Title: Spherical functions on spherical varieties I. Abstract: I will introduce the Gross-Prasad conjecture and its enhanced form- Ichino-Ikeda conjecture. I will then introduce the general period conjecture made by Sakellaridis-Venkatesh, we will see that the unramified Plancherel formula proved by Sakellaridis plays an important role in the formulation of the conjecture, which is a consequence of his study […]
Title: Bulk-edge correspondence and an alternative proof of Bott periodicity theorem via Quot schemesAbstract: The study of topological matter is one example of the application of algebraic topology in physics. The bulk-edge correspondence, which asserts that the bulk index of a topological matter coincides with its edge index, has been formulated mathematically in various forms […]
Title: Condensed abelian groups II Abstract: In the previous talk, we learned about how k-Condensed Abelian Groups are abelian categories of an especially nice sort. In this talk, I will discuss how this fixes issues in some of our motivating problems and endows additional structure on the category that we will use later. Time permitting, […]
Title: Rigid-analytic analogue of Artin-Grothendieck vanishing Abstract: The classical theorem of Andreotti and Frankel says that any Stein complex manifold has homotopy type of a CW complex of real dimension ≤ dimX. In particular, this implies that the cohomology groups H^i(X, A) vanish for any abelian group A and any i > dimX. This vanishing […]
Title: Decategorified Heegaard Floer theory and actions of both E and FAbstract: I will outline a relative of decategorified bordered sutured Heegaard Floer homology in which the vector spaces assigned to surfaces S with corners have actions of the full psl(1|1) (with both E and F) for smooth circles in the boundary of S, and […]
Title: Carleson ε^2 conjecture in higher dimensionsAbstract: I will talk about a joint work with Xavier Tolsa and Michele Villa on a higher dimensional analogue of the Carleson ε^2 conjecture. In this work, we characterise tangent points of certain domains in Euclidean space via a novel "spherical" square function which measures whether the common boundary of the […]
Title: The mathematics of quantum mechanics The development of quantum mechanics in the early 20th century was a revolution in physics, and had a transformational impact on mathematics, as well. Even as we still grapple with the interpretations of the theory, its mathematical foundations are rigorous, and rooted in simple concepts of linear algebra. I […]
Title: CohomologyAbstract: Our goal in this talk is to recover the classical sheaf cohomology on topological spaces from related condensed set and condensed abelian groups. I will also discuss some infinity category theory, because from some point of view, that is where the actual story of derived category and cohomology happens.
Title: Igusa stacks and cohomology of Shimura varieties Abstract: Cohomology of Shimura varieties has drawn interest from number theory and representation theory. Following ideas developed by Caraiani-Scholze, Koshikawa, Santos, Hamann-Lee, I will explain how vanishing type of results for the generic part in the cohomology of Shimura varieties with torsion coefficients can be obtained, using […]