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August 5, 2020

Open Systems in Classical Mechanics

Posted by John Baez

I think we need a ‘compositional’ approach to classical mechanics. A classical system is typically built from parts, and we describe the whole system by describing its parts and then saying how they are put together. But this aspect of classical mechanics is typically left informal. You learn how it works in a physics class by doing lots of homework problems, but the rules are never completely spelled out, which is one reason physics is hard.

I want an approach that makes the compositionality of classical mechanics formal: a category (or categories) where the morphisms are open classical systems—that is, classical systems with the ability to interact with the outside world—and composing these morphisms describes putting together open systems to form larger open systems.

Posted at 2:22 AM UTC | Permalink | Followups (9)

August 3, 2020

Octonions and the Standard Model (Part 4)

Posted by John Baez

Last time we saw what we can do by choosing a square root of 1-1 in the octonions. They become a 4-dimensional complex vector space, and their automorphisms fixing this square root of 1-1 form the group SU(3)\mathrm{SU}(3). This is the symmetry group of the strong force —and even better, its representation on the octonions matches the one we see for one quark and one lepton in the Standard Model.

What happens if we play the same game for some larger structures built from octonions? For example 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}), the space of 3×33 \times 3 self-adjoint matrices with octonion entries?

Maybe some of you can guess where I’m going with this, but I think I should start at the beginning and go slow, so more people can jump aboard the train!

Posted at 1:03 PM UTC | Permalink | Followups (10)

July 27, 2020

Linear Logic Flavoured Composition of Petri Nets

Posted by John Baez

guest post by Elena Di Lavore and Xiaoyan Li

Petri nets are a mathematical model for systems in which processes, when activated, consume some resources and produce others. They can be used to model, among many others, business processes, chemical reactions, gene activation or parallel computations. There are different approaches to define a categorical model for Petri nets, for example, Petri nets are monoids, nets with boundaries and open Petri nets.

This first post of the Applied Category Theory Adjoint School 2020 presents the approach of Carolyn Brown and Doug Gurr in the paper A Categorical Linear Framework for Petri Nets, which is based on Valeria de Paiva’s dialectica categories. The interesting aspect of this approach is the fact that it combines linear logic and category theory to model different ways of composing Petri nets.

Posted at 12:20 AM UTC | Permalink | Followups (10)

July 24, 2020

Octonions and the Standard Model (Part 3)

Posted by John Baez

Now I’ll finally explain how a quark and a lepton fit together into an octonion — in the very simplified picture where we treat these particles merely as representations of SU(3)\mathrm{SU}(3), the symmetry group of the strong force. I’ll say just enough about physics for mathematicians to get a sense of what this means. (The most substantial part of this post will be a quick intro to ‘basic triples’, a powerful technique for working with octonions.)

Posted at 12:17 PM UTC | Permalink | Followups (7)

July 22, 2020

Octonions and the Standard Model (Part 2)

Posted by John Baez

My description of the octonions in Part 1 raised enough issues that I’d like to talk about it a bit more. I’ll show you a prettier formula for octonion multiplication in terms of 3\mathbb{C} \oplus \mathbb{C}^3… and also a very similar-looking formula for it in terms of 7\mathbb{R} \oplus \mathbb{R}^7.

Posted at 7:36 PM UTC | Permalink | Followups (9)

July 17, 2020

Octonions and the Standard Model (Part 1)

Posted by John Baez

I want to talk about some attempts to connect the Standard Model of particle physics to the octonions. I should start out by saying I don’t have any big agenda here. It’d be great if the octonions — or for that matter, anything — led to new insights in particle physics. But I don’t have such insights, and for me particle physics is just a hobby. I’m not trying to come up with a grand unified theory. I just want to explain some patterns linking the Standard Model to the octonions.

Understanding these patterns requires knowing a bit of physics and a bit of math. I’ll focus on the math side of things: mainly, I’ll be polishing up some existing ideas and trying to make them more pretty. I’ll assume you either know the physics or can fake it: either way, it won’t be the main focus.

In writing this first post, my attempt to explain an octonionic description of the strong force led me to a construction of the octonions that makes them look very much like the quaternions. I don’t know if it’s new, but I’d never seen it before. The basic idea is that octonions are to 3\mathbb{C}^3 as quaternions are to 3\mathbb{R}^3.

Posted at 1:34 AM UTC | Permalink | Followups (20)

July 8, 2020

Self-Referential Algebraic Structures

Posted by John Baez

Any group acts as automorphisms of itself, by conjugation. If we differentiate this idea, we get that any Lie algebra acts as derivations of itself. We can then enhance this in various ways: for example a Poisson algebra is both a Lie algebra and a commutative algebra, such that any element acts as derivations of both these structures.

Why do I care?

In my paper on Noether’s theorem I got excited by how physics uses structures where each element acts to generate a one-parameter group of automorphisms of that structure. I proved a super-general version of Noether’s theorem based on this idea. It’s Theorem 8, in case you’re curious.

But the purest expression of the idea of a “structure where each element acts as an automorphism of that structure” is the concept of “rack”.

Posted at 7:28 PM UTC | Permalink | Followups (24)

July 2, 2020

Congratulations, John!

Posted by Tom Leinster

Our own John Baez is famous for inspiring people all around the world through the magic of the internet, but what’s it like to actually be one of his grad students? Fantastic, apparently! The University of California at Riverside has just given him the Doctoral Dissertation Advisor/Mentoring Award, one of just two given by the university. It “celebrates UCR faculty who have demonstrated an outstanding and long history of mentorship of graduate students”.

Forgive a completely irrelevant digression, but partway through writing that paragraph, while regretting that more details of John’s prize weren’t available, something rather extraordinary forced me to stop writing…

Posted at 2:19 AM UTC | Permalink | Followups (8)

June 29, 2020

Getting to the Bottom of Noether’s Theorem

Posted by John Baez

Most of us have been staying holed up at home lately. I spent the last month holed up writing a paper that expands on my talk at a conference honoring the centennial of Noether’s 1918 paper on symmetries and conservation laws. This made my confinement a lot more bearable. It was good getting back to this sort of mathematical physics after a long time spent on applied category theory. It turns out I really missed it.

While everyone at the conference kept emphasizing that Noether’s 1918 paper had two big theorems in it, my paper is just about the easy one—the one physicists call Noether’s theorem:

Posted at 10:21 PM UTC | Permalink | Followups (30)

June 27, 2020

ACT2020 Program

Posted by John Baez

Applied Category Theory 2020 is coming up soon! After the Tutorial Day on Sunday July 6th, there will be talks from Monday July 7th to Friday July 10th.

Here is the program—click on it to download a more readable version:

All talks will be live on Zoom. Recorded versions should appear on YouTube later.

And here’s a list of the talks….

Posted at 10:06 PM UTC | Permalink | Post a Comment

June 17, 2020

ACT2020 Tutorial Day

Posted by John Baez

If you’re wanting to learn some applied category theory, register for the tutorials that are taking place on July 5, 2020 as part of ACT2020!

More details follow….

Posted at 8:20 PM UTC | Permalink | Followups (3)

June 11, 2020

Categorical Statistics Group

Posted by John Baez

As a spinoff of the workshop Categorical Probability and Statistics, Oliver Shetler has organized a reading group on category theory applied to statistics. The first meeting is Saturday June 27th at 17:00 UTC.

You can sign up for the group here, and also read more about it there. We’re discussing the group on the Category Theory Community Server, so if you want to join the reading group should probably also join that.

Posted at 8:14 PM UTC | Permalink | Followups (1)

June 7, 2020

Jordan Algebras

Posted by John Baez

I’ve learned a fair amount about Jordan algebras by now, but I still don’t have a clear conceptual understanding of the Jordan algebra axioms, and it’s time to fix that.

A Jordan algebra is a vector space with a commutative bilinear operation \circ that obeys

(xy)(xx)=x(y(xx)) (x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x))

That’s how Wikipedia defines it. This axiom is an affront to my mathematical sense of taste. It looks like a ridiculously restricted version of the associative law, plucked from dozens of variants one could imagine. There has to be a better way to understand what’s going on here!

Posted at 6:32 PM UTC | Permalink | Followups (8)

June 1, 2020

Categorical Probability and Statistics 2020

Posted by Tom Leinster

Guest post by Tobias Fritz, Rory Lucyshyn-Wright, and Paolo Perrone

As many of you will already know, we are organizing the workshop Categorical Probability and Statistics, which will take place online over the upcoming weekend, June 5–8.

The goal is to provide a platform for exchange of results and ideas between the various communities who pursue categorical approaches to probability and statistics. The talks will be hosted on Zoom and accompanied by discussions in a chat forum. The final schedule of the talks is available under the link above.

Posted at 11:23 PM UTC | Permalink | Followups (4)

May 30, 2020

Online Magnitude Talk by Mark Meckes

Posted by Simon Willerton

For any magnitude fans out there, Mark Meckes is giving a Zoom talk at the Online Asymptotic Geometric Analysis Seminar next Saturday, June 6, 11:30AM (New York time) 4:30PM (Sheffield time).

  • Magnitude and intrinsic volumes of convex bodies.

Abstract: Magnitude is an isometric invariant of metric spaces with origins in category theory. Although it is very difficult to exactly compute the magnitude of interesting subsets of Euclidean space, it can be shown that magnitude, or more precisely its behavior with respect to scaling, recovers many classical geometric invariants, such as volume, surface area, and Minkowski dimension. I will survey what is known about this, including results of Barcelo-Carbery, Gimperlein-Goffeng, Leinster, Willerton, and myself, and sketch the proof of an upper bound for the magnitude of a convex body in Euclidean space in terms of intrinsic volumes.

Posted at 6:19 PM UTC | Permalink | Followups (1)