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March 22, 2023

Azimuth Project News

Posted by John Baez

I blog here and also on Azimuth. Here I tend to talk about pure math and mathematical physics. There I talk about the Azimuth Project.

Let me say a bit about how that’s been going. My original plans didn’t work as expected. But I joined forces with other people who came up with something pretty cool: a rather general software framework for scientific modeling, which explicitly uses abstractions such as categories and operads. Then we applied it to epidemiology.

This is the work of many people, so it’s hard to name them all, but I’ll talk about some.

Posted at 8:03 PM UTC | Permalink | Followups (1)

March 17, 2023

Jeffrey Morton

Posted by John Baez

When he was my grad student, Jeffrey Morton worked on categorifying the theory of Feynman diagrams, and describing extended topological quantum field theories using double categories.

He got his PhD in 2007. Later he did many other things. For example, together with Jamie Vicary, he did some cool work on categorifying the Heisenberg algebra using spans of spans of groupoids. This work still needs to be made fully rigorous—someone should try!

But this is about something else.

Posted at 8:56 PM UTC | Permalink | Followups (2)

March 9, 2023

Cloning in Classical Mechanics

Posted by John Baez

Everyone likes to talk about the no-cloning theorem in quantum mechanics: you can’t build a machine where you drop an electron in the top and two electrons in the same spin state as that one pop out below. This is connected to how the category of Hilbert spaces, with its usual tensor product, is non-cartesian.

Here are two easy versions of the no-cloning theorem. First, if the dimension of a Hilbert space HH exceeds 1 there’s no linear map that duplicates states:

Δ: H HH ψ ψψ \begin{array}{cccl} \Delta \colon & H & \to & H \otimes H \\ & \psi & \mapsto & \psi \otimes \psi \end{array}

Second, there’s also no linear way to take two copies of a quantum system and find a linear process that takes the state of the first copy and writes it onto the second, while leaving the first copy unchanged:

F: HH HH ψϕ ψψ \begin{array}{cccl} F \colon & H \otimes H & \to & H \otimes H \\ & \psi \otimes \phi & \mapsto & \psi \otimes \psi \end{array}

But what about classical mechanics?

Posted at 5:50 AM UTC | Permalink | Followups (17)

March 7, 2023

This Week’s Finds (101–150)

Posted by John Baez

Here’s another present for you!

I can’t keep cranking them out at this rate, since the next batch is 438 pages long and I need a break. Tim Hosgood has kindly LaTeXed all 300 issues of This Week’s Finds, but there are lots of little formatting glitches I need to fix — mostly coming from how my formatting when I initially wrote these was a bit sloppy. Also, I’m trying to add links to published versions of all the papers I talk about. So, it takes work — about two weeks of work for this batch.

So what did I talk about in Weeks 101–150, anyway?

Posted at 7:33 PM UTC | Permalink | Followups (12)

March 6, 2023

Philosophical Perspectives on Category Theory

Posted by David Corfield

This is the title of an online talk I’m giving to the Topos Institute this Thursday (17:00 UTC), 9 March. Brush up on your Fermat primes and you can join the Zoom meeting.

It’s a good opportunity to reflect on the many years devoted to the cause of promoting the philosophical significance of category theory. As storm clouds gather over the Humanities Division here at Kent, and inducements are offered for us to leave, the brighter future I envisage may come too late for me. But I don’t doubt that the first thrill of encountering category theory around 30 years ago was the intimation of a profound way of thinking.

For my most recent views on what we should make of the rise of category theory in mathematics, see Thomas Kuhn, Modern Mathematics and the Dynamics of Reason.

But perhaps it will be successes in Applied Category Theory that will prove to be unignorable, carried out by

a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social science, systems, linguistics and other subjects using category-theoretic tools. ACT2023

Posted at 4:06 PM UTC | Permalink | Followups (2)

An Invitation to Geometric Higher Categories

Posted by David Corfield

Guest post by Christoph Dorn

While the term “geometric higher category” is new, its underlying idea is not: coherences in higher structures can be derived from (stratified) manifold topology. This idea is central to the cobordism hypothesis (and to the relation of manifold singularities and dualizability structures as previously discussed on the nn-Category Café), as well as to many other parts of modern Quantum Topology. So far, however, this close relation of manifold theory and higher category theory hasn’t been fully worked out. Geometric higher category theory aims to change that, and this blog post will sketch some of the central ideas of how it does so. A slightly more comprehensive (but blog-length-exceeding) version of this introduction to geometric higher categories can be found here:

Today, I only want to focus on two basic questions about geometric higher categories: namely, what is the idea behind the connection of geometry and higher category theory? And, what are the first ingredients needed in formalizing this connection?

What is geometric about geometric higher categories?

I would like to argue that there is a useful categorization of models of higher structures into three categories. But, I will only give one good example for my argument. The absence of other examples, however, can be taken as a problem that needs to be addressed, and as one of the motivations for studying geometric higher categories! The three categories of models that I want to consider are “geometric”, “topological” and “combinatorial” models of higher structures. Really, depending on your taste, different adjectives could have been chosen for these categories: for instance, in place of “combinatorial”, maybe you find that the adjectives “categorical” or “algebraic” are more applicable for what is to follow; and in place of “geometric”, maybe saying “manifold-stratified” would have been more descriptive.

Posted at 10:32 AM UTC | Permalink | Followups (3)

March 3, 2023

Special Relativity and the Mercator Projection

Posted by John Baez

When you look at an object zipping past you at nearly the speed of light, it looks not squashed but rotated.

This phenomenon is well known: it’s called Terrell rotation. But this paper puts a new spin on it:

Posted at 8:06 PM UTC | Permalink | Followups (7)

February 21, 2023

This Week’s Finds (51–100)

Posted by John Baez

Grab a copy of this:

These are the second fifty issues of my column, from April 23, 1995 to March 23, 1997. They discuss quantum gravity, topological quantum field theory and other topics in mathematics and physics. They were typeset in 2020 by Tim Hosgood. Since then I’ve edited them more and changed most references to preprints into references to published papers and/or the arXiv. If you see typos or other problems please report them. (I already know the cover page looks weird).

By the way, I’ve also done a lot more editing of the first 50 issues, with help from Fridrich Valach, so please grab a new copy of this even if you already have it:

But let me say a bit more about what’s in issues 51–100.

Posted at 9:41 PM UTC | Permalink | Followups (11)

February 14, 2023

Category Theory Outreach Panel

Posted by John Baez

They just don’t quit! Besides their Joy of Abstraction book club, the Topos Institute also has another way for you to start learning category theory. It’s called the CT Outreach Panel, and it’s happening on March 16, 2023 at 17:00 UTC.

Some of the best explainers of category theory in the world—Emily Riehl, Eugenia Cheng, Tai-Danae Bradley, Paul Dancstep and Oliver Lugg—will explain their approaches to the subject and answer questions.

You can submit questions here:

For more details, read on….

Posted at 5:17 PM UTC | Permalink | Post a Comment

February 12, 2023

Talk on the Tenfold Way

Posted by John Baez

There are ten ways that a substance can have symmetry under time reversal, switching particles and holes, both or neither. But this fact turns out to extend far beyond condensed matter physics! It’s built into the fabric of mathematics in a deep way.

I gave a talk on this at Nicohl Furey’s seminar Algebra, Particles and Quantum Theory, and you can see a video of my talk here.

You can also watch another version, where I explain this stuff to my friend James Dolan.

Posted at 7:19 AM UTC | Permalink | Followups (15)

February 10, 2023

The Joy of Abstraction

Posted by John Baez

The Topos Institute is excited to announce a book club for The Joy of Abstraction by Eugenia Cheng.

Posted at 5:32 AM UTC | Permalink | Post a Comment

February 5, 2023

Applied Category Theory 2023

Posted by John Baez

You can now submit a paper if you want to give a talk here:

This event will be hybrid, so you can give a talk even if you can’t attend in person.

Posted at 7:57 PM UTC | Permalink | Post a Comment

January 26, 2023

Mathematics for Humanity

Posted by John Baez

I mentioned this earlier, but now it’s actually happening! I hope you can think of good workshops and apply to run them in Edinburgh.

Posted at 9:42 PM UTC | Permalink | Followups (6)

January 23, 2023

Question on Condensed Matter Physics

Posted by John Baez

The tenfold way is a mathematical classification of Hamiltonians used in condensed matter physics, based on their symmetries. Nine kinds are characterized by choosing one of these 3 options:

  • antiunitary time-reversal symmetry with T 2=1T^2 = 1, with T 2=1T^2 = -1, or no such symmetry.

and one of these 3 options:

  • antiunitary charge conjugation symmetry with C 2=1C^2 = 1, with C 2=1C^2 = -1, or no such symmetry.

(Charge conjugation symmetry in condensed matter physics is usually a symmetry between particles - e.g. electrons or quasiparticles of some sort - and holes.)

The tenth kind has unitary “SS” symmetry, a symmetry that simultaneously reverses the direction of time and interchanges particles and holes. Since it is unitary and we’re free to multiply it by a phase, we can assume without loss of generality that S 2=1S^2 = 1.

What are examples of real-world condensed matter systems of all ten kinds?

Posted at 7:43 PM UTC | Permalink | Followups (9)

January 17, 2023

The Tenfold Way (Part 8)

Posted by John Baez

Last time I explained a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.

This time I’ll do something different. I’ll explain a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.

Yes, it’s different! Not only will the details of the construction look very different, it gives a different correspondence! And I hope you can help me figure out what’s going on.

I thank Claude Schochet for pointing out that these two constructions don’t match.

Posted at 11:05 PM UTC | Permalink | Followups (4)