Abstracts of upcoming talks at the MPIM. For an overview see also the calendar.

## Recent developments in Quantum Topology -- Cancelled --

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

## -- 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 -- 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.

## -- 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 --

## -- CANCELLED --

## -- 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 $p$.

## -- CANCELLED --

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |