Azimuth: A Tour

John Carlos Baez blogs at Azimuth, the official blog of the Azimuth Project, which “is a group effort to study the mathematical sciences for ‘saving the planet.'” Anna and Evelyn mentioned Azimuth in previous posts on this blog (such as “Planet Math” and “Solidarity with Scientists”). Please join me on a tour of a few of his newer posts.

“Open Systems: A Double Categorical Perspective (Part 1)”

This post is about the Ph.D. thesis “Open Systems: A Double Categorical Perspective” by Kenny Courser, who is one of Baez’s students. Baez notes that Courser “has been the driving force behind a lot of work on open systems and networks at U.C. Riverside.”

The post includes nice, easy-to-follow pictures. Baez wrote:

His thesis unifies a number of papers:

• Kenny Courser, A bicategory of decorated cospans, Theory and Applications of Categories 32 (2017), 995–1027.

• John Baez and Kenny Courser, Coarse-graining open Markov processes, Theory and Applications of Categories 33 (2018), 1223–1268. (Blog article here.)

• John Baez and Kenny Courser, Structured cospans. (Blog article here.)

• John Baez How to turn a Petri net into a category where the morphisms say what the Petri net can do., Kenny Courser and Christina Vasilakopoulou, Structured versus decorated cospans.

The last, still being written, introduces the new improved decorated cospans and proves their equivalence to structured cospans under some conditions. For now you’ll have to read Kenny’s thesis to see how this works!

Linear Logic and Petri Nets”

This is Baez’s summary of “Linear Logic Flavoured Composition of Petri Nets” by Elena Di Lavore and Xiaoyan Li, who wrote the piece as a guest post for The n-Category Café.

“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,” Di Lavore and Li wrote.

Baez’s post uses an example to show what Petri nets are. He then shows three ways to form categories using Petri nets. He wrote that he and Jade Master have focused on the first two, while the post by Di Lavore and Li describes the third approach.

Getting to the Bottom of Noether’s Theorem”

In this June 29 post, Baez wrote “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.”

He wrote that his paper focuses on just one of the two theorems from Noether’s 1918 paper: Noether’s theorem. Furthermore, he noted that his paper “studies the theorem algebraically, without mentioning Lagrangians.”

“In talking about Noether’s theorem I keep using an interlocking trio of important concepts used to describe physical systems: ‘states’, ‘observables’ and `generators,'” Baez wrote. After explaining what these concepts are, including differences between observables and generators, he wrote:

When we can identify observables with generators, we can state Noether’s theorem as the following equivalence:

The generator a generates transformations that leave the
observable b fixed.

\Updownarrow

The generator b generates transformations that leave the observable a fixed.

In this beautifully symmetrical statement, we switch from thinking of a as the generator and b as the observable in the first part to thinking of b as the generator and a as the observable in the second part. Of course, this statement is true only under some conditions, and the goal of my paper is to better understand these conditions. But the most fundamental condition, I claim, is the ability to identify observables with generators.

In the rest of the post, he explains more about what that means and how it relates to his paper.

Finally, Baez has shared a diary containing many of his tweets and Google+ posts about math, physics, his travels and more. It’s more than 2,000 pages long and includes content from 2003 to July 2020.

Want to reach out with ideas or feedback? Leave a comment below or reach out on Twitter (@writesRCrowell).

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