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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.

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