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February 7, 2019

Applied Category Theory 2019

Posted by John Baez

I hope to see you at this conference!

Here’s some information about it, such as how to submit papers.

Posted at 7:42 AM UTC | Permalink | Post a Comment

February 4, 2019

Jacobi Manifolds

Posted by John Baez

Here at the conference Foundations of Geometric Structures of Information 2019, Aïssa Wade of Penn State gave a talk about Jacobi manifolds. She got my attention with these words: “Poisson geometry is a good framework for classical mechanics, while contact geometry is the right framework for classical thermodynamics. Jacobi manifolds are a natural bridge between these.”

So what’s a Jacobi manifold?

Posted at 7:40 PM UTC | Permalink | Followups (8)

January 28, 2019

Symposium on Compositional Structures 3

Posted by John Baez

One of the most lively series of conferences on applied category theory is ‘SYCO’: the Symposium on Compositional Structures. And the next one is coming soon!

Symposium on Compositional Structures 3, University of Oxford, 27-28 March, 2019.

Posted at 11:03 PM UTC | Permalink | Followups (1)

January 27, 2019

On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Posted by Simon Willerton

guest post by Heiko Gimperlein and Magnus Goffeng

This fourth and final blog post concludes our discussion of the magnitude function of Euclidean domains. The previous post and this one discuss the following prototypical open problems that connect magnitude to other fields of mathematics:

  • Topology: In what sense (i.e. in which topology) is the magnitude function continuous in the Euclidean domain?
  • Geometry: Can one “magnitude the shape of a drum”?
  • Analysis: Is convexity detected by the poles of the magnitude function?

Below the fold we address the geometry and analysis of magnitude. Topological problems, examples and counterexamples were discussed in our third blog post.

Posted at 11:38 AM UTC | Permalink | Followups (15)

January 25, 2019

Magnitude Workshop, Edinburgh, July 2019

Posted by Tom Leinster

There are always many reasons to visit Edinburgh, especially in the summer (with the world’s largest arts festival). But there are more reasons than ever this summer: not only are we offering Category Theory 2019, but we’re also running a workshop on magnitude. It’s on Thursday 4 and Friday 5 July 2019, which are the two weekdays immediately before the category theory meeting.

Aerial view of Edinburgh

Click for incredible full image. Photo by Alfred G. Buckham, c. 1920.

Posted at 5:43 PM UTC | Permalink | Followups (1)

January 23, 2019

On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective (a)

Posted by Simon Willerton

guest post by Heiko Gimperlein and Magnus Goffeng

In this third in a series of blog posts on the magnitude function of Euclidean domains, we shall state from a more general perspective and discuss some new problems that we think might lead to a better understanding of magnitude. The starting point for these problems is in the analogy that our work draws between magnitudes and geometric analysis, an analogy we briefly explained in our first and second blogposts.

While working on magnitudes several questions arose that we have not been able to answer (yet). As they might lead to interesting connections between magnitudes and other fields of mathematics, we hope that they could be of interest to the readers of this blog. The questions come in three different flavors: topological continuity properties, geometric content and (particularly non-asymptotic) analytic results. We list some prototypical problems:

  • Topological: In what sense (i.e. in which topology) is the magnitude function continuous in the Euclidean domain?
  • Geometric: Can one “magnitude the shape of a drum”?
  • Analytic: Is convexity detected by the poles of the magnitude function?

Below you find out more about topological problems, examples and counterexamples. A forthcoming post looks into the geometry and analysis of magnitude.

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

January 21, 2019

The Passing of Michael Atiyah and Andrew Ranicki

Posted by Simon Willerton

guest post by Bruce Bartlett

Michael Atiyah, the celebrated geometer, passed away on Friday 11 January. He was 89 years old. He achieved mathematical fame for the Index Theorem which he proved with Isadore Singer in 1963, his work with Hirzebruch on K-theory, the “Woods Hole” fixed point theorem with Raoul Bott, his classic paper “The Yang-Mills equations over Riemann surfaces” also with Bott, and many other things. He famously also “poached” Edward Witten from theoretical physics into mathematics, and the subjects have never been the same since.

Posted at 12:20 PM UTC | Permalink | Followups (2)

January 5, 2019

Applied Category Theory 2019 School

Posted by John Baez

Dear scientists, mathematicians, linguists, philosophers, and hackers:

We are writing to let you know about a fantastic opportunity to learn about the emerging interdisciplinary field of applied category theory from some of its leading researchers at the ACT2019 School. It will begin February 18, 2019 and culminate in a meeting in Oxford, July 22–26. Applications are due January 30th; see below for details.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our community’s members is as varied as the systems being studied.

The goal of the ACT2019 School is to help grow this community by pairing ambitious young researchers together with established researchers in order to work on questions, problems, and conjectures in applied category theory.

Posted at 10:44 PM UTC | Permalink | Post a Comment

December 25, 2018

HoTT 2019

Posted by John Baez

The first International Conference on Homotopy Type Theory, HoTT 2019, will take place from August 12th to 17th, 2019 at Carnegie Mellon University in Pittsburgh, USA. Here is the organizers’ announcement:

Posted at 6:21 AM UTC | Permalink | Post a Comment

December 24, 2018

Category Theory 2019

Posted by Tom Leinster

As announced here previously, the major annual category theory meeting is taking place next year in Edinburgh, on 7-13 July. And after a week in the city of Arthur Conan Doyle, James Clerk Maxwell, Dolly the Sheep and the Higgs Boson, you can head off to Oxford for the Applied Category Theory 2019.

We’re now pleased to advertise our preliminary list of invited speakers, together with key dates for others who’d like to give talks.

Posted at 3:44 PM UTC | Permalink | Followups (2)

December 23, 2018

Monads and Lawvere Theories

Posted by John Baez

guest post by Jade Master

I have a question about the relationship between Lawvere theories and monads.

Posted at 1:08 AM UTC | Permalink | Followups (2)

November 18, 2018

Modal Types Revisited

Posted by David Corfield

We’ve discussed the prospects for adding modalities to type theory for many a year, e.g., here at the Café back at Modal Types, and frequently at the nLab. So now I’ve written up some thoughts on what philosophy might make of modal types in this preprint. My debt to the people who helped work out these ideas will be acknowledged when I publish the book.

This is to be the fourth chapter of a book which provides reasons for philosophy to embrace modal homotopy type theory. The book takes in order the components: types, dependency, homotopy, and finally modality.

The chapter ends all too briefly with mention of Mike Shulman et al.’s project, which he described in his post – What Is an n-Theory?. I’m convinced this is the way to go.

PS. I already know of the typo on line 8 of page 4.

Posted at 9:14 AM UTC | Permalink | Followups (9)

November 15, 2018

Magnitude: A Bibliography

Posted by Tom Leinster

I’ve just done something I’ve been meaning to do for ages: compiled a bibliography of all the publications on magnitude that I know about. More people have written about it than I’d realized!

This isn’t an exercise in citation-gathering; I’ve only included a paper if magnitude is the central subject or a major theme.

I’ve included works on magnitude of ordinary, un-enriched, categories, in which context magnitude is usually called Euler characteristic. But I haven’t included works on the diversity measures that are closely related to magnitude.

Enjoy! And let me know in the comments if I’ve missed anything.

Posted at 12:31 AM UTC | Permalink | Followups (3)

November 12, 2018

A Well Ordering Is A Consistent Choice Function

Posted by Tom Leinster

Well orderings have slightly perplexed me for a long time, so every now and then I have a go at seeing if I can understand them better. The insight I’m about to explain doesn’t resolve my perplexity, it’s pretty trivial, and I’m sure it’s well known to lots of people. But it does provide a fresh perspective on well orderings, and no one ever taught me it, so I thought I’d jot it down here.

In short: the axiom of choice allows you to choose one element from each nonempty subset of any given set. A well ordering on a set is a way of making such a choice in a consistent way.

Posted at 1:59 AM UTC | Permalink | Followups (23)

November 2, 2018

More Papers on Magnitude

Posted by Simon Willerton

I’ve been distracted by other things for the last few months, but in that time several interesting-looking papers on magnitude (co)homology have appeared on the arXiv. I will just list them here with some vague comments. If anyone (including the author!) would like to write a guest post on any of them then do email me.

For years a standing question was whether magnitude was connected with persistent homology, as both had a similar feel to them. Here Nina relates magnitude homology with persistent homology.

In both mine and Richard’s paper on graphs and Tom Leinster and Mike Shulman’s paper on general enriched categories, it was magnitude homology that was considered. Here Richard introduces the dual theory which he shows has the structure of a non-commutative ring.

I haven’t looked at this yet as I only discovered it last night. However, when I used to think a lot about gerbes and Deligne cohomology I was a fan of Kiyonori Gomi’s work with Yuji Terashima on higher dimensional parallel transport.

This is the write-up of some results he announced in a discussion here at the Café. These results answered questions asked by me and Richard in our original magnitude homology for graphs paper, for instance proving the expression for magnitude homology of cyclic graphs that we’d conjectured and giving pairs of graphs with the same magnitude but different magnitude homology.

Posted at 11:05 AM UTC | Permalink | Followups (1)