## 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

such surfaces by Euler characteristic in a cusped hyperbolic 3-manifolds. Our proofs use PL methods (ideal triangulations, normal and almost normal surfaces), simple isotopies, hyperbolic geometry, come with a computer-implementation, illustrate phenomena for triangulated 3-manifolds with at most 18 ideal tetrahedra and connect with quantum 3-manifold invariants (the q-series of the 3D-index of Dimofte-Gaiotto-Gukov). Joint work with Nathan Dunfield and Hyam Rubinstein.

## 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

connectedness of $M$ will be discussed. These are the resuts of several joint works of the speaker with

M.Jardim, D.Markushevich, A.Ivanov, C.Almeida and others.

## 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.

We generalize the notion of regular, quasi-regular and induced representations to arbitrary

infinite-dimensional groups. In order to do so, we replace the non-existing Haar measure on an

infinite-dimensional group by a suitable quasi-invariant measure on an appropriate completion

of the initial group or on the completion of a homogeneous space.

## 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. It seems to underline some seemingly unrelated duality phenomena: Matlis Duality and Riemann-Hilbert Correspondence are examples. In the first half of the talk we will review Comodule-Contramodule Correspondence on a cakewalk example of a DG-coalgebra. In the second half we will discuss an approach to it from the viewpoint of model categories, It is a work in progress, joint with K. Hristova and J. D, S, Jones.

## 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 (skewsymmetrized) punctures.

In my talk I explain that the latter ribbon graph complex controls also universal deformations of the Goldman-Turaev Lie bialgebra structure on the free loop space of a genus g Rieman surface with n+1 boundaries. Hence every element of grt

gives us a universal and highly non-trivial deformation of the latter.

The construction can be made rather transparent. We show explicitly the first non-trivial contribution to the standard

Goldman-Turaev bracket and co-bracket coming from Kontsevich’s tetrahedron class in grt.

The talk is based on a work in progress.

## New guests at the MPIM

## Extending commutative multiplications and André-Quillen cohomology

## Vorlesung Topology of 4-Manifolds

## Vorlesung Topology of 4-Manifolds

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