Sophie Maclean Counting Polyominoes and Janet Axelrud Making New Math

It's really hard to know you've done these counting problems right, as I recall. There's a useful thing called called the Orbit-Stabilizer Theorem of a group action, but it's still easy to make a mistake. I guess that in these we're only talking about polyominoes made of squares, not ones made of equilateral triangles or hexagons, so it's not that complicated.

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Daniel Tubbenhauer has done a series of lectures on the Analytic Theory of Monoidal Categories, which is something he made up himself, I think. It's about the curious fact that some of these asymptotic formulae are far easier to prove than you might expect, based on how hard the underlying counting problems are. This is lecture one, which is actually the only part I even vaguely understand! The other four parts are in this playlist.

Part II where he talks about sums and products and dimensions, ... I got lost about half-way through this one:

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Mathematics is kind of the story of how you start out making some arbitrary rules and follow their inevitable consequences and you quickly become bewildered by how complicated everything gets! In the end you get cats eating chocolate in space and people making strange sea creatures and then Rene Descartes appears, and all hell breaks loose.

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Something similar happens with Homotopy Equivalence in Category Theory. You end up reasoning about pigs eating food from a trough and Universal properties of pullbacks:

That talk is only fifty minutes or so, the rest is Q&A. What she is describing is a mathematical theory of transformations on proofs, which is also a mathematical theory of transformations of programs and algorithms. 

1:02:03 There's an interesting question of Valeria de Paiva about classical and linear logic and the Curry Howard correspondence.

1:20:34 Steve Awodey comments on this.

Valeria de Paiva on Gödel's Dialectica Interpretation of PA in Heyting Arithmetic in System-T. In her PhD under Martin Hyland she found a monoidal closed category model for Girard's Linear Logic:

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