John C. Baez's Blog, page 16

Modes (Part 8)

In Part 6 and Part 7 we used George Russell’s theories about the Lydian mode to illuminate some beautiful patterns in the modes of the major scale.

Now let’s make those patterns more precise using a little group theory. We’ll see that the 84-element group

acts as symmetries on the set of all modes of the major scale, in all keys! This group combines the symmetries of the 12-tone scale of black and white keys on the piano (the chromatic scale) and the 7-tone scale of just white keys (the di...

Agent-Based Models (Part 2)

Some news! Nathaniel Osgood, Evan Patterson, Kris Brown, Xiaoyan Li, Sean Wu, William Waites and I are going to work together at the International Centre for Mathematical Sciences for six weeks starting on May 1st, 2024. We’re going to use category theory to design better software for agent-based models.

Needless to say, we’re getting started on the math now, since 6 weeks is not enough for that.

Here’s a bit more about our plan.

Introduction

An agent-based model (ABM) is a computational cons...

8 and 24

On 8/24 I’m giving a talk about the numbers 8 and 24.

• Two of my favorite numbers: 8 and 24

Abstract. The numbers 8 and 24 play special roles in mathematics. The number 8 is special because of Bott periodicity, the octonions and the E8 lattice, while 24 is special for many reasons, including the binary tetrahedral group, the 3rd stable homotopy group of spheres, and the Leech lattice. The number 8 does for superstring theory what the number 24 does for bosonic string theory. In this tal...

Modes (Part 7)

If you start at any note of the piano and keep going up fifths until you’ve got a total of 7 notes, you get the Lydian mode. I explained this last time.

Above I show how it works starting with the note C. If we take these 7 notes and list them in increasing order starting with C, we get

C D E F♯ G A B

which is C Lydian. Great! But please remember that C is an arbitrary choice of starting note. If we’d started at any other note we’d still get a Lydian scale.

Lydian is mostly made from ...

Modes (Part 6)

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The notes in the major scale are spaced in a funny way. Look at the white keys on the piano: some have black keys between them, other don’t. But you can understand the major scale using the circle of fifths. Start with C. Go up a fifth and you get G. Go up another fifth and you get D. Go up another and you get A. Go up another and you get E. Go up another and you get B. Go up another and you get F. And these are the notes in the C major scale!

That sounds like satisfying explanatio...

Norton’s Dome

Can you roll a ball with exactly enough energy to reach the top of a dome, and have it reach the top in a finite amount of time?

I’m going to idealize the hell out of this problem so we can easily study it using math. So: no friction, no air resistance… in fact, none of the sneaky stuff you’re probably thinking about!

The problem is still tricky. For an ordinary dome the answer is no. If the ball has just enough energy to make it to the top, it rolls slower and slower as it gets near the to...

Teaching Systems Dynamics

“Stock and flow diagrams” are a nice graphical tool for modeling systems, part of a tradition called System Dynamics. Our new software based on category theory lets teams build models using these diagrams:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams, to appear in Mathematics for Public Health, Springer.

But here’s something cool Nathaniel just told me: people have had success teaching stock an...

Software for Compositional Modeling in Epidemiology

Here are the slides for my talk at Applied Category Theory 2023, in case anyone wants a look at them before I give the talk—or during, or after:

• Software for compositional modeling in epidemiology.

The talk is on Wednesday August 2nd, 14:00 UTC, which is 10 am Eastern Time. Eventually a video should appear on YouTube. Here’s the abstract:

Abstract. Mathematical models of disease are important and widely used, but building and working with these models at scale is challenging. Many epidemi...

ACT2023 – Network Theory Talks

It’s not too late to register to join talks on Zoom at Applied Category Theory 2023. It starts on Monday July 31, 2023! But if you miss out, you can watch talks later on YouTube.

Here’s where you can register:

https://www.eventbrite.com/e/applied-category-theory-2023-registration-643212404617

and here’s the program of talks:

https://act2023.github.io/programme.html

Besides talks there will also be tutorials:

Monday (simultaneously):
• David Jaz Myers, Lenses
• Paolo Perrone, Markov categories...

Skara Brae

I don’t like to fly around much anymore. But since I’m spending half a year in Scotland I figured it wasn’t too bad to take a ferry up to the Orkney Islands, at the northern end of this country, to see a bunch of Neolithic, Bronze Age and Iron Age sites.

I spent all my time on the largest island, which curiously is called the ‘Mainland’ though it’s far smaller than the main island of the British isles, to the south.

A fascinating thing about the Orkney Islands is that they hold the earliest me...