## On a relation between the spectra of the Laplacian and the Dirichlet-to-Neumann map on manifolds

The Dirichlet-to-Neumann operator is a first order elliptic pseudodifferential operator. It acts on smooth functions on the boundary of a Riemannian manifold and maps a function to the normal derivative of its harmonic extension. The eigenvalues of the Dirichlet-to-Neumann map are also called Steklov eigenvalues. It has been known that the geometry of the boundary has a strong influence on the Steklov eigenvalues. In this talk, we show that for every $k$, the $k$th Steklov eigenvalue is comparable to the square root of the $k$th Laplace eigenvalue.

## New guests at the MPIM

## What are Masur-Veech and Weil-Peterson volumes?

## Tautological Rings of Fibrations

In this talk we study the analogue of tautological rings of fibre bundles in the context of fibrations with Poincare fibre, i.e. the ring of obtained by fibre integrating powers of the fibrewise Euler class. I will discuss this object and an approach to compute it via rational homotopy theory, as well as some interesting properties and applications.

## Counting incompressible surfaces in 3-manifolds

A 3-manifold is small, smallish or very large if it contains no embedded incompressible surfaces (up to isotopy), finitely many, or infinitely many. We will prove a structure theorem for counting

## Rational homotopy theory, III

Rational homotopy theory is a branch of homotopy theory focused on the study of spaces “modulo torsion”. Roughly, this consists of two steps. First, one “kills” all torsion phenomena of a given space in a nice way – by a process called the “rationalization”, or “localization at the empty set of primes”. Second, one finds algebraic models that faithfully capture the homotopy type of this rationalization, and which are amenable to computations.

## New results in geography and geometry of moduli spaces of stable rank 2 sheaves on projective space

In this talk, we will give an overview of recent results on the geography and geometry of the Gieseker-Maruyama

moduli scheme $M = M(c_1,c_2,c_3)$ of rank 2 stable coherent sheaves with first Chern class $c_1 = 0$ or $-1$,

second Chern class $c_2$, and third Chern class $c_3\ge0$ on the projective space $\mathbb{P}^3$.

We will enumerate all currently known irreducible components of $M$ for small values of $c_2$ and $c_3\ge0$.

We then present the constructions of new series of components of $M$ for arbitrary $c_2$. The problem of

## Steklov problem, II

In the last few years there has been growing interest in the Steklov problem - this is

an eigenvalue problem with the spectral parameter in the boundary conditions. In this

talk we will discuss some recent advances and open questions.

## Steklov problem, I

In the last few years there has been growing interest in the Steklov problem - this is

an eigenvalue problem with the spectral parameter in the boundary conditions. In this

talk we will discuss some recent advances and open questions.

## Gorenstein duality for topological modular forms with level structure

The notion of Gorenstein duality introduced by Dwyer, Greenlees and Iyengar

allows one to view a number of dualities such as Poincaré duality for manifolds,

Gorenstein duality for commutative rings and Benson-Carlson duality for

cohomology rings of finite groups as instances of a single phenomenon. In this

talk I will set up an equivariant generalization of the framework and look at

examples coming from derived algebraic geometry, namely - the ring spectra of

topological modular forms with level structure.

## tba

## Higher-rank Bohr sets and multiplicative diophantine approximation

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. This talk is about joint work with Sam Chow where we provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known in the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation.

## Regular, Quasi-regular and Induced representations of infinite-dimensional groups

Almost all harmonic analysis on locally compact groups is based on the existence (and uniqueness)

of the Haar measure. If the group is not locally compact there is no Haar measure on it. The aim

of the talk is to give a systematic development, by example, of noncommutative harmonic

analysis on infinite-dimensional (non-locally compact) matrix groups.

## tba

## New guests at the MPIM

## The algebraic index theorem

## Comodule-Contramodule Correspondence, Higher Geometry Viewpoint

Both comodules and contramodules were introduced in 1960-s by Eilenberg and Moore. While the comodules spread widely, the contramodules remained a poor cousin. The numbers of MathSciNet hits for them are 1327 and 14 correspondingly. In fact, the contramodules do not appear in the literature at all between 1970 and 2007, when Positselski resurrected interest to them by discovering what is now known as Comodule-Contramodule Correspondence but should probably be called Positselski Duality.

## Signatures of Lefschetz fibrations

In his Ph. D. thesis, Burak Ozbagci described an algorithm computing signatures of Lefschetz fibrations where the input is a factorization of the monodromy into a product of Dehn twists. I will talk about a reformulation of Ozbagci's algorithm which becomes much easier to implement. Our main tool will be Wall's non-additivity formula applied to what I call partial fiber sum decomposition of a Lefschetz fibrations over disk. This is a joint work with A. Cengel.

## A noncommutative geometry model for Dyson--Schwinger equations

In this talk I plan to show the importance of Noncommutative Geometry to mathematically describe quantum motions in Quantum Field Theory. At the first step, I explain the construction of a new class of spectral triples with respect to the information of Dyson--Schwinger equations. At the second step, I explain the structure of a new noncommutative differential geometry machinery derived from the BPHZ renormalization of Dyson--Schwinger equations.

## Ribbon graph complexes, the Grothendieck-Teichmueller group and Goldman-Turaev Lie bialgebra

Using recent results of Chan-Galatius-Payne “Tropical curves, graph complexes, and top weight cohomology of $cM_g$” (preprint arXiv:1805.10186) it is not hard to show that the Grothendieck-Teichmueller Lie algebra grt injects into the cohomology Lie algebra of the ribbon graph complex introduced in the paper of Merkulov-Willwacher “Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves” (preprint arXiv:1511.07808) in the context of a study of the totality of cohomology groups of moduli spaces of algebraic curves with (skewsym

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