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August 31, 2024
The Space of Physical Frameworks (Part 1)
Posted by John Baez
Besides learning about individual physical theories, students learn different frameworks in which physical theories are formulated. I’m talking about things like this:
- classical statics
- classical mechanics
- quantum mechanics
- thermodynamics
- classical statistical mechanics
- quantum statistical mechanics
A physical framework often depends on some physical constants that we can imagine varying, and in some limit one framework may reduce to another. This suggests that we should study a ‘moduli space’ or ‘moduli stack’ of physical frameworks. To do this formally, in full generality, we’d need to define what counts as a ‘framework’, and what means for two frameworks to be equivalent. I’m not ready to try that yet. So instead, I want to study an example: a 1-parameter family of physical frameworks that includes classical statistical mechanics — and, I hope, also thermodynamics!
August 25, 2024
Stirling’s Formula from Statistical Mechanics
Posted by John Baez
Physicists like to study all sorts of simplified situations, but here’s one I haven’t seen them discuss. I call it an ‘energy particle’. It’s an imaginary thing with no qualities except energy, which can be any number .
I hate it when on Star Trek someone says “I’m detecting an energy field” — as if energy could exist without any specific form. That makes no sense! Yet here I am, talking about energy particles.
Earlier on the n-Café, I once outlined a simple proof of Stirling’s formula using Laplace’s method. When I started thinking about statistical mechanics, I got interested in an alternative proof using the Central Limit Theorem, mentioned in a comment by Mark Meckes. Now I want to dramatize that proof using energy particles.
August 16, 2024
Bernoulli Numbers and the Harmonic Oscillator
Posted by John Baez
I keep wanting to understand Bernoulli numbers more deeply, and people keep telling me stuff that’s fancy when I want to understand things simply. But let me try again.
August 15, 2024
Galois Theory
Posted by Tom Leinster
I’ve just arXived my notes for Edinburgh’s undergraduate Galois theory course, which I taught from 2021 to 2023.
I first shared the notes on my website some time ago. But it took me a while to arXiv them, because I wanted to simultaneously make public most of the other course materials. I now have, which means the following are now available to all:
Notes forming a complete, self-contained account of the part of Galois theory that we covered.
About 40 short explanatory videos.
A large collection of problems.
Nearly 500 multiple choice questions.
August 14, 2024
Confluence in Graph Rewriting
Posted by John Baez
guest post by Anna Matsui and Innocent Ob
In this blog post we discuss the main (‘algebraic’) ideas of term rewriting and how they can be applied to term graph and arbitrary graph (i.e. string diagrams) rewriting. Our group was interested in the application of critical pairs for string diagram rewriting. We read Detlef Plump’s Critical Pairs in Term Graph Rewriting and given the task before us wanted to understand the following question: if Plump needed a new decision procedure for critical pairs in graph rewriting, how would we know what we might need for string diagram rewriting? In the this blog post we start at the root (pun intended) and work our way up from term rewriting to rewriting on arbitrary graphs. The goal is to understand, for example, why we can’t just apply, without modification, the ideas in Plump’s paper to rewriting in the λ-calculus. This post is aimed at an audience unfamiliar with term rewriting and presents foundational ideas and results to set the stage for (old and more) recent categorical applications.
August 13, 2024
Introduction to Categorical Probability
Posted by Tom Leinster
Guest post by Utku Boduroğlu, Drew McNeely, and Nico Wittrock
When is it appropriate to completely reinvent the wheel? To an outsider, that seems to happen a lot in category theory, and probability theory isn’t spared from this treatment. We’ve had a useful language to describe probability since the 17th century, and it works. Why change it up now?
This may be a tempting question when reading about categorical probability, but we might argue that this isn’t completely reinventing traditional probability from the ground up. Instead, we’re developing a set of tools that allows us to work with traditional probability in a really powerful and intuitive way, and these same tools also allow us define new kinds of uncertainty where traditional probability is limited. In this blog post, we’ll work with examples of both traditional finite probability and nondeterminism, or possibility to show how they can be unified in a single categorical language.
August 12, 2024
Skew Monoidal Categories Through Triangulations and Examples
Posted by Tom Leinster
Guest post by Thea Li and Pablo S. Ocal
The study of monoidal categories and their applications is an essential part of the research and applications of category theory. However, on occasion the coherence conditions of these categories turns out to be too strong for what one might want (or one might simply be curious as to what happens when their axioms are not all met), leading to the development of various weakenings of these notions. In our case at hand, we consider a weakening in which neither the associator nor the unitors are invertible, which is known as a skew monoidal category. A particularity of these categories is that Mac Lane’s coherence theorem does not apply, whereas there may be parallel arrows formed by structural maps that are not equal.
August 10, 2024
Prismatic Category Theory
Posted by Tom Leinster
Guest post by C.B. Aberlé and Rubén Maldonado
Fibrations are a fundamental concept of category theory and categorical logic that have become increasingly relevant to the world of applied category theory thanks to their prominent use in applications such as the categorical semantics of dependent type theory, the study of dynamical systems, etc. With the increasing occurrence of higher-categorical structures in these applications, there is an evident need for both pure and applied category theorists to develop and refine higher-categorical analogues of the ordinary theory of fibrations.
This blog post aims to sketch the basis of a general framework for the study of higher-categorical fibrations. We begin with a recap of the fundamental building-blocks of the theory of Grothendieck fibrations, including Cartesian arrows and the Grothendieck construction. In doing so, we develop a simple graphical framework for studying properties of functors. An analysis of this framework then reveals that it can be interpreted in the internal language of the topos of simplicial sets, which paves the way for studying fibrations on higher-categorical structures such as double categories.
August 8, 2024
Dimensional Analysis in Algebra and Geometry
Posted by John Baez
Let’s take a look at how various mathematicians over the ages have dealt with the idea that quantities have ‘dimensions’, in the sense of dimensional analysis.