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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)