School of Mathematics

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.

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.

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". 



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.

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.

Time quasi-periodic gravity water waves in finite depth

Massimiliano Berti
International School for Advanced Studies
November 8, 2017
We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water waves solutions, namely periodic and even in the space variable $x$, of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a set of asymptotically full measure. This is a small divisor problem.

Topological and arithmetic intersection numbers attached to real quadratic cycles

Henri Darmon
McGill University
November 8, 2017

Abstract:  I will discuss a recent conjecture formulated in an ongoing project with Jan Vonk relating the intersection numbers of one-dimensional topological cycles on certain Shimura curves to the arithmetic intersections of associated real multiplication points on the Drinfeld p-adic upper half-plane.  Numerical experiments carried out with Vonk and James Rickards supporting the conjecture will be described.