Title: Topological Expanders.
Abstract:
A classical result of Boros-Furedi (for d=2) and Barany (for d>=2) from the 80's, asserts that given any n points in R^d, there exists a point in R^d which is covered by a constant fraction (independent of n) of all the geometric (=affine) d-simplices defined by the n points. In 2010, Gromov strengthen this result, by allowing to take topological d-simplices as well, i.e. drawing continuous lines between the n points, rather then straight lines and similarly continuous simplices rather than affine.

A very old question in additive number theory is: how large can a subset of Z/NZ be which contains no three-term arithmetic progression? An only slightly younger problem is: how large can a subset of (Z/3Z)^n be which contains no three-term arithmetic progression? The second problem was essentially solved in 2016, by the combined work of a large group of researchers around the world, touched off by a brilliantly simple new idea of Croot, Lev, and Pach.

Over a decade ago, Welschinger defined invariants of real symplectic manifolds of complex dimensions 2 and 3, which count pseudo-holomorphic disks with boundary and interior point constraints. Since then, the problem of extending the definition to higher dimensions has attracted much attention.

Abstract:
Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

Title: Positional games
Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science.

Liouville's rigidity theorem (1850) states that a map $f:\Omega\subset
R^d \to R^d$ that satisfies $Df \in SO(d)$ is an affine map. Reshetnyak
(1967) generalized this result and showed that if a sequence $f_n$
satisfies $Df_n \to SO(d)$ in $L^p$, then $f_n$ converges to an affine
map.
In this talk I will discuss generalizations of these theorems to mappings
between manifolds and sketch the main ideas of the proof (using techniques
from the calculus of variations and from harmonic analysis).

Abstract: Knot Floer homology is an invariant for knots, defined using methods from symplectic geometry. This invariant contains topological information about the knot, such as its Seifert genus; it can be used to give bounds on the unknotting number; and it can be used to shed light on the structure of the knot concordance group. I will outline the construction and basic properties of knot Floer. Knot Floer homology was originally defined in collaboration with Zoltan Szabo, and independently by Jacob Rasmussen.

I’ll talk about the advances and open questions in three dimensional
Ricci flow. Topics include the finiteness of the number of surgeries,
the long-time behavior and flowing through singularities. No prior
knowledge of Ricci flow will be assumed.

A compact Riemann surface gives rise to several families of vector spaces, associated to divisors on the Riemann surface. A finite group G of automorphisms acts on the spaces associated with invariant divisors, and a natural question is to characterize the resulting representations of G. We show how a very simple normalization for the invariant divisors can help in answering this question in a very direct manner, and if time permits present some applications.

Abstract: The development of flexible and rigid sides of symplectic and contact topology towards each other shaped this subject since its inception, and continues shaping its modern development. In the talk I will discuss the history of this struggle and describe recent breakthroughs on the flexible side.

Abstract:

We approach the formalism of quantum mechanics from the logician point of view and treat the canonical commutation relations and the conventional calculus based on it as an algebraic syntax of quantum mechanics. We then aim to establish a geometric semantics of this syntax. This leads us to a geometric model, the space of states with the action of time evolution operators, which is a limit of finite models. The finitary nature of the space allows us to give a precise meaning and calculate various classical quantum mechanical quantities.

Title: Extremes of logarithmically correlated fields
Abstract: The general theory of Gaussian processes gives a recipe for estimating the maximum of a random field,
which is neither easy to compute nor sharp enough for obtaining the law of the maximum. In recent years, much effort was invested in understanding the extrema of logarithmically correlated fields, both Gaussian and non-Gaussian. I will explain the motivation, and discuss some of the recent results and the techniques that have been involved in proving them.

This is part of our ongoing effort to develop what we call "High-dimensional combinatorics". We equate a permutation with its permutation matrix, namely an nxn array of zeros and ones in which every line (row or column) contains exactly one 1. In analogy, a two-dimensional permutation is an nxnxn array of zeros and ones in which every line (row, column or shaft) contains exactly one 1. It is not hard to see that a two-dimensional permutation is synonymous with a Latin square. It should be clear what a d-dimensional permutation is, and those are still very partially understood.

To a linear differential equation on the projective line with finitely many points of singularities, is associated a monodromy group; when the singularities are "reguar singular", then the monodromy group gives more or less complete information about the (asymptotics of the ) solutions.
The cases of interest are the hypergeometric differential equations, and there is much recent work in this area, centred around a question of Peter Sarnak on the arithmeticity/thin-ness of these monodromy groups. I give a survey of these recent results.