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Scattering amplitudes and vertex algebras

Abstract: Scattering amplitudes are fundamental quantities in quantum field theory. Vertex algebras are well-developed algebraic objects which encode the structure of two-dimensional conformal field theories. I will describe a surprising relationship between these two topics, which goes by the name “celestial holography”. This relationship gives new techniques for computing scattering amplitudes of Yang-Mills theory. No prior knowledge of quantum field theory will be required for this talk.

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Speaker: Kevin Costello (Perimeter Institute)

(some) Things you always wanted to know about Mathematical General Relativity

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Speaker: Igor Rodnianski (Princeton University)

The André–Grothendieck period conjecture over function fields

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Speaker: Jacob Tsimerman (University of Toronto)

Vanishing theorems in algebraic geometry

Abstract: Vanishing theorems in complex geometry, such as the Kodaira vanishing theorem, play an important role in understanding the structure of complex algebraic varieties. Ultimately, they rely on critical input from Hodge theory. After recalling this story, I will survey some recent and ongoing work giving analogous results in mixed characteristic as well as applications to birational geometry; the key input now comes from p-adic Hodge theory.

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Speaker: Bhargav Bhatt (IAS/Princeton University and University of Michigan)

The work of Bhargav Bhatt

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Speaker: Peter Scholze (University of Bonn)

The work of Søren Galatius and Oscar Randal-Williams

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Speaker: Ulrike Tillmann (University of Oxford and INI)

The works of Buckmaster, Isett and Vicol in incompressible fluid dynamics

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Speaker: Camillo De Lellis (IAS)

Gauge theory and the analytic approach to geometric Langlands

Abstract: Recently P. Etingof, E. Frenkel, and D. Kazhdan, following earlier contributions by R. Langlands and J. Teschner, described an “analytic” approach to the geometric Langlands correspondence, in which the main ingredients are quantum states and operators acting on them rather than categories and functors. In this talk, I will review the gauge theory approach to the “categorical” version of geometric Langlands, and then, following the paper arXiv:2107.01732 with D. Gaiotto, I will explain how the same ingredients can be arranged differently to give a gauge theory interpretation of the “analytic” version of geometric Langlands.

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Speaker: Edward Witten (IAS)

Convex integration and synthetic turbulence

Abstract: In the past decade convex integration has been established as a powerful and versatile technique for the construction of weak solutions of various nonlinear systems of partial differential equations arising in fluid dynamics, including the Euler and Navier-Stokes equations. The existence theorems obtained in this way come at a high price: solutions are highly irregular, non-differentiable, and very much non-unique as there is usually infinitely many of them. Therefore this technique has often been thought of as a way to obtain mathematical counterexamples in the spirit of Weierstrass’ non-differentiable function, rather than advancing physical theory; “pathological”, “wild”, “paradoxical”, “counterintuitive” are some of the adjectives usually associated with solutions obtained via convex integration. In this lecture I would like to draw on some recent examples to show that there are many more sides to the story, and that, with proper usage and interpretation, the convex integration toolbox can indeed provide useful insights for problems in hydrodynamics.

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Speaker: László Székelyhidi (IAS, Leipzig)

The work of Wei Zhang

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Speaker: Christopher Skinner (Princeton)

On the conjectures of Gan, Gross, and Prasad

Abstract: This is a talk aimed at a general mathematical audience, in which I hope to explain the conjectures we made, some over twenty-five years ago. These conjectures translate central questions in modern number theory, such as the determination of local epsilon factors and the special values of L-functions, into central questions in representation theory, such as the restriction of irreducible representations of classical groups and the periods of automorphic forms. I will motivate these conjectures with some results on the representations of compact Lie groups, then review the relevant number theory, and then discuss the Langlands correspondence (which makes the translation possible). I will review the progress that has been made on these conjectures over the past ten years, and end with an arithmetic conjecture in the same style, which generalizes the formula I found with Zagier.

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Speaker: Benedict Gross (UCSD)

First order rigidity of high-rank arithmetic groups

Abstract: The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.

A number of remarkable results about them have been proven including; Weil local rigidity, Mostow strong rigidity, Margulis Super rigidity and the Schwartz-Eskin-Farb Quasi-isometric rigidity.

We will add a new type of rigidity: “first order rigidity”. Namely if D is such a non-uniform characteristic zero arithmetic group and L a finitely generated group which is elementary equivalent to it then L is isomorphic to D.

This stands in contrast with Zlil Sela’s remarkable work which implies that the free groups, surface groups and hyperbolic groups (many of which are low-rank arithmetic groups) have many non isomorphic finitely generated groups which are elementary equivalent to them.

Joint work with Nir Avni and Chen Meiri (Invent. Math. 217 (219-240) 2019).

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Speaker: Alex Lubotzky (HUJ)