# School of Mathematics

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

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.

## Automorphic forms and motivic cohomology I

In the lectures I will formulate a conjecture asserting that there is a hidden action of certain motivic cohomology groups on the cohomology of arithmetic groups. One can construct this action, tensored with $\mathbb C$, using differential forms. Also one can construct it, tensored with $\mathbb Q_p$, by using a derived version of the Hecke algebra (or a derived version of the Galois deformation rings).

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

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

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)'

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)

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

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.