## -- CANCELLED -- The icosahedron, the Rogers-Ramanujan identities, and beyond

In view of the situation with the coronavirus the lecture on March 17 and for the immediate future will not take place.

It will be resumed when the situation allows it, perhaps around April 20, which is the planned start of the summer semester.

You will be informed as soon as this is known.

## -- CANCELLED -- Parabolicity conjecture of F-isocrystals

We will present recent developments in the theory of overconvergent F-isocrystals, the p-adic analogue of ell-adic lisse sheaves. For the most part of the talk, we will explain a new result on the algebraic monodromy groups of these objects. At the end, we will mention an application of the theorem to the finiteness of separable p-torsion points of an abelian variety.

## -- CANCELLED -- Specialization of Néron-Severi groups in positive characteristic

Given a family Y------> X of smooth projective varieties over a field k, we study the locus X^{ex} of closed points x in X where the rank of the Neron-Severi group of the fiber of Y------> X at x is bigger then the rank of the generic one. As simple examples show, the properties of X^{ex} depend on the arithmetic of k. We prove that if the characteristic of k is positive and k is infinite finitely generated then this locus is "small", extending previous results in characteristic zero of André and Cadoret-Tamagawa.

## -- CANCELLED -- Classifying spaces for commutativity in groups, II

Given a topological group G, we can think of the space ofhomomorphisms $\hom(\mathbb{Z}^n, G)$ as the space of

$n$-tuples of elements of $G$ that commute pairwise. These spaces are more subtle than one might think, and even basic

invariants such as the number of connected components can lead to surprising results. Fixing $G$ and varying $n$ we can

construct what is known as the classifying space for commutativity in $G$. I will survey what is known about these

classifying spaces, whose study is still young.

## -- CANCELLED -- Classifying spaces for commutativity in groups, I

Given a topological group G, we can think of the space ofhomomorphisms $\hom(\mathbb{Z}^n, G)$ as the space of

$n$-tuples of elements of $G$ that commute pairwise. These spaces are more subtle than one might think, and even basic

invariants such as the number of connected components can lead to surprising results. Fixing $G$ and varying $n$ we can

construct what is known as the classifying space for commutativity in $G$. I will survey what is known about these

classifying spaces, whose study is still young.

## -- CANCELLED -- Hilbert modular double octic Calabi-Yau 3-fold

I will discuss modularity of a Calabi-Yau threefold $X$ with Hodge numbers of $H^3(X)$ equal to (1,1,1,1).

The restriction of the Galois representation on $H^3(X)$ decomposes over $Q(\sqrt{2})$ into the direct sum of

the Galois representation for a Hilbert modular form of weight [4,2] and its conjugate.

The Calabi-Yau threefold $X$ is defined as a resolution of singularities of a double covering of $P^3$ branched

along a union of eight planes. The proof is based on a careful study of geometry of $X$, which allows us to find a

## -- CANCELLED -- New guests at the MPIM

## Optimality of the logarithmic upper-bound sieve, with explicit estimates

(joint work with Clara Aldana, Emanuel Carneiro, Carlos Andres Chirre Chavez and Julian Mejia Cordero)

At the simplest level, an upper bound sieve of Selberg type is a choice of $\rho(d),$ $d\le D$, with $\rho(1)=1$, such that $$S = \sum_{n\leq N}\left(\sum_{d\mid n}\mu(d) \rho(d)\right)^2$$

## -- CANCELLED -- Stability of polynomials modulo primes

For a polynomial $f\in\mathbb{K}[X]$ over some field $\mathbb{K}$ we define the sequence of polynomials

$$

f^0(X)=X, \quad \text{and} \quad f^{(n)}(X)=f(f^{(n-1)}(X)), \quad n=1,2,\dots

$$

The polynomial $f$ is said to be *stable* if all iterates $f^{(n)}$ are irreducible.

It is conjectured, that for a quadratic polynomial $f\in\mathbb{Z}[X]$, its reduction $f_p\in\mathbb{F}_p[X]$ modulo $p$ can be stable just for finitely many primes $p$.

## The icosahedron, the Rogers-Ramanujan identities, and beyond

The first part of this lecture series is meant to illustrate the dictum that all suÿciently beautiful mathematical objects are connected. The two objects we choose to illustrate this are the icosahedron, the most subtle of the Platonic solids, and the Rogers-Ramanujan identities, often considered the most beautiful formulas in all of mathematics.

## -- CANCELLED -- On the Skolem Problem for parametric families of linear recurrence sequences and some G.C.D. problems

In this talk we discuss a parametric version of the Skolem Problem about decidability of the existence of a zero in a linear recurrence sequence. We show that in some natural parametric families for all but finitely many values of the parameter in the algebraic closure of the rational numbers it can be effectively solved. We then connect this problem to studying the greatest common divisor of two linear recurrence sequences of polynomials. Also, as an application we obtain an explicit version of a result of F. Amoroso, D. Masser and U.

## Models of Lubin-Tate spectra via real bordism theory

In this talk, I will present models of Lubin-Tate theories at $p=2$ and all heights. These models come with explicit formulas for some finite subgroups of the Morava stabilizer groups on the coefficient rings. The construction utilizes equivariant formal group laws and are based on techniques of Hill-Hopkins-Ravenel. I will also talk about implications of the theorem, such as periodicity and differentials in spectral sequences. This is joint work with Beaudry, Hill and Shi.

## The Poisson cohomology of $\mathrm{sl}_2(\mathbb{C})$

To a Poisson manifold $(M,\pi)$ one can naturally associated a cohomology called Poisson cohomology. Although Poisson cohomology is important for questions such as linearization and deformations of poisson structures, it is in general quite difficult to compute. In this talk we look at the Poisson structure on the dual of a Lie algebra $(g, [~,~])$. We look at the relation of Poisson cohomology with the linearization problem and outline the general ideas behind the calculations for the case of $\mathrm{sl}_2(\mathbb{C})$. This is based on joint work with Ioan Marcut.

## Approximating foliations by contact structures

Although their definitions are in some sense opposite, contact structures and foliations display many similarities. This is especially clear in the $3$-dimensional theory of confoliations which unites both structures in a single framework. A famous theorem by Eliashberg and Thurston states that, with a single exception ($\mathbb{S}^1 \times \mathbb{S}^2$ foliated by spheres), any (con)foliation on a $3$-manifold can be approximated by contact structures.

## Koszul duality and deformation theory

Koszul duality between Lie algebras and cocommutative coalgebras constructed by Hinich is the basis for formal deformation theory, at least in characteristic zero. In this talk I explain, following Manetti, Pridham and Lurie, how Koszul duality, combined with Brown representability theorem from homotopy theory leads to representability of a formal deformation functor up to homotopy. Sometimes a formal deformation functor has a `noncommutative structure', meaning that it is defined on a suitable homotopy category of associative algebras.

## Global group laws and the equivariant Quillen theorem

Quillen's theorem that the complex bordism ring carries the universal formal group law is a fundamental result in stable homotopy theory. In this talk I will discuss equivariant versions of this result, both over a fixed abelian compact Lie group and in a global equivariant/stack setting.

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