School of Mathematics

Decomposition theorem for semisimple algebraic holonomic D-modules

Takuro Mochizuki
Kyoto University
November 13, 2017

Decomposition theorem for perverse sheaves on algebraic varieties, proved by Beilinson-Bernstein-Deligne-Gabber, is one of the most important and useful theorems in the contemporary mathematics. By the Riemann-Hilbert correspondence, we may regard it as a theorem for regular holonomic D-modules of geometric origin. Rather recently, it was generalized to the context of semisimple holonomic D-modules which are not necessarily regular.

Learning models: connections between boosting, hard-core distributions, dense models, GAN, and regularity I

Russell Impagliazzo
University of California, San Diego
November 13, 2017

A theme that cuts across many domains of computer science and mathematics is to find simple representations of complex mathematical objects such as graphs, functions, or distributions on data. These representations need to capture how the object interacts with a class of tests, and to approximately determine the outcome of these tests.

Zagier's conjecture on zeta(F,4)

Alexander Goncharov
Yale University; Member, School of Mathematics
November 10, 2017

Abstract: We prove that the weight 4 Beilinson's regulator map can be expressed via the classical n-logarithms, $n \leq 4$.


This plus Borel's theorem implies Zagier's conjecture, relating the value of the Dedekind zeta functions at $s=4$ and the classical tetralogarithm.  Another application is to the values of L-functions of elliptic curves over $Q$ at $s=4$.


One of the new tools is a connection between cluster varieties and polylogarithms, generalising our work with V. Fock relating cluster varieties and the dilogarithm.

Solvable descent for cuspidal automorphic representations of GL(n)'

Laurent Clozel
Universite Paris-Sud; Member, School of Mathematics
November 10, 2017

Abstract: A rather notorious mistake occurs p. 217 in the proof of Lemma 6.3 of the book "Simple algebras, base change, and the advanced theory of the trace formula" (1989) by J. Arthur, L. Clozel.  E. Lapid and J. Rogawski (1998) proposed a proof of this lemma based on what they called "Theorem A". 

 

 

Pseudorepresentations and the Eisenstein ideal

Preston Wake
University of California, Los Angeles
November 9, 2017

Abstract:  In his ladmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences.  We'll also discuss some partial results in the composite-level case.  This is joint work with Carl Wang-Erickson.

A derived Hecke algebra in the context of the mod $p$ Langlands program

Rachel Ollivier
University of British Columbia
November 8, 2017

Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of smooth representations of G and the category of H-modules. When the field of coefficients has characteristic zero this link is well understood by work of Bernstein and Borel.