Title: Oscillatory integrals on manifolds and related Kakeya and Nikodym problems. Abstract: This talk is about oscillatory integrals on manifolds and their connections to Kakeya and Nikodym problems on manifolds. There are two types of manifolds that are particularly interesting: manifolds of constant sectional curvature and manifolds satisfying Sogge's chaotic curvature conditions. I will discuss these two […]
Title: Light condensed setsAbstract: This talk will introduce us to the light condensed setting, the new way of dealing with set-theoretic technicalities in the condensed setup. Particular emphasis will be on the changes, differences, and simplifications compared to the "old" approach, which we discussed last semester.
Title: A moduli-theoretic approach to heights on stacks.Abstract: A theory of heights of rational points on stacks was recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about counting problems over global fields. In this talk I will present a moduli theoretic approach to heights on stacks over […]
Title: Restricted Arithmetic Quantum Unique ErgodicityAbstract: The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms. I […]
Title: TBA
Title: Arithmetic Quantum Field Theory? Abstract: Mathematical structures suggested by quantum field theory have revolutionised important areas of algebraic geometry, differential geometry, as well as topology in the last three decades. This talk will introduce a few of the recent ideas for applying structures inspired by physics to arithmetic geometry.
Title: Maximal Subellipticity Abstract: The theory of elliptic PDE stands apart from many other areas of PDE because sharp results are known for very general linear and fully nonlinear elliptic PDE. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDE leans heavily on […]
Title: Solid modules Abstract: For this talk I would like to define and study solid abelian groups, whose construction is motivated in topology/analysis, by the desire to single out (metrizable) topological vector spaces that are complete.
Title: The Quillen-Lichtenbaum dimension of an algebraic variety and the integral Hodge conjecture.Abstract: I will explain a numerical invariant of complex varieties born out of the difference between algebraic and topological K-theory. In some cases, it is a birational invariant which is weaker than some known ones coming from unramified cohomology. However, it is also a derived […]
Title: Compatibility of canonical ell-adic local systems on Shimura varieties Abstract: In his 1979 Corvallis paper, Deligne uses a modular interpretation to construct canonical models of Shimura varieties of Hodge type, and suggested a dream that all Shimura varieties with rational weight should be moduli spaces of motives over number fields. Outside of the abelian […]
Title: The stability of irrotational shocks and the Landau law of decay. Abstract: We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, […]