James Arthur: The Langlands program: arithmetic, geometry and analysis

Abstract:
As the Abel Prize citation points out, the Langlands program represents a grand unified theory of mathematics. We shall try to explain in elementary terms what this means. We shall describe an age old question concerning the arithmetic prime numbers, together with a profound generalization of the problem that lies at the heart of algebraic geometry. We shall then discuss the tenets of the Langlands program that resolve these questions in terms of harmonic analysis. Finally, we shall say something of Langlands' many fundamental contributions to the program, with the understanding that there is still much to be done.
James Arthur is a Canadian mathematician working on automorphic forms, and former President of the American Mathematical Society. He is a Mossman Chair and University Professor at the University of Toronto Department of Mathematics.
This lecture was held at The University of Oslo, May 23, 2018 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.
Program for the Abel Lectures 2018
1. "On the Geometric Theory" by Abel Laureate Robert P. Langlands, Institute for Advanced Study, Princeton University
2. "The Langlands Program: Arithmetic, Geometry and Analysis" by Professor James Arthur, University of Toronto
3. "Langlands Program and Unification" by Professor Edward Frenkel, UC Berkley

[@Achrononmaster]

@9:10 physicists are in fact looking at mathematical symmetries, in the Lie groups. The "unification" is not truly well-defined because no one knows what the final structure of elementary particles must be, but since we have all the low energy broken symmetries we know what possibilities there are for larger groups. The groups are nothing but the instructions for how one particle "rotates" into another (in a topological space sense, not a literal spacetime sense). The proper unification in physics is to unite gravity with the SM of QM. That's also about groups too, since we'd need to know if gravity is purely global and has no need for a graviton, if it's all just gravitons then it requires a special difficult sort of theory (it is non-renormalizable, so cannot be "computational," and no one knows of a way around that, except by postulating maybe gravity is more global/classical and there are no gravitons). Note, if there are no gravitons then it'd be a good thing, since then the SM and gravity are already as united as they can get. Then spin-2 fields get left out of the Lorentzian path integral. Elementary particles can be in superposition, but not spacetime itself. There is a super cool unification then, the parameters in the group for GR vacuum have 36 gauge fields, they are particle-less (have no propagating modes) so are pure pure vacuum. BUt if the Hiiggs particle is actually composite, so not elementary, then these 36 zero-dimension fields would explain precisely why there are three generations of fermions. _Precisely._ I take this as a decent hint gravity should not be re-quantized (it is already a quantum theory, just not for gravitons). However, having a few gravitons is no problem, provided they are particles, not fields and so few of them there is no effective field theory for them. In this case graviton do not mess up the prediction of three generations, since the graviton is not a fermion. This is all just to say, if any mathematician reading this want a far simpler puzzle than Langlands, talk to me! The physics prizes are deeply mathematical. All about Lie groups. Contrary to popular myth, quantum mechanics is _more_ continuous than classical mechanics, not more "discrete". The discrete arises in QM because of topology/homology, but it is still smooth spacetime.

[@kamilziemian995]

Very enjoyable talk.

[@lachenmann]

13:41 "There's gonna be two, you can see what to do..." I thought he was going for a little rhyme there. That would've been nice. In all seriousness though, great talk, I enjoyed it very much. Amazing channel.

[@dickybannister5192]

I find it odd that the description of the Langlands work is so simplistic and over-arching yet, for may years, mathematics has abandoned notions that are such. A for-instance would be the concept of 'remarkable points'. before the introduction of Algebraic Geometry to describe what is 'interesting' about curves, remarkable points would be anything that was not the most straightforward notion of curvature. hence, points of inflection would be 'remarkable'. but, for Algebraic Geometry, they are differentiable and give no information, and are hence not 'remarkable'. even the idea of spirality, the point wherein a curve must either have a remarkable point (if, when following a curve, the trajectory as a straight line intersects the part of the curve already described) is not interesting because infinite spirals cannot be Algebraic. but the proof of this highlights, to me, anyway, one of the fundamental results of early work on curves, Cramer's Theorem. Certainly for combinatorical questions and possibilities this gives rise to an interesting set of questions which can be asked in either domain analogously. these pertain mainly to convexity which is a question of the straightforward notion of curvature which has been lost to time.

[@michaelgonzalez9058]

Authorship what does it mean

[@juan-fernandogomez-molina645]

Logic is here part of arithmetic? Russell?

[@manueldelrio7147]

So... if the two pillars are 'motives' (from Algebraic Geometry) and 'Automorphic Forms'(from Harmonic Analysis) where does the Arithmetic actually enter?

[@ludvigm]

One way it enters from the algebraic geometry side is the following: The example Arthur brings up of the motive of an elliptic curve tells you about the number of solutions to an equation like y^2+y=x^3-x^2 modulo primes, using the Grothendieck-Lefschetz trace formula. This bounds the number of honest integer solutions (i.e. the left hand side is equal to the right hand side and not only to the right hand side plus some multiple of a prime), as any integer solution gives you a solution modulo every prime.

[@johnhughes1556]

@@ludvigm n mm

[@genecat]

Like most mathematicians, James Artur apologizes to mathematicians for having to put the subject's ideas in laymans' terms and then proceeds to cover much of the topic in terms only mathematicians will understand. It's a common disability (or inside joke) among that group.

[@leon_noel1687]

😂😂😂

[@magnuswahlstrom766]

I don't think you understand just how technical this talk would have been if he'd allowed himself to use notions he'd expect every mathematician, but not every layperson, to know.

[@michaelgonzalez9058]

Think he is referring to graphite

[@lcyken]

The noise at 41:15 gives me a heart attack

[@jackozeehakkjuz]

ikr it's horrible

[@diktakt1187]

14 15

[@peasant12345]

math is more like 'music/art'. I get it

[@jamesbarton4331]

' motives ' & " motifs " convey his general perceptual ' abstract inuendos ' but definition wise they mean discretely different things....if I were going to describe the visuals of an afterdeath experience and stated there exist no.proper language to adequately describe....the motive/motif dichotomy would best represent the divide WITH the qualification that one understand from maths perspective THAT of the implied " motive "

[@PopeLando]

Was he breathing helium?

[@Gringohuevon]

What a load of rubbish..two mathematians construct two different ivory towers and try to get from one to to the other without going all the way down and climbing up the other..even a cursory analysis shows that There is an infinite number of possible towers and no one is more relevatory than the other. This was shown by Godel over a century ago

[@zeonive1173]

Can you tell me what Gödel's work has to do with this? Sure, some theorems might be undecidable but why would it not be worth it to investigate whether some interesting conjectures are true?

[@hygvbug]

Academic hater

[@dankurth4232]

@Gringohuevon You mistake Gödel for having shown ( or having intended to show) that mathematics is futile, but he has shown that mathematics essentially is incomplete and not omniscient. That leaves an infinite space of research and challenges for mathematicians to tackle. Langlands program is fascinating as it unveils intrinsic correspondences or dualities of otherwise - or at least traditionally seen as - unrelated mathematical structures

[@leon_noel1687]

This might be the most disrespectful comment about fundamental science ever written.

[@SpencerTwiddy]

..you need make more than a cursory analysis of the Langlands Program