March 25, 2020

MIT Categories Seminar

Posted by John Baez

The MIT Categories Seminar is an informal teaching seminar in category theory and its applications, with the occasional research talk. This spring they are meeting online each Thursday, 12 noon to 1pm Eastern Time.

The talks are broadcast over YouTube here, with simultaneous discussion on the Category Theory Community Server. (To join the channel, click here.) Talks are recorded and remain available on the YouTube channel.

Category Theory Community Server

Posted by John Baez

My student Christian Williams has started a community server for category theory, computer science, logic, as well as general science and industry. In just a few days, it has grown into a large and lively place, with people of many backgrounds and interests. Please feel free to join!

Join here:

https://categorytheory.zulipchat.com/join/52grbi4jw3b989fywh56pull/

and from then on you can just go here:

http://categorytheory.zulipchat.com

Both the ACT@UCR seminar and the MIT categories seminar will have discussions on this server.

March 24, 2020

ACT@UCR Seminar

Posted by John Baez

Coronavirus is forcing massive changes on the academic ecosystem, and here’s another:

We’re having a seminar on applied category theory at U. C. Riverside, organized by Joe Moeller and Christian Williams.

It will take place on Wednesdays at 5 pm UTC, which is 10 am in California or 1 pm on the east coast of the United States, or 6 pm in England. It will be held online via Zoom, here:

https://ucr.zoom.us/j/607160601

We will have discussions online here:

https://categorytheory.zulipchat.com/

The first two talks will be:

• Wednesday April 1st, John Baez: Structured cospans and double categories.

Abstract. One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce “structured cospans” as a way to study networks with inputs and outputs. Given a functor $L \colon A \to X$, a structured cospan is a diagram in $X$ of the form $L(a) \to x \leftarrow L(b).$ If $A$ and $X$ have finite colimits and $L$ is a left adjoint, we obtain a symmetric monoidal category whose objects are those of $A$ and whose morphisms are certain equivalence classes of structured cospans. However, this arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans, not equivalence classes thereof. We explain the mathematics and illustrate it with an example from chemistry.

• Wednesday April 8th, Prakash Panangaden: A categorical view of conditional expectation.

Abstract. This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones — which are abstract versions of the familiar cones in vector spaces — of measures and related categories cones of $L_p$ functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

Posted at 5:29 AM UTC | Permalink | Followups (9)

Applied Category Theory 2020 (Part 2)

Posted by John Baez

Due to the coronavirus outbreak, many universities are moving activities online. This is a great opportunity to open up ACT2020 to a broader audience, with speakers from around the world.

The conference will take place July 6-10 online, coordinated by organizers in Boston USA. Each day there will be around six hours of live talks, which will be a bit more spaced out than usual to accommodate the different time zones of our speakers. All the talks will be both live streamed and recorded on YouTube. We will also have chat rooms and video chats in which participants can discuss various themes in applied category theory.

We will give more details as they become available and post updates on our official webpage:

http://act2020.mit.edu

March 19, 2020

Michael Harris on Virtues of Priority

Posted by David Corfield

Michael Harris has an interesting new article on the arXiv today - Virtues of Priority. He wrote it for an edition of a philosophy journal on virtues in mathematics, but, as he explains in the footnote on the first page, it has ended up being published on the arXiv rather than in that journal. I think it provides interested philosophers of mathematics with excellent material to think through issues concerning the role of the virtues in intellectual lives.

Abstract: The conjecture that every elliptic curve with rational coefficients is a so-called modular curve – since 2000 a theorem due in large part to Andrew Wiles and, in complete generality, to Breuil-Conrad-Diamond-Taylor – has been known by various names: Weil Conjecture, Taniyama-Weil Conjecture, Shimura-Taniyama-Weil Conjecture, or Shimura-Taniyama Conjecture, among others. The question of the authorship of this conjecture, one of whose corollaries is Fermat’s Last Theorem, has been the subject of a priority dispute that has often been quite bitter, but the principles behind one attribution or another have (almost) never been made explicit. The author proposes a reading inspired in part by the “virtue ethics” of Alasdair MacIntyre, analyzing each of the attributions as the expression of a specific value, or virtue, appreciated by the community of mathematicians.

Posted at 1:41 PM UTC | Permalink | Followups (10)

March 14, 2020

The Hardest Math Problem

Posted by John Baez

Not about coronavirus… just to cheer you up:

Puzzle. What math problem has taken the longest to be solved? It could be one that’s solved now, or one that’s still unsolved.

Let’s start by looking at one candidate question. Can you square the circle with compass and straightedge? After this question became popular among mathematicians, it took at least 2296 years to answer it!

Posted at 9:55 PM UTC | Permalink | Followups (8)

March 3, 2020

Applied Category Theory 2020 (Part 1)

Posted by John Baez

Here’s the big annual conference on applied category theory:

• ACT2020, 2020 July 6-10, online worldwide. Organized by Brendan Fong and David Spivak.

This happens right after the applied category theory school, which will be held June 29 – July 3. There will also be a tutorial day on Sunday July 5, with talks by Paolo Perrone, Emily Riehl, David Spivak and others.

Posted at 1:06 AM UTC | Permalink | Followups (15)

March 2, 2020

String Diagrams in Computation, Logic, and Physics

Posted by John Baez

A workshop:

String diagrams are a powerful tool for reasoning about processes and composition. Originally developed as a convenient notation for the arrows of monoidal and higher categories, they are increasingly used in the formal study of digital circuits, control theory, concurrency, programming languages, quantum and classical computation, natural language, logic and more. String diagrams combine the advantages of formal syntax with intuitive aspects: the graphical nature of terms means that they often reflect the topology of systems under consideration. Moreover, diagrammatic reasoning transforms formal arguments into dynamic, moving images, thus building domain specific intuitions, valuable both for practitioners and pedagogy.

Posted at 8:10 PM UTC | Permalink | Followups (2)