Manchester Building (Hall 2), Hebrew University Jerusalem
It has been known for almost a hundred years that most polynomials with integral coefficients are irreducible and have a big Galois group. For a few dozen years, people have been interested in whether the same holds when one considers sparse families of polynomials—notably, polynomials with plus-minus 1 coefficients. In particular, “some guy on the street” conjectures that the probability for a random plus-minus 1 coefficient polynomial to be irreducible tends to 1 as the degree tends to infinity (a much earlier conjecture of Odlyzko-Poonen is about the 0-1 coefficients model).
Probabilistic limit theorems for (distance expanding and hyperbolic) dynamical systems is a well studied
topic. In this talk I will present conditions guaranteeing that a local central limit theorem holds true for certain families of distance
expanding random dynamical systems. If time permits, I will also discuss a version of the Berry-Esseen theorem.
Joint work with Yuri Kifer.
Given two Hamiltonian isotopic curves in a surface, one would like to tell whether they are "close" or "far apart". A natural way to do that is to consider Hofer's metric which computes mechanical energy needed to deform one curve into the other. However due to lack of tools the large-scale Hofer geometry is only partially understood. On some surfaces (e.g. S^2) literally nothing is known.
First we am going to recall first basic facts about vector bundles on smooth projective curves. Then we will talk about moduli "spaces" of vector bundles on curves. If time permits, we will also talk about related "spaces" like Hecke stacks and moduli "spaces" of shtukas.
Key words: Riemann-Roth theorem for curves, vector bundles on curves, degree.
Abstract: I will discuss applications of algebraic results to combinatorics, focusing in particular on Lefschetz theorem, Decomposition theorem and Hodge Riemann relations. Secondly, I will discuss proving these results combinatorially, using a technique by McMullen and extended by de Cataldo and Migliorini. Finally, I will discuss Lefschetz type theorems beyond positivity.
Recommended prerequisites: basic commutative algebra