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February 20, 2019
Tychonoffication
Posted by John Baez
Joshua Meyers is a grad student in my real analysis class. We had an interesting conversation about topology and came up with some conjectures. Maybe someone has already proved them. I just want to write them down somewhere.
Sporadic SICs and Exceptional Lie Algebras I
Posted by John Baez
guest post by Blake C. Stacey
Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy. I am grateful to John for giving me the opportunity to discuss one such confluence, where quantum information theory comes together with geometry, root systems and even the octonions.
In what follows, I will be divvying up these notes into blog posts. The first step is to introduce the geometrical statement of the SIC problem. Then, we’ll establish some basics about quantum theory, which may be fairly standard if you learned out of Mike and Ike while being a little un-standard outside of quantum information.
February 7, 2019
Applied Category Theory 2019
Posted by John Baez
I hope to see you at this conference!
- Applied Category Theory 2019, July 15-19, 2019, Oxford, UK.
Here’s some information about it, such as how to submit papers.
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?
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.
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.
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.
Click for incredible full image. Photo by Alfred G. Buckham, c. 1920.
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.
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.
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.
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:
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.
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.
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.
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.