Daniel Tubbenhauer on Algebraic Geometry and Edward Frenkel on The Langlands Program ...

... and the Analytic Langlands Program.

See Analytic Langlands correspondence for PGL(2) on P^1 with parabolic structures over local fields.

I've just kind of got the point of this whole thing, and I think what I'm supposed to be studying is algebraic geometric algebra and geometric algebraic geometry, and showing that they are really the same thing!

Here's Daniel's Playlist giving an overview of the subject:

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Here's Edward Frenkel's part II discussion with Curt Jaimungal on The Langlands Program.

See Part I here: Peter Voit and Edward Frenkel on Unification in Physics and Mathematics. See also Curt Jaimungal Talking With Roger Penrose.

34:52 On the ability to make good mistakes. This puzzled me for a long time. I first came across the phenomenon in the story about Jacques Herbrand who wrote a paper called "On the consistency of arithmetic" which was published posthumously because he died in a climbing accident shortly after pointing out that Goedel's Incompleteness Theorem did not contradict his result which he obtained through a reduction of First order predicate calculus to propositional logic. It turned out that the paper had serious flaws which were corrected by someone else (that is just hearsay, so I don't have any references for it.) Weirdly, Herbrand's other contribution to mathematics was the Herbrand–Ribet theorem. What puzzled me was how intuition, in the touchy-feely sense, could arrive at formally correct results. I've since come to the view that it is because we invent formal languages with the expressed purpose of capturing such intuitive ideas in a form where we can communicate them precisely, so it's kind of obvious how it happens. I don;t know if that's how Taniyama worked like that though, so I may be wrong.

47:33 When they start talking about Reductive Groups, it got a bit too technical for me. It seems like a reductive group is some group that be constructed out of primitive irreducible representations by a pairing operation which applies the group operators pairwise, in the obvious way. See Group representation and Direct Sum. Then down the Rabbit Hole: there are five exceptional finite Dynkin Diagrams. So we are talking about groups which act on vector spaces and there are exactly five Platonic Solids and Évariste Galois was studying Euclid's Elements when he came up with the notion of a group in the first place, ... so that's worth remembering. It's curious that there are, I think, five places in the elements where Euclid makes a mistake with different directions of edges, as in Proposition 26 of Book I.

That is from J.L Heiberg's translation, which was produced after Galois (and Euclid) had died though! And Galois' paper on groups was also published posthumously. Mathematics, clearly, is a dead subject. 😇

Fibonacci: 1+1+2+3+5=13 - 1!

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Here's Toby doing a proof-story with algebra:

See Hopf fibration.

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58:11 Frenkel talking about GL(2) reminds me of my experiments with geometric series implemented using a lazy (normal order) lambda calculator program I wrote in Bolivia. What it was was Hindley-Milner type inference implementing a polymorphically typed lambda calculus. See Talking About Computation and https://prooftoys.org/ian-grant/hm/. All I did was implement a term Ω which was basically YI, the Y-combinator applied to the Identity Function I=λx.x and by applying this "infinity" operator to any Church Numeral N=λfx.f^N x (where f^N x is f applied N times to x) I could iterate any function an arbitrary number of times. Then I put little matrices into the resulting function and I played around with finding their (rational) eigenvalues by iteration. I used it to implement Newton's method to find the square root of 2, and then to find φ, the Golden Ratio. I remember that I wrote an email to Thomas Forster about this and shortly afterwards everything went to hell! Which reminds me that he's published an interesting thing recently: Term models for weak set theories with a universal set.

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The Ugly Side of Mathematics:

See Norman Wildberger's Box Arithmetic.

See Jade on The Halting Problem and Question About the Alan Turing Archives at King's College.

I got to CA too late to get interviewed. Here's Jess' interview instead.

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I was thinking of getting a job at a cinema, ...

See Everyman in Unit 42 of the Grand Arcade, Cambridge.

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But I only got as far as 1:25:00 of that discussion, ... watch this space, David! I hope you're OK man.

Thinking about types of Ω=YI and things (talking absolute crap, by the way!):

On S.W.P. Steen's Mathematical Logic

But maybe I got the year wrong? Perhaps Steen wrote that book after he had Roger Penrose attending his course. See Curt Jaimungal Talking With Roger Penrose.

On Conway's view of computational machinery

See John H. Conway on Weird Programming Languages which has bit-rotted. I talked about this after I met the Dog King in Mexico: Ian Grant's Weather Report 8/24/22. Conway's lecture he gave at Berkeley is here:

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At 2:05:15 they say there will be a part II, and that if you watched this far then it's for you. Well, I don't know about that. I have spent a whole day listening to this and it's like watching someone show you their stamp collection, and it's a much bigger stamp collection than you could ever hope to have because they've been doing it for years and have an enormous amount of money to spend on stamps, and they get to go to all the best stamp-collectors meetings, so they know what the best stamps are to get, and they have all of them! I wrote to Steve Awodey once and asked him about HoTT. I was interested in the motivation behind it, and he just said "Because we can prove lots of theorems". He also said "Sorry, who are you? I've never heard your name before" and I said I wasn't surprised, and that I lived in Bolivia and didn't get to go to conferences and stuff.

Here's Daniel Tubbenhauer explaining what Homotopy Type Theory is. It starts with a crazy-looking definition by Henri Poincaré when he was thinking about path-equivalence (and probably variational methods):

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Here's Vladimir Voevodsky doing his "big waffle" on the subject. Actually, I think this is more like a sketch!

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I think it comes up like this:

See the video description:

Trying to get at the question "What is a 'physical system', really?" The answer is that it is an abstract representation of [some situation], but what kinds of abstraction count as faithful representations?

Thierry Coquand on Logic and topology

At 17:27 he talks about Martin Löf's notion of an equality type expressed in terms of paths and a transport function:

At 20:00 he shows Voevodsky's contribution:

And then at 27:28 Martin Löf's logical tweak:

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YouTube recommends: Per Martin Löf doing Real Greek Lego:

How did 'judgement' come to be a term of logic? It seems to have crept in to the translations of The Organon of Aristotle from an unknown Arabic source: