## An informal THH session for topologists

## Multiplicity in restriction and tempered $L$-packets for $p$-adic groups

Studying the behavior of complex representations of a $p$-adic group restricted to its closed subgroup with the same derived group, we can derive useful information to establish the local Langlands correspondence for the subgroup. In this context, after reviewing the correspondences for $p$-adic inner forms of $SL(n), Sp(4),$ and others, we shall discuss the multiplicity of tempered representations of $p$-adic groups in the restriction, and relate it to its counterpart of corresponding irreducible representations of finite groups in the Langlands dual side, under some assumptions.

## Open problems in low-dimensional topology

## Filtrations of the knot concordance group

## On localizing rings, categories, and weight structures

(joint work with V. Sosnilo)

## The K-theory of schemes

## Formal solution to the $\hbar-$KP hierarchy

We find all formal solutions to the $\hbar$-dependent Kadomtsev-Petviashvili (KP) hierarchies. For $\hbar=1$ and $\hbar=0$ they are solutions of standard KP hierarchy and dispersionless KP hierarchy respectively. The solutions are characterized by certain Cauchy-like data that are functions of one variable. The solutions are found in a form of formal series for the tau-functions of the hierarchies and for its logarithms. An explicit combinatorial description of the coefficients of the series is provided. The talk is based on join work with A.Zabrodin.

## New guests at the MPIM

## The K-theory of exact categories

## A Chebotarev density theorem for families, and an application to class groups

This talk will present a new effective Chebotarev theorem that holds for all but a possible zero-density subfamily of certain families of number fields of fixed degree. For certain families, this work is unconditional, and in other cases it is conditional on the strong Artin conjecture and certain conjectures on counting number fields. As an application, we obtain nontrivial average upper bounds on l-torsion in the class groups of the families of fields. This talk is on joint work with Lillian Pierce and Melanie Matchett Wood.

## Endoscopic classification of supercuspidal representations for p-adic unitary groups

Given a supercuspidal representation of a p-adic unitary

group, we can associate to it a Langlands parameter by the endoscopic

classification of Arthur and Mok. In this talk, we explain how to use

the internal structure of this parameter to construct an underlying

type, in the sense of Bushnell-Kutzko and Stevens, of this

representation.

## A proof of the prime number theorem IV

The prime number theorem asserts that the $n$-th largest prime has approximate size $n \log n$.

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## A proof of the prime number theorem III

The prime number theorem asserts that the $n$-th largest prime has approximate size $n \log n$.

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## A proof of the prime number theorem II

The prime number theorem asserts that the $n$-th largest prime has approximate size $n \log n$.

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## A proof of the prime number theorem I

We shall give the proof of Iwaniec in his recent AMS book on the Riemann zeta function.

These lectures are at the level of beginning graduate students.

## The fundamental local equivalence for quantum Geometric Langlands: comparing Kac-Moody modules, Whittaker sheaves, and modules for the quantum group

## Gauge theory and knot concordance

## Fibred knots and applications

## Higher algebraic K-theory: Q construction and group completion

## D-modules in birational geometry

I will give an overview of techniques based on the theory of mixed Hodge modules, which lead to a number of applications of a rather elementary nature in birational and complex geometry. The key point I will emphasize is the use of vanishing and positivity theorems in the context of filtered D-modules of Hodge theoretic origin.

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