The works of Buckmaster, Isett and Vicol in incompressible fluid dynamics
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Speaker: Camillo De Lellis (IAS)
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Speaker: Camillo De Lellis (IAS)
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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.
Speaker: Edward Witten (IAS)
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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.
Speaker: László Székelyhidi (IAS, Leipzig)
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Speaker: Christopher Skinner (Princeton)
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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.
Speaker: Benedict Gross (UCSD)
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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).
Speaker: Alex Lubotzky (HUJ)
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Abstract: This will be a broad survey talk on interactions between geometry and representation theory, with a focus on representations in positive characteristic (“modular representation theory”). I will outline several basic questions (e.g. for modular representations of the symmetric group) which appear very difficult, and have resisted direct algebraic approaches for over a hundred years. Over the last two decades, a new approach has emerged via geometric representation theory. It turns out that subtle questions concerning torsion in cohomology control these problems, and this allows some progress to be made. Many open questions remain, but at this point there can be no doubt that a fascinating and deep theory awaits us.
Speaker: Geordie Williamson (Sydney)
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Abstract: For many mathematical structures the collection of all objects having that structure may naturally be assembled into a space, either because the objects can be deformed or because they have symmetries: this is a moduli space. The most naive “classification” of such objects would be a description of the path-components of this moduli space, or in other words its zeroth cohomology, but to understand families of such objects one is led to investigate its higher cohomology.
It is hopeless to say anything in this generality, but topologists have developed some broad principles for studying certain kinds of moduli spaces. I will survey some of these techniques, focussing on moduli spaces of Riemann surfaces and their natural generalisation—from the point of view of differential topology—to higher dimensional smooth manifolds
Speaker: Oscar Randal-Williams (Cambridge)
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Speaker: Andrei Okounkov (Columbia)
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Speaker: John Pardon (Princeton)
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Speaker: Stanislav Smirnov (Geneva)
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Speaker: Wendelin Werner (ETH)