## New guests at the MPIM

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## A short survey of Prismatic Cohomology (after P. Scholze and B. Bhatt), II

We will discuss the recently suggested definition of prismatic cohomology that unifies several integral cohomology theories in p-adic geometry. We briefly start with an introduction of prisms as generalization of perfectoid rings, and then describe available constructions of a prismatic site:

1) a variant of the complex $\Delta_{R/A}$$ (2016);

2) construction via topological Hochschild homology (2018).

It requires as input a highly non-trivial calculation from homotopy theory

due to Bockstedt.

## A short survey of Prismatic Cohomology (after P. Scholze and B. Bhatt), I

This is the first of the two talks that will be given in this seminar. It is an attempt to understand the work of

Bhargav Bhatt and Peter Scholze. We will discuss the motivations and give an overview of p-adic Hodge

theory. In the process, we will introduce some of the basic notions such as delta-rings, Witt vectors,

prisms and prismatic site.

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## Lower bounds for Mahler measure that depend on the number of monomials

We prove a new lower bound for the Mahler measure of a polynomial in one and

in several variables that depends on the complex coefficients and the number of

monomials, but not on the degree. In one variable the lower bound generalizes a

classical inequality of Mahler. In M variables the inequality depends on Z^M as an

ordered group, and in general the lower bound depends on the choice of ordering. In

one variable the proof is elementary. In M variables the proof exploits an idea used

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## Translation surfaces of infinite-type, IV

Compact translation surfaces are intimately linked to the geometry

of Teichmüller spaces and the billiard dynamics in rational polygons.

During this is two week mini-course (4 x 2h30) our aim is to introduce

translation surfaces of infinite type (that is non-compact) both through

geometric and dynamical points of view.

We will first present several examples that motivate the study of

translation surfaces of infinite type, in particular several models

of billiards. The next lectures will go through the topological,

## Translation surfaces of infinite-type, III

Compact translation surfaces are intimately linked to the geometry

of Teichmüller spaces and the billiard dynamics in rational polygons.

During this is two week mini-course (4 x 2h30) our aim is to introduce

translation surfaces of infinite type (that is non-compact) both through

geometric and dynamical points of view.

We will first present several examples that motivate the study of

translation surfaces of infinite type, in particular several models

of billiards. The next lectures will go through the topological,

## Translation surfaces of infinite-type, II

Compact translation surfaces are intimately linked to the geometry

of Teichmüller spaces and the billiard dynamics in rational polygons.

During this is two week mini-course (4 x 2h30) our aim is to introduce

translation surfaces of infinite type (that is non-compact) both through

geometric and dynamical points of view.

We will first present several examples that motivate the study of

translation surfaces of infinite type, in particular several models

of billiards. The next lectures will go through the topological,

## Translation surfaces of infinite-type, I

of Teichmüller spaces and the billiard dynamics in rational polygons.

During this is two week mini-course (4 x 2h30) our aim is to introduce

translation surfaces of infinite type (that is non-compact) both through

geometric and dynamical points of view.

We will first present several examples that motivate the study of

translation surfaces of infinite type, in particular several models

of billiards. The next lectures will go through the topological,

## Fundamental groups and rational points

One powerful tool in the study of rational solutions to equations comes from various theories of fundamental groups of varieties, most notably the etale fundamental group developed by Alexander Grothendieck. In this talk, we will outline how one can apply such algebro-topological tools to the determination of solutions to equations, both in terms of Grothendieck's famous *Section Conjecture* and (time permitting) the non-abelian Chabauty method of Minhyong Kim.

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## Homogeneous spaces, algebraic K-theory and cohomological dimension of fields

In 1986, Kato and Kuzumaki stated a set of conjectures which aimed at giving a Diophantine characterization of the cohomological dimension of fields in terms of Milnor K-theory and points on projective hypersurfaces of small degree. Those conjectures are known to be wrong in general.

In this talk, we will prove a variant of Kato and Kuzumaki's conjectures in which projective hypersurfaces of small degree are replaced by homogeneous spaces. This is joint work with Giancarlo Lucchini Arteche.

## New guests at the MPIM

## Integral structure of vector-valued Siegel modular forms

We show how one can obtain almost all vector-valued Siegel modular forms from scalar-valued ones of higher degree by using appropriate differential operators, more precisely, only Siegel type Eisenstein series are needed in this procedure. In this way, one gets an integral structure on vector-valued modular forms from a classical result on rational Fourier coefficients (with bounded denominators) of such Eisenstein series. This procedure works for arbitrary congruence subgroups. We also report on applications of this result in the theory of p-adic vector-valued modular forms.

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