## Friedrich Hirzebruch Lecture

The annual Friedrich Hirzebruch Lecture is a series of lectures started in 2007 on the occasion of the 80th birthday of Prof. Friedrich Hirzebruch. The lectures address a general audience and aim at illustrating the relation between mathematics and art, society and other fields.

## MPI-Oberseminar

The Oberseminar is a very long running seminar at MPI (‘Ober‘ standing for 'upper'). Its idea is that the guests of the MPI speak in this seminar (hopefully early in their stay) and get the chance to explain their work to the other guests.

This often leads to further mathematical interaction, and in any case it is very interesting to know what one's colleagues are working on.

This implies two things:

- When you speak at the Oberseminar you should try to make sure that your talk is understandable and interesting to everyone, not just to the people in the same field. (We have many specialized seminars where you can present your work at a more technical level.)
- Please always attend the Oberseminar, even if the title of the talk seems technical, because you know that the speaker is going to do a good job. The only reason for absence is that the talk is in your field and thus will be too easy for you.

We hope to see you at the Oberseminar!

Christian Kaiser (organizer)

The directors:

Prof. Ballmann

Prof. Faltings

Prof. Harder

Prof. Hirzebruch †

Prof. Manin

Prof. Teichner

Prof. Zagier

**Upcoming talks**

### Don, 24 Okt 2019

**Past talks**

For the abstracts click on the titles or see the list of abstracts.

### Don, 17 Okt 2019

### Don, 10 Okt 2019

### Don, 26 Sep 2019

### Don, 19 Sep 2019

### Don, 12 Sep 2019

### Don, 05 Sep 2019

### Don, 29 Aug 2019

### Don, 22 Aug 2019

### Don, 15 Aug 2019

### Don, 08 Aug 2019

### Don, 25 Jul 2019

### Don, 18 Jul 2019

### Don, 04 Jul 2019

### Don, 27 Jun 2019

### Mit, 19 Jun 2019

### Don, 13 Jun 2019

### Don, 23 Mai 2019

### Don, 16 Mai 2019

### Don, 02 Mai 2019

### Don, 25 Apr 2019

### Don, 18 Apr 2019

### Don, 11 Apr 2019

### Don, 04 Apr 2019

### Don, 28 Mär 2019

### Don, 21 Mär 2019

### Don, 14 Mär 2019

### Don, 07 Mär 2019

### Don, 21 Feb 2019

### Don, 14 Feb 2019

### Don, 07 Feb 2019

### Don, 24 Jan 2019

### Don, 17 Jan 2019

### Don, 10 Jan 2019

### Don, 13 Dez 2018

### Don, 06 Dez 2018

### Don, 29 Nov 2018

### Don, 15 Nov 2018

### Don, 08 Nov 2018

### Don, 25 Okt 2018

### Don, 18 Okt 2018

### Don, 11 Okt 2018

### Don, 04 Okt 2018

### Don, 27 Sep 2018

### Don, 20 Sep 2018

### Don, 13 Sep 2018

### Don, 30 Aug 2018

### Don, 23 Aug 2018

### Don, 16 Aug 2018

### Don, 09 Aug 2018

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## Reading group on geometric group theory

## Seminar "Arithmetic of Algebraic Groups"

Seminar "Arithmetic of Algebraic Groups"

covering all of the winter term. The topics dealt with are centered around arithmetic groups, their cohomology groups [in various disguises] and their relation with the theory of automorphic forms. Some of the results obtained have interesting applications in the latter theory, and, more generally, in number theory. Since arithmetic groups have a very distinctive geometric flavour [via their action on suitable symmetric spaces or other geometric objects] we will also look at the resulting quotient spaces from a geometric point of view, for example, by studying the construction of geometric cycles and related questions.

## Reading group on "Integer points in polyhedra"

Website: https://sites.google.com/view/integerpointsonpolyhedra/home

We will introduce the algebra of (indicator functions) of polyhedra and study its properties. Linear forms on this algebra produce functions on the set of polytopes (*i.e.* bounded polyhedra), that are compatible with set-theoretic decompositions of polyhedra. For polytopes, the lattice point count or the volume are examples of such linear forms.

## IMPRS seminar on various topics: Gamma-spaces and partial abelian monoids

## IMPRS Thementag/Lectures by IMPRS students

## A study group on Milnor invariants

## IMPRS seminar on various topics: infinity-categories

## Seminar on Weighted Hurwitz numbers and topological recursion

## IMPRS seminar on various topics: operads

## PLeaSANT (Participative learning seminar analytic number theory)

## Seminar on tilting characters of reductive groups

## Geometric recursion learning seminar

### Content

Geometric Recursion (GR) is a fairly new technique that extends the usual Topological Recursion (TR) theory by means of Teichmüller theory, and relates to several results of Maryam Mirzakhani. It sits in the interplay between many areas of mathematics as mathematical physics, algebraic geometry and category theory.

The first part of the learning seminar aims to define and introduce GR. The second part of the seminar is more open and it will be tailored during the first weeks according to the taste of the participants towards open research questions.

### Seminar web page

### References

- J.E. Andersen, G. Borot, N. Orantin: Geometric recursion https://arxiv.org/abs/1711.04729
- B. Eynard: A short overview of the "Topological recursion" https://arxiv.org/abs/1412.3286
- G. Borot: Lecture notes on topological recursion and geometry https://arxiv.org/abs/1705.09986
- M. Mirzakhani: Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces https://link.springer.com/content/pdf/10.1007/s00222-006-0013-2.pdf

## DyGIT Seminar: (Dynamics, Geometry and Interactions on Tuesdays)

## Seminar Geometry and Dynamics

## Working group on Grothendieck-Teichmüller groups

The idea of this working group is to have an exchange of points of views

concerning a tool which connects both geometric and arithmetic areas.

People who do not necessarily want to give a talk, are also welcome.

## Learning seminar on deformation theory

The goal of the seminar is to rigorously understand the statement ''Every deformation problem in characteristic zero is controlled by a differential graded Lie algebra". This statement has long been a philosophy/guiding principle when studying deformations of algebraic or geometric structures. By the end of the seminar we aim to understand the statement and proof of its following modern incarnation:**Theorem **(Lurie, Pridham)

There is an equivalence of $\infty$-categories between the $\infty$-category of formal moduli problems and the $\infty$-category of dgLa's over a field of characteristic zero.

Everyone is welcome, whether to give a talk or simply attend. If you think you'd like to give a talk please come to the first meeting or get in touch with one of the organizers.

#### Contents

In the first part we will see some deformation problems that naturally give rise to dgLa's, and that can also be encoded in deformation functors (also called formal moduli problems). We will see that the two are related by the Maurer-Cartan equation.

In the second part we will study how to construct a deformation functor out of a dgLa using the Maurer-Cartan equation. Conversely, we will build a dgLa out of a deformation functor. For that, we will need to understand some categorical properties of the $\infty$-category of dgLa's - roughly, that we can describe it in terms of generators and relations. This will be done making use of the Chevalley-Eilenberg complex of a dgLa, so that we can work in differential graded local Artinian rings (dgArt) instead.

In the third part we will see that the construction of a dgLa out of a deformation functor is an equivalence of $\infty$-categories between formal moduli problems and dgLa's. Finally, we will see that the inverse of this equivalence is given by the Maurer-Cartan construction.

#### Talks

- Motivation and examples
- Deformation Problems and Moduli Problems (Notes by Alex)
- Deformation functors - Modern approach and the MC equation (Notes by Joao)
- The Chevalley-Eilenberg complex $C^*$, and how it is related to the Maurer-Cartan equation (Notes by Joost)
- The model category dgLa
- Koszul duality I - $C^*$ and its adjoint
- Koszul duality II - $C^*$ is an equivalence (sometimes) (Notes by Sylvain)
- Proof of the equivalence in the Main Theorem, part 1(Notes by Christian)
- Proof of the equivalence in the Main Theorem, part 2 (Notes by David)
- The inverse of the equivalence is given by Maurer-Cartan

#### References

- Jacob Lurie, DAG X: Formal moduli problems

http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf - Bertrand Toën, Problèmes de modules formels

https://perso.math.univ-toulouse.fr/btoen/files/2012/ 04/Bourbaki-Toen-2016-final1. pdf

- Vladimir Hinich, Descent of Deligne groupoids

https://arxiv.org/abs/alg-geom/9606010 - Vladimir Hinich, DG coalgebras as formal stacks

https://arxiv.org/abs/math/9812034 - Marco Manetti, Deformation theory via differential graded Lie algebras

https://arxiv.org/abs/math/0507284 - Damien Calaque, Julien Grivaux, Formal moduli problems and formal derived stacks

https://arxiv.org/abs/1802.09556 - Michael Schlessinger, Functors of Artin rings

https://www.ams.org/journals/tran/1968-130-02/S0002-9947-1968-0217093-3/S0002-9947-1968-0217093-3.pdf - Daniel Quillen, Rational homotopy theory

https://www.jstor.org/stable/1970725 - W. G. Dwyer, J. Spalinski, Homotopy theories and model categories

hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf

**On classical/motivating examples:**

- Murray Gerstenhaber - On the Deformation of Rings and Algebras

https://www.jstor.org/stable/1970484?seq=1#page_scan_tab_ contents - Marco Manetti - Lectures on deformations of complex manifolds

https://arxiv.org/abs/math/0507286 - Albert Nijenhuis, R. W. Richardson - Cohomology and deformations in graded Lie algebras

https://projecteuclid.org/euclid.bams/1183527432

**Historical:**

- Deligne's letter to Millson (For the philosophy "Every deformation problem in characteristic zero is controlled by a dgla") https://publications.ias.edu/
sites/default/files/millson. pdf

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