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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 ℳ X(R):=Mag(X,R⋅d)\mathcal{M}_X(R) := \mathrm{Mag}(X,R \cdot \mathrm{d}) of a convex body X⊂ℝ nX \subset \mathbb{R}^n with Euclidean distance d\mathrm{d} captures classical geometric information about XX:

ℳ X(R)= 1n!ω nvol n(X)R n+12(n−1)!ω n−1vol n−1(∂X)R n−1+⋯+1 = 1n!ω n∑ j=0 nc j(X)R n−j\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 ℝ n\mathbb{R}^n.

Even more basic geometric questions have remained unknown, including:

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

Continue reading “Grothendieck Conference” …
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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.

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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.

Continue reading “Shannon Entropy from Category Theory” …
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