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May 17, 2022

The Magnitude of Information

Posted by Tom Leinster

Guest post by Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca

The magnitude of a metric space (X,d)(X,d) does not require further introduction on this blog. Two of the hosts, Tom Leinster and Simon Willerton, conjectured that the magnitude function X(R):=Mag(X,Rd)\mathcal{M}_X(R) := \mathrm{Mag}(X,R \cdot \mathrm{d}) of a convex body X nX \subset \mathbb{R}^n with Euclidean distance d\mathrm{d} captures classical geometric information about XX:

X(R)= 1n!ω nvol n(X)R n+12(n1)!ω n1vol n1(X)R n1++1 = 1n!ω n j=0 nc j(X)R nj\begin{aligned} \mathcal{M}_X(R) =& \frac{1}{n! \omega_n} \mathrm{vol}_n(X)\ R^n + \frac{1}{2(n-1)! \omega_{n-1}} \mathrm{vol}_{n-1}(\partial X)\ R^{n-1} + \cdots + 1 \\ =& \frac{1}{n! \omega_n} \sum_{j=0}^n c_j(X)\ R^{n-j} \end{aligned}

where c j(X)=γ j,nV j(X)c_j(X) = \gamma_{j,n} V_j(X) is proportional to the jj-th intrinsic volume V jV_j of XX and ω n\omega_n is the volume of the unit ball in n\mathbb{R}^n.

Even more basic geometric questions have remained unknown, including:

  • What geometric content is encoded in X\mathcal{M}_X?
  • What can be said about the magnitude function of the unit disk B 2 2B_2 \subset \mathbb{R}^2?

We discuss in this post how these questions led us to possible relations to information geometry. We would love to hear from you:

  • Is magnitude an interesting invariant for information geometry?
  • Is there a category theoretic motivation, like Lawvere’s view of a metric space as an enriched category?
  • Does the magnitude relate to notions studied in information geometry?
  • Do you have interesting questions about this invariant?
Posted at 11:44 AM UTC | Permalink | Followups (10)

May 14, 2022

Grothendieck Conference

Posted by John Baez

There’s a conference on Grothendieck’s work coming up soon here in Southern California!

  • Grothendieck’s approach to mathematics, May 24-28, 2022, Chapman University, Orange, California. Organized by Peter Jipsen (Mathematics), Alexander Kurz (Computer Science), Andrew Mosher (Mathematics and Computer Science), Marco Panza (Mathematics and Philosophy), Ahmed Sebbar (Physics and Mathematics), Daniele Struppa (Mathematics).

To attend in person register here. To attend via Zoom go here. The talks will be recorded, and I hear they will be made available later on YouTube.

Posted at 9:49 PM UTC | Permalink | Post a Comment

May 9, 2022

Communicating Mathematics Conference

Posted by Emily Riehl

Communicating Mathematics is a 4-day workshop for mathematicians at all career stages who are interested in exploring how we share our research and interests with fellow mathematicians, students, and the public.

The workshop takes place August 8-11 at Cornell University and will run concurrently online over zoom.

Planned sessions and workshops include:

  • Mathematics for the common good (social and civil justice issues)

  • Engaging the public in mathematical discourse

  • Inclusivity and communication in the classroom

  • Communicating to policymakers

  • Community outreach: communicating mathematics to young people (e.g. math circles)

  • Advocating for your department (communicating to university administration)

  • Communicating with fellow mathematicians:

    • What makes an engaging research talk?

    • How should we be communicating our work to each other?

    • Succinctly describing your research to both a specialist and non-specialist.

We will also have structured breakout/lunchtime discussions on specific issues related to improving communication and dissemination.

Posted at 11:04 PM UTC | Permalink | Post a Comment

May 2, 2022

Shannon Entropy from Category Theory

Posted by John Baez

I’m giving a talk at Categorical Semantics of Entropy on Wednesday May 11th, 2022. You can watch it live on Zoom if you register, or recorded later. Here’s the idea:

Shannon entropy is a powerful concept. But what properties single out Shannon entropy as special? Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function. Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous. This is joint work with Tom Leinster and Tobias Fritz.

You can see the slides now, here.

Posted at 1:37 AM UTC | Permalink | Followups (7)

April 20, 2022

Categorical Semantics of Entropy

Posted by John Baez

There will be a symposium on the categorical semantics of entropy at the CUNY Grad Center in Manhattan on Friday May 13th, organized by John Terilla. Tai-Danae Bradley and I will give a tutorial on this subject on Wednesday May 11th. For both events you need to register to attend, either in person or via Zoom. All the talks will be recorded and made available later.

Details are below…

Posted at 6:13 PM UTC | Permalink | Post a Comment

April 10, 2022

Holomorphic Gerbes (Part 2)

Posted by John Baez

Thanks to some help from Francis, I’ve converted some of my conjectures on the classification of holomorphic nn-gerbes into theorems!

Posted at 7:08 AM UTC | Permalink | Followups (1)

March 30, 2022

Bicategories, Categorification and Quantum Theory

Posted by Tom Leinster

Guest post by Nicola Gambino

We are pleased to announce that pre-registration for the London Mathematical Society Research School “Bicategories, categorification and quantum theory”, to be held 11th-15th July 2022 at the University of Leeds, is now open. 

The school will include mini-courses on Bicategories (Richard Garner, Macquarie University), Monoidal categories like that of Hilbert Spaces (Chris Heunen, University of Edinburgh), Categorification (Marco Mackaay, University of Algarve) and Hopf Algebras (Sonia Natale, Universidad National de Córdoba), as well as tutorials.

Posted at 11:48 PM UTC | Permalink | Followups (4)

March 27, 2022

Holomorphic Gerbes (Part 1)

Posted by John Baez

I have some guesses about holomorphic gerbes. But I don’t know much about them; what I know is a small fraction of what’s in here:

I recently blogged about the classification of holomorphic line bundles. Since gerbes are a lot like line bundles, it’s easy to guess some analogous results for holomorphic gerbes. I did that, and then looked around to see what people have already done. And it looks like I’m on the right track, though I still have lots of questions.

Posted at 6:57 AM UTC | Permalink | Followups (12)

March 19, 2022

Line Bundles on Complex Tori (Part 2)

Posted by John Baez

Last time I explained how the job of classifying holomorphic line bundles LL on a complex torus XX breaks into two parts:

  • the ‘discrete part’: the underlying topological line bundle of LL is classified by an element of a finitely generated free abelian group called the Néron–Severi group NS(X)\mathrm{NS}(X).

  • the ‘continuous part’: the holomorphic line bundles with a given underlying topological line bundle are classified by a complex torus called the Jacobian Jac(X)\mathrm{Jac}(X).

Today I want to talk more about the discrete part: the Néron–Severi group. Studying examples of this leads to beautiful pictures like this one by Roice Nelson:

Posted at 2:08 AM UTC | Permalink | Followups (2)

March 13, 2022

Line Bundles on Complex Tori (Part 1)

Posted by John Baez

A complex abelian variety is a group in the category of smooth complex projective varieties. They’re called that because — wonderfully — they turn out to all be abelian! I’ve been studying holomorphic line bundles on complex abelian varieties, which is a really nice topic with fascinating connections to quantum physics, Jordan algebras and number theory. This is the book that’s helped me the most so far:

  • Christina Birkenhake and Herbert Lange, Complex Abelian Varieties, Springer, Berlin, 2004.

But the subject is so rich that it can be hard to see the forest for the trees! So for my own benefit I’d like to describe the classification of holomorphic line bundles on an abelian variety — or more generally, any ‘complex torus’.

A complex torus is the same as the quotient of a finite-dimensional complex vector space by a lattice. Every abelian variety is a complex torus, but not every complex torus is an abelian variety: you can’t make them all into projective varieties.

I will avoid saying a lot of things people usually say about this subject, in order to keep things short.

Posted at 10:59 PM UTC | Permalink | Followups (13)

March 8, 2022

Compositional Thermostatics (Part 3)

Posted by John Baez

guest post by Owen Lynch

This is the third part (Part 1, Part 2) of a blog series on this paper:

In the previous two posts we talked about what a thermostatic system was, and how we think about composing them. In this post, we are going to back up from thermostatic systems a little bit, and talk about operads: a general framework for composing things! But we will not yet discuss how thermostatic systems use this framework — we’ll do that in the next post.

Posted at 9:32 PM UTC | Permalink | Followups (2)

March 5, 2022

Topos Institute Research Associates

Posted by John Baez

Come spend the summer at the Topos Institute! For early-career researchers, we’re excited to open up applications for our summer research associate (RA) program.

Summer RAs are an important part of life at Topos — they help explore new directions relevant to Topos projects, and they bring new ideas, energy, and expertise to our research groups. This year we’ll welcome a new cohort of RAs to our offices in Berkeley, CA, with the program running from June to August.

Posted at 1:50 AM UTC | Permalink | Post a Comment

February 28, 2022

Hardy, Ramanujan and Taxi No. 1729

Posted by John Baez

In his book Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, G. H. Hardy tells this famous story:

He could remember the idiosyncracies of numbers in an almost uncanny way. It was Littlewood who said every positive integer was one of Ramanujan’s personal friends. I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Namely,

10 3+9 3=1000+729=1729=1728+1=12 3+1 3 10^3 + 9^3 = 1000 + 729 = 1729 = 1728 + 1 = 12^3 + 1^3

But there’s more to this story than meets the eye.

Posted at 12:55 AM UTC | Permalink | Followups (12)

February 25, 2022

Applied Category Theory 2022

Posted by John Baez

The Fifth International Conference on Applied Category Theory, ACT2022, will take place at the University of Strathclyde from 18 to 22 July 2022, preceded by the Adjoint School 2022 from 11 to 15 July. This conference follows previous events at Cambridge (UK), Cambridge (MA), Oxford and Leiden.

Applied category theory is important to a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social science, linguistics and other subjects using category-theoretic tools. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, strengthen the applied category theory community, disseminate the latest results, and facilitate further development of the field.

Posted at 5:09 PM UTC | Permalink | Followups (14)

February 15, 2022

Questions About the Néron–Severi Group

Posted by John Baez

A friend of mine with good intuitions sometimes says things without proof, and sometimes I want to know why — or even whether — these things are true.

Here are some examples from algebraic geometry.

Posted at 8:16 PM UTC | Permalink | Followups (6)