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January 7, 2022

Optimal Transport and Enriched Categories IV: Examples of Kan-type Centres

Posted by Simon Willerton

Last time we were thinking about categories enriched over ¯ +\bar{\mathbb{R}}_+, the extended non-negative reals; such enriched categories are sometimes called generalized or Lawvere metric spaces. In the context of optimal transport with cost matrix kk, thought of as a ¯ +\bar{\mathbb{R}}_+-profunctor k:𝒮k\colon \mathcal{S}\rightsquigarrow\mathcal{R} between suppliers and receivers, we were interested in the centre of the ‘Kan-type adjunction’ between enriched functor categories, which is the following:

In this post I want to give some examples of the Kan-type centre in low dimension to try to give a sense of what they look like over ¯ +\bar{\mathbb{R}}_+. Here’s the simplest kind of example we will see.


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Posted by John Baez

The Topos Institute has a new seminar:

The talks will be streamed and also recorded on YouTube.

It’s a new seminar series on the mathematics of interacting systems, their composition, and their behavior. Split in equal parts theory and applications, we are particularly interested in category-theoretic tools to make sense of information-processing or adaptive systems, or those that stand in a ‘bidirectional’ relationship to some environment. We aim to bring together researchers from different communities, who may already be using similar-but-different tools, in order to improve our own interaction.

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January 2, 2022

Adjoint School 2022

Posted by John Baez

Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the Adjoint School.

You can apply to be a student at the 2022 Adjoint School now, and applications are due January 29th! Go here:

Read on for more about how this works!

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December 28, 2021

Spaces of Extremal Magnitude

Posted by Tom Leinster

Mark Meckes and I have a new paper on magnitude!

Tom Leinster and Mark Meckes, Spaces of extremal magnitude. arXiv:2112.12889, 2021.

It’s a short one: 7 pages. But it answers two questions that have been lingering since the story of magnitude began.

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December 25, 2021

The Binary Octahedral Group

Posted by John Baez

It’s been pretty quiet around the nn-Café lately! I’ve been plenty busy myself: Lisa and I just drove back from DC to Riverside with stops at Roanoke, Nashville, Hot Springs, Okahoma City, Santa Rosa (a small town in New Mexico), Gallup, and Flagstaff. A lot of great places! Hot Springs claims to have the world’s shortest street, but I’m curious what the contenders are. Tomorrow I’m supposed to talk with James Dolan about hyperelliptic curves. And I’m finally writing a paper about the number 24.

But for now, here’s a little Christmas fun with Platonic solids and their symmetries. For more details, see:

All the exciting animations in my post here were created by Greg. And if you click on any of the images in my post here, you’ll learn more.

Posted at 3:04 AM UTC | Permalink | Followups (20)

December 4, 2021

Surveillance Publishing

Posted by John Baez

Björn Brembs recently explained how

“massive over-payment of academic publishers has enabled them to buy surveillance technology covering the entire workflow that can be used not only to be combined with our private data and sold, but also to make algorithmic (aka ‘evidenceled’) employment decisions.”

Reading about this led me to this article:

It’s all about what publishers are doing to make money by collecting data on the habits of their readers. Let me quote a bunch!

Posted at 11:49 PM UTC | Permalink | Followups (12)

December 1, 2021

Mysterious Triality

Posted by David Corfield

When we started this blog back in 2006 my co-founders were both interested in higher gauge theory. Their paths diverged as Urs looked to adapt these constructions to formulate the elusive M-theory.

Over the years I’ve been following this work, which has taken up a proposal by Hisham Sati in Framed M-branes, corners, and topological invariants, Sec 2.5 that M-theory be understood in terms of 4-cohomotopy, culminating in what they call Hypothesis H. I even chipped in sufficiently to one article to be included with them as an author:

Philosophically speaking, I’ve been intrigued by the idea that the novel mathematical framework of twisted equivariant differential cohomology theory, required for Hypothesis H, may be formulated via modal homotopy type theory. This is the line of thought I mentioned a few weeks ago in Dynamics of Reason Revisited.

But I’ve also been thinking that you can’t add something important to fundamental physics without it causing ripples through mathematics. So I was interested to see appear yesterday:

Posted at 8:04 AM UTC | Permalink | Followups (17)

November 23, 2021

Compositional Thermostatics

Posted by John Baez

At the Topos Institute this summer, a group of folks started talking about thermodynamics and category theory. It probably started because Spencer Breiner and my former student Joe Moeller, both working at the National Institute of Standards and Technology, were talking about thermodynamics with some people there. But I’ve been interested in thermodynamics for quite a while now — and Owen Lynch, a grad student visiting from the University of Utrecht, wanted to do his master’s thesis on the subject. He’s now working with me. Sophie Libkind, David Spivak and David Jaz Myers also joined in: they’re especially interested in open systems and how they interact.

Prompted by these conversations, a subset of us eventually wrote a paper on the foundations of equilibrium thermodynamics:

Posted at 6:44 PM UTC | Permalink | Followups (16)

November 17, 2021

Large Sets: The Movie

Posted by Tom Leinster

Earlier this year, I wrote a series of blog posts on large sets — or large cardinals, if you prefer — in categorical set theory. Thinking about large sets in Glasgow’s beautiful green spaces, writing those posts, and chatting about them with people here at the Café was one of the highlights of my summer.

Juan Orendain at the Universidad Nacional Autónoma de México was kind enough to invite me to give a talk in their category theory seminar, which I did today. I chose to speak about large sets, first giving a short introduction to categorical set theory, and then explaining some of the key points from this summer’s blog posts.

You can watch the video or read the slides.

Posted at 10:48 PM UTC | Permalink | Followups (8)

November 8, 2021

Causality in Machine Learning

Posted by David Corfield

Back when we started the Café in 2006, I was working as a philosopher embedded with a machine learning group in the Max Planck Institute in Tübingen. Here I am reporting on my contribution to a NIPS workshop, held amongst the mountains of Whistler, on how one may still be able to learn when the distributions from which data is drawn for training and testing purposes differ. My proposal was that background knowledge, much of it causal, had to be deployed. It turns out that a video of the talk is still available – links to this and the resulting book chapter, Projection and Projectability, are here.

I was reminded of this work recently after seeing the strides taken by the machine learning community to integrate causal graphical models with their statistical techniques in Towards Causal Representation Learning and Causality for Machine Learning. Who knows? Perhaps my talk, which was after all addressed to some of these people, sowed a seed.

Posted at 10:15 AM UTC | Permalink | Followups (4)

October 30, 2021

Firoozbakht’s Conjecture

Posted by John Baez

The Iranian mathematician Farideh Firoozbakht made a strong conjecture in 1982: the nnth root of the nnth prime keeps getting smaller as we make nn bigger! For example:

21>32>53>74>115> \sqrt[1]{2} \; > \; \sqrt[2]{3} \; > \; \sqrt[3]{5} \; > \; \sqrt[4]{7} \; > \; \sqrt[5]{11} \; > \; \cdots

It’s been checked for all primes up to about 18 quintillion, but nobody has a clue how to prove it. In fact, some experts think it’s probably false.

Posted at 2:35 AM UTC | Permalink | Followups (9)

October 28, 2021

Learn Applied Category Theory!

Posted by John Baez

Do you like the idea of learning applied category theory by working on a project, as part of a team led by an expert? If you’re an early career researcher you can apply to do that now!

After working with your team online, you’ll take an all-expenses-paid trip to a conference center in upstate New York for a week in the summer. There will be a pool, bocci, lakes with canoes, woods to hike around in, campfires at night… and also whiteboards, meeting rooms, and coffee available 24 hours a day to power your research!

Later you’ll get invited to the 2023 Joint Mathematics Meetings in Boston.

Posted at 6:43 PM UTC | Permalink | Followups (3)

October 25, 2021

The Kuramoto–Sivashinsky Equation (Part 2)

Posted by John Baez

I love the Kuramoto–Sivashinsky equation, beautifully depicted here by Thien An, because it’s one of the simplest partial differential equations that displays both chaos and a visible ‘arrow of time’. Now we’ll see that it’s also invariant under ‘Galilean transformations’ — that is, transformations into a moving frame of reference.

Posted at 3:07 AM UTC | Permalink | Followups (2)

October 22, 2021

The Kuramoto–Sivashinsky Equation (Part 1)

Posted by John Baez

I love this movie showing a solution of the Kuramoto–Sivashinsky equation, made by Thien An. If you haven’t seen her great math images on Twitter, check them out!

I hadn’t known about this equation, and it looked completely crazy to me at first. But it turns out to be important, because it’s one of the simplest partial differential equations that exhibits chaotic behavior and an ‘arrow of time’: that is, a difference between the future and past.

Posted at 4:07 AM UTC | Permalink | Followups (1)

October 20, 2021

What is the Uniform Distribution?

Posted by Tom Leinster

Today I gave the Statistics and Data Science seminar at Queen Mary University of London, at the kind invitation of Nina Otter. There I explained an idea that arose in work with Emily Roff. It’s an answer to this question:

What is the “canonical” or “uniform” probability distribution on a metric space?

You can see my slides here, and I’ll give a lightning summary of the ideas now.

Posted at 4:06 PM UTC | Permalink | Followups (39)