Title: On stable transfer operators and functorial transfer kernels Abstract: Langlands introduced stable transfer operators as a fundamental part of his proposal of Beyond Endoscopy. They are intended to be used in comparisons of his proposed refinements of stable trace formulas, in an analogous role to that of endoscopic transfer operators in the theory of […]
Title: Number-theoretic methods to produce counterexamples for questions motivated by PDE’s.Abstract: In 1980 Carleson posed a question in PDE’s: how “well-behaved” must an initial data function be, to guarantee pointwise convergence of the solution of the linear Schrödinger equation (as time goes to zero)? After being studied by many authors over nearly 40 years, this celebrated […]
Title: Number-theoretic methods to produce counterexamples for questions motivated by PDE’s.Abstract: In 1980 Carleson posed a question in PDE’s: how “well-behaved” must an initial data function be, to guarantee pointwise convergence of the solution of the linear Schrödinger equation (as time goes to zero)? After being studied by many authors over nearly 40 years, this celebrated […]
Title: Automorphic Galois representations for classical groupsAbstract: It is a theorem due to many people, most recently by Harris-Lan-Taylor-Thorne and Scholze, that there exist Galois representations associated with regular cuspidal automorphic representations of GL(n) over totally real or CM fields. This may be thought of as one direction of the global Langlands correspondence for GL(n). I […]
Title: Automorphic Galois representations for classical groupsAbstract: It is a theorem due to many people, most recently by Harris-Lan-Taylor-Thorne and Scholze, that there exist Galois representations associated with regular cuspidal automorphic representations of GL(n) over totally real or CM fields. This may be thought of as one direction of the global Langlands correspondence for GL(n). I […]
Title: Categorical diagonalization.Abstract: We give an introduction to the circle of ideas known as categorification, guided by the following question: what might it mean to diagonalize a functor?Given a category, one can forget most of the information and just remember skeletal information about the objects up to isomorphism, a process known as decategorification. For example, the […]
Title: Categorical diagonalization.Abstract: We give an introduction to the circle of ideas known as categorification, guided by the following question: what might it mean to diagonalize a functor?Given a category, one can forget most of the information and just remember skeletal information about the objects up to isomorphism, a process known as decategorification. For example, the […]
Title: Decomposition problem for topological Azumaya algebras with an involution of the first kindAbstract: Topological Azumaya algebras (TAAs) over a space are topological shadows of more complicated algebraic Azumaya algebras defined over schemes. Tensor product is a well-defined operation on TAAs. Hence given a TAA $mathcal{A}$ of degree $mn$ over $X$, where $m$ and $n$ […]
Title: A fourth-order Escobar-Yamabe problem on a half-ballAbstract: The celebrated Yamabe problem asks us to make a conformal change on a compact Riemannian manifold such that the scalar curvature becomes constant. The (Type-II) Escobar-Yamabe problem is to make a conformal change on a compact Riemannian manifold with boundary so that the scalar curvature vanishes and […]
Title: Local-global compatibility over function fields Abstract: We show that V. Lafforgue's global Langlands correspondence is compatible with Fargues–Scholze's semisimplified local Langlands correspondence. As a result, we canonically lift Fargues–Scholze's construction to a non-semisimplified local Langlands correspondence for fields of characteristic ≥ 5. We also deduce that Fargues–Scholze's construction agrees with that of Genestier–Lafforgue, answering a question of Fargues–Scholze, Hansen, […]
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Title: Harris-Venkatesh plus Stark Abstract: The class number formula describes the behavior of the Dedekind zeta function at s=0 and s=1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s=0 and s=1 in terms of units and class numbers. The Harris-Venkatesh conjecture describes […]