptotic variant of Leinster and Willerton’s conjecture, as well as an asymptotic inclusion-exclusion principle. Starting from ideas by Juan Antonio Barceló and Tony Carbery, our approach connects the magnitude function with ideas from spectral geometry, heat kernels and the Atiyah-Singer index theorem.

We will also address the location of the poles in the complex plane of the magnitude function. For example, here is a plot of the poles and zeros of the magnitude function of the 2121-dimensional ball.

poles and zeros of the magnitude function of the 21-dim ball

We thank Simon for inviting us to write this post and also for his paper on the magnitude of odd balls as the computations in it rescued us from some tedious combinatorics.

The plan for the three café posts is as follows:

  1. State the recent results on the asymptotic behaviour as a metric space is scaled up and on the meromorphic extension of the magnitude function.

  2. Discuss the proof in the toy case of a compact domain XX\subseteq \mathbb{R} and indicate how it generalizes to arbitrary odd dimension.

  3. Consider the relationship of the methods to geometric analysis and potential ramifications; also state some open problems that could be interesting.

Posted at 5:11 PM UTC | Permalink | Followups (13)

December 23, 2017

An M5-Brane Model

Posted by John Baez

When you try to quantize 10-dimensional supergravity theories, you are led to some theories involving strings. These are fairly well understood, because the worldsheet of a string is 2-dimensional, so string theories can be studied using 2-dimensional conformal quantum field theories, which are mathematically tractable.

When you try to quantize 11-dimensional supergravity, you are led to a theory involving 2-branes and 5-branes. People call it M-theory, because while it seems to have magical properties, our understanding of it is still murky — because it involves these higher-dimensional membranes. They have 3- and 6-dimensional worldsheets, respectively. So, precisely formulating M-theory seems to require understanding certain quantum field theories in 3 and 6 dimensions. These are bound to be tougher than 2d quantum field theories… tougher to make mathematically rigorous, for example… but even worse, until recently people didn’t know what either of these theories were!

In 2008, Aharony, Bergman, Jafferis and Maldacena figured out the 3-dimensional theory: it’s a supersymmetric Chern–Simons theory coupled to matter in a way that makes it no longer a topological quantum field theory, but still conformally invariant. It’s now called the ABJM theory. This discovery led to the ‘M2-brane mini-revolution’, as various puzzles about M-theory got solved.

The 6-dimensional theory has been much more elusive. It’s called the (0,2) theory. It should be a 6-dimensional conformal quantum field theory. But its curious properties got people thinking that it couldn’t arise from any Lagrangian — a serious roadblock, given how physicists normally like to study quantum field theories. But people have continued avidly seeking it, and not just for its role in a potential ‘theory of everything’. Witten and others have shown that if it existed, it would shed new light on Khovanov duality and geometric Langlands correspondence! The best introduction is here:

Posted at 3:42 PM UTC | Permalink | Followups (54)

December 21, 2017

Arithmetic Gauge Theory

Posted by David Corfield

Around 2008-9 we had several exchanges with Minhyong Kim here at the Café, in particular on his views of approaching number theory from a homotopic perspective, in particular in the post Kim on Fundamental Groups in Number Theory. (See also the threads Afternoon Fishing and The Elusive Proteus.)

I even recall proposing a polymath project based on his ideas in Galois Theory in Two Variables. Something physics-like was in the air, and this seemed a good location with two mathematical physicists as hosts, John having extensively written on number theory in This Week’s Finds.

Nothing came of that, but it’s interesting to see Minhyong is very much in the news these days, including in a popular article in Quanta magazine, Secret Link Uncovered Between Pure Math and Physics.

The Quanta article has Minhyong saying:

“I was hiding it because for many years I was somewhat embarrassed by the physics connection,” he said. “Number theorists are a pretty tough-minded group of people, and influences from physics sometimes make them more skeptical of the mathematics.”

Café readers had an earlier alert from an interview I conducted with Minhyong, reported in Minhyong Kim in The Reasoner. There he was prepared to announce

The work that occupies me most right now, arithmetic homotopy theory, concerns itself very much with arithmetic moduli spaces that are similar in nature and construction to moduli spaces of solutions to the Yang-Mills equation.

Now his articles are appearing bearing explicit names such as ‘Arithmetic Chern-Simons theory’ (I and II), and today, we have Arithmetic Gauge Theory: A Brief Introduction.

Posted at 11:47 AM UTC | Permalink | Followups (3)

December 19, 2017

On Writing Short Papers

Posted by Mike Shulman

In the old days, when mathematics journals were all published on paper, there were hard budgetary constraints on the number of pages available in any issue, so long papers were naturally a much harder sell than short ones. But now that the primary means of dissemination of papers is electronic, this should no longer be the case. So journals that still impose draconian page constraints (I’m looking at you, CS conference proceedings), or reject papers because they are too long, are just a holdover from the past, an annoyance to be put up with until they die out.

At least, that’s what I used to believe.

Posted at 11:24 PM UTC | Permalink | Followups (24)

SageMath and 3D Models in Webpages

Posted by Simon Willerton

I want to write a few posts (which means at least one!) on things I’ve done around SageMath, not necessarily about SageMath, but using that as a springboard. Here I’ll say how I used it to help visualization – for both the students and me! – in the differential geometry course I’ve been teaching this semester.

SageMath (formerly SAGE) is a computer algebra system like Mathematica, Maple and MATLAB. However, unlike those other systems, it doesn’t start with an ‘M’. More importantly though, it is an open source project which, amongst other things, provides a unified ‘front-end’ for many other pieces of open source mathematical software such as Maxima, PARI and GAP. Having been using Maple since I was a PhD student, I started to make the switch to SageMath a couple of years ago, which was not that easy as the biggest problem with SageMath is the lack of introductory material and documentation, although this is now definitely improving, see for instance the book Mathematical Computation with SageMath, originally available in French.

Now onto visualization, here is a static picture of a catenoid surface.

catenoid_static.png

Beneath the fold I’ll explain two ways in which you can use SageMath to embed a rotatable model of the surface in a webpage. All being well, in the main body of the post you should be able to play with the catenoid yourself.

This is, in some sense, a follow-on from one of the first posts I wrote here on using blender for creating 3d models of surface diagrams, nearly eight years ago.

Posted at 12:29 PM UTC | Permalink | Followups (3)

December 16, 2017

Entropy Modulo a Prime (Continued)

Posted by Tom Leinster

In the comments last time, a conversation got going about pp-adic entropy. But here I’ll return to the original subject: entropy modulo pp. I’ll answer the question:

Given a “probability distribution” mod pp, that is, a tuple π=(π 1,,π n)(/p) n \pi = (\pi_1, \ldots, \pi_n) \in (\mathbb{Z}/p\mathbb{Z})^n summing to 11, what is the right definition of its entropy H p(π)/p? H_p(\pi) \in \mathbb{Z}/p\mathbb{Z}?

Posted at 4:53 PM UTC | Permalink | Followups (19)

December 14, 2017

Entropy Modulo a Prime

Posted by Tom Leinster

In 1995, the German geometer Friedrich Hirzebruch retired, and a private booklet was put together to mark the occasion. That booklet included a short note by Maxim Kontsevich entitled “The 1121\tfrac{1}{2}-logarithm”.

Kontsevich’s note didn’t become publicly available until five years later, when it was included as an appendix to a paper on polylogarithms by Philippe Elbaz-Vincent and Herbert Gangl. Towards the end, it contains the following provocative words:

Conclusion: If we have a random variable ξ\xi which takes finitely many values with all probabilities in \mathbb{Q} then we can define not only the transcendental number H(ξ)H(\xi) but also its “residues modulo pp” for almost all primes pp !

Kontsevich’s note was very short and omitted many details. I’ll put some flesh on those bones, showing how to make sense of the sentence above, and much more.

Posted at 11:00 PM UTC | Permalink | Followups (12)

December 11, 2017

From the Icosahedron to E8

Posted by John Baez

Here’s a draft of a little thing I’m writing for the Newsletter of the London Mathematical Society. The regular icosahedron is connected to many ‘exceptional objects’ in mathematics, and here I describe two ways of using it to construct E 8 \mathrm{E}_8. One uses a subring of the quaternions called the ‘icosians’, while the other uses Du Val’s work on the resolution of Kleinian singularities. I leave it as a challenge to find the connection between these two constructions!

(Dedicated readers of this blog may recall that I was struggling with the second construction in July. David Speyer helped me a lot, but I got distracted by other work and the discussion fizzled. Now I’ve made more progress… but I’ve realized that the details would never fit in the Newsletter, so I’m afraid anyone interested will have to wait a bit longer.)

You can get a PDF version here:

From the icosahedron to E8.

But blogs are more fun.

Posted at 1:13 AM UTC | Permalink | Followups (16)

December 4, 2017

The 2-Dialectica Construction: A Definition in Search of Examples

Posted by Mike Shulman

An adjunction is a pair of functors f:ABf:A\to B and g:BAg:B\to A along with a natural isomorphism

A(a,gb)B(fa,b). A(a,g b) \cong B(f a,b).

Question 1: Do we get any interesting things if we replace “isomorphism” in this definition by something else?

  • If we replace it by “function”, then the Yoneda lemma tells us we get just a natural transformation fg1 Bf g \to 1_B.
  • If we replace it by “retraction” then we get a unit and counit, as in an adjunction, satisfying one triangle identity but not the other.
  • If AA and BB are 2-categories and we replace it by “equivalence”, we get a biadjunction.
  • If AA and BB are 2-categories and we replace it by “adjunction”, we get a sort of lax 2-adjunction (a.k.a. “local adjunction”)

Are there other examples?

Question 2: What if we do the same thing for multivariable adjunctions?

A two-variable adjunction is a triple of functors f:A×BCf:A\times B\to C and g:A op×CBg:A^{op}\times C\to B and h:B op×CAh:B^{op}\times C\to A along with natural isomorphisms

C(f(a,b),c)B(b,g(a,c))A(a,h(b,c)). C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c)).

What does it mean to “replace ‘isomorphism’ by something else” here? It could mean different things, but one thing it might mean is to ask instead for a function

A(a,h(b,c))×B(b,g(a,c))C(f(a,b),c). A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c).

Even more intriguingly, if A,B,CA,B,C are 2-categories, we could ask for an ordinary two-variable adjunction between these three hom-categories; this would give a certain notion of “lax two-variable 2-adjunction”. Question 2 is, are notions like this good for anything? Are there any natural examples?

Now, you may, instead, be wondering about

Question 3: In what sense is a function A(a,h(b,c))×B(b,g(a,c))C(f(a,b),c) A(a,h(b,c)) \times B(b,g(a,c)) \to C(f(a,b),c) a “replacement” for isomorphisms C(f(a,b),c)B(b,g(a,c))A(a,h(b,c)) C(f(a,b),c) \cong B(b,g(a,c)) \cong A(a,h(b,c)) ?

But that question, I can answer; it has to do with comparing the Chu construction and the Dialectica construction.

Posted at 10:26 PM UTC | Permalink | Followups (4)

November 23, 2017

Real Sets

Posted by John Baez

Good news! Janelidze and Street have tackled some puzzles that are perennial favorites here on the nn-Café:

  • George Janelidze and Ross Street, Real sets, Tbilisi Mathematical Journal, 10 (2017), 23–49.

Abstract. After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions:

• what is a set with half an element?

• what is a set with π elements?

The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories series monoidal and conclude by only briefly mentioning the non-commutative possibility called ω-monoidal. We include some remarks on sets having cardinalities in [,][-\infty,\infty].

Posted at 6:52 AM UTC | Permalink | Followups (21)

November 22, 2017

Internal Languages of Higher Categories II

Posted by Emily Riehl

Guest post by Chris Kapulkin

Two years ago, I wrote a post for the n-Cafe, in which I sketched how to make precise the claim that intensional type theory (and ultimately HoTT) is the internal language of higher category theory. I wrote it in response to a heated discussion on Michael Harris’ blog Mathematics Without Apologies about what (if anything) HoTT is good for, with the goal of making certain arguments mathematically precise, rather than taking a side in the discussion. Back then, I had very little hope that we will be able to carry out the work necessary to prove these conjectures in any foreseeable future, but things have changed and the purpose of this post is to report on the recent progress.

First off, a tl;dr version.

  • Shortly after my post, Mike Shulman posted a paper describing a new class of models of the Univalence Axiom, this time in categories of simplicial presheaves over what he calls EI-categories.
  • Peter Lumsdaine and I figured out how to equip the category of type theories with a left semi-model structure and were able to give a precise statement of the internal language conjectures.
  • Mike Shulman proposed a tentative definition of an elementary \infty-topos, which, conjecturally, gives the higher-categorical counterpart of HoTT.
  • A few weeks ago, Karol Szumiło and I proved a version of the first of the conjectures, relating type theories with Id\mathrm{Id}- and Σ\Sigma-types with finitely cocomplete \infty-categories.
  • And maybe the most surprising of all: Michael Harris and I are organizing a conference together (and you should attend it!).
Posted at 2:39 PM UTC | Permalink | Followups (9)

November 17, 2017

Star-autonomous Categories are Pseudo Frobenius Algebras

Posted by Mike Shulman

A little while ago I talked about how multivariable adjunctions naturally form a polycategory: a structure like a multicategory, but in which codomains as well as domains can involve multiple objects. Now I want to talk about some structures we can define inside this polycategory MVarMVar.

What can you define inside a polycategory? Well, to start with, a polycategory has an underlying multicategory, consisting of the arrows with unary target; so anything we can define in a multicategory we can define in a polycategory. And the most basic thing we can define in a multicategory is a monoid object — in fact, there are some senses in which this is the canonical thing we can define in a multicategory.

So what is a monoid object in MVarMVar?

Posted at 7:28 PM UTC | Permalink | Followups (4)

November 13, 2017

HoTT at JMM

Posted by Mike Shulman

At the 2018 U.S. Joint Mathematics Meetings in San Diego, there will be an AMS special session about homotopy type theory. It’s a continuation of the HoTT MRC that took place this summer, organized by some of the participants to especially showcase the work done during and after the MRC workshop. Following is the announcement from the organizers.

Posted at 10:03 PM UTC | Permalink | Followups (2)

November 11, 2017

Topology Puzzles

Posted by John Baez

Let’s say the closed unit interval [0,1][0,1] maps onto a metric space XX if there is a continuous map from [0,1][0,1] onto XX. Similarly for the Cantor set.

Puzzle 0. Does the Cantor set map onto the closed unit interval, and/or vice versa?

Puzzle 1. Which metric spaces does the closed unit interval map onto?

Puzzle 2. Which metric spaces does the Cantor set map onto?

The first one is easy; the second two are well-known… but still, perhaps, not well-known enough!

Posted at 4:38 AM UTC | Permalink | Followups (26)

November 9, 2017

The 2-Chu Construction

Posted by Mike Shulman

Last time I told you that multivariable adjunctions (“polyvariable adjunctions”?) form a polycategory MVarMVar, a structure like a multicategory but in which codomains as well as domains can involve multiple objects. This time I want to convince you that MVarMVar is actually (a subcategory of) an instance of an exceedingly general notion, called the Chu construction.

As I remarked last time, in defining multivariable adjunctions we used opposite categories. However, we didn’t need to know very much about the opposite of a category AA; essentially all we needed is the existence of a hom-functor hom A:A op×ASethom_A : A^{op}\times A \to Set. This enabled us to define the representable functors corresponding to multivariable morphisms, so that we could then ask them to be isomorphic to obtain a multivariable adjunction. We didn’t need any special properties of the category SetSet or the hom-functor hom Ahom_A, only that each AA comes equipped with a map hom A:A op×ASethom_A : A^{op}\times A \to Set. (Note that this is sort of “half” of a counit for the hoped-for dual pair (A,A op)(A,A^{op}), or it would be if SetSet were the unit object; the other half doesn’t exist in CatCat, but it does once we pass to MVarMVar.)

Furthermore, we didn’t need any cartesian properties of the product ×\times; it could just as well have been any monoidal structure, or even any multicategory structure! Finally, if we’re willing to end up with a somewhat larger category, we can give up the idea that each AA should be equipped with A opA^{op} and hom Ahom_A, and instead allow each objects of our “generalized MVarMVar” to make a free choice of its “opposite” and “hom-functor”.

Posted at 12:09 PM UTC | Permalink | Followups (6)