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.

Serre's thesis and its aftermath rolled in a golden age of algebraic topology which led to the impression that we can really understand (necessarily highly nonlinear) maps from one space to another. With the work of Thom on cobordism and Smale on immersions and the Poincare conjecture, a paradigm developed where geometric problems would be solved by reduction to algebraic topological ones.

The Globally Valued Fields (GVF) project is a joint effort with E. Hrushovski to understand (standard and) non-standard global fields - namely fields in which a certain abstraction of the product formula holds. One possible motivation is to give a model-theoretic framework
for various asymptotic distribution results in global fields.
Formally, a GVF is a field together with a "valuation" in the additive group of an L^1 space, such that the integral of v(a) vanishes for every non-zero a .

Abstract: We will discuss the geometry of CAT(0) (non-positively curved) cube complexes and groups which act on them. In particular, we will focus on "hyperbolic-like" behavior in such spaces.

I will describe some recent advances in the study of
infinite and finite groups, related to probability,
growth and complexity.
I will start with the celebrated Tits alternative
for linear groups, and present extensions and variations,
including a joint work with Larsen on a probabilistic Tits alternative. This is related to the notion of probabilistic
identities, and related results and open problems will be
mentioned.
I will then discuss approximate subgroups, an important
result by Breuillard-Green-Tao and Pyber-Szabo, and

Title: Riesz bases of exponentials.
Abstract:

The subject of phase transitions and critical phenomena in statistical mechanics is a rich source of interesting and difficult mathematical problems. There has been considerable success in solving such problems for systems in spatial dimension 2, or in high dimensions, but not in dimension 3. This lecture is intended to provide an introduction to recent work that employs a renormalisation group method to study spin systems and self-avoiding walk in dimension 4 (joint with Bauerschmidt and Brydges), as well as long-range versions of these models in dimensions 1,2,3 via an "epsilon expansion."

Abstract:
A classical problem in the theory of automorphic forms is to count the
number of eigenvalues of the Laplacian acting on cuspidal functions on
the quotient of the upper half plane by a lattice R. For R=SL(2,Z) (or a
congruence subgroup thereof) the answer is given by Selberg's Weyl law
while for the higher rank situation it was established by Mueller and
Lindenstrauss-Venkatesh.
Additional to the Laplacian there is another large family of operators,
namely the Hecke operators attached to SL(2,Z). Sarnak proved that the

I will discuss, for differentiable dynamical systems, three ways to capture dynamical complexity:
hyperbolicity, which measures the sensitivity of dependence on initial conditions.
entropy, which measures the predictability of future dynamical events in the sense of information theory.
the speed of correlation decay or equivalently the rate at which memory is lost.

Topological ideas have at various times played an important role in condensed matter physics. This year's Nobel Prize recognized the origins of a particular application of great current interest: the classification of phases of a quantum mechanical system. Mathematically, we would like describe them as path components of a moduli space, but that is not rigorously defined as of now. In joint work with Mike Hopkins we apply
stable homotopy theory (Adams spectral sequence) to compute the group of
topological phases of "invertible" systems. We posit a continuum field

Title: Ultra-Products and Chromatic Homotopy Theory.
Abstract:
The category of spectra is one of the most important constructions in modern algebraic topology,

Given some class of "geometric spaces", we can make a ring as follows.
1. (additive structure) When U is an open subset of such a space X, [X] = [U] + [(X \ U)]
2. (multiplicative structure)} [X x Y] = [X] [Y].
In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.

We discuss recent developments concerning ”Golden Gates” which are number theoretic generators of PU(2) ,their application to the construction of optimally efficient universal quantum gates ,and some closely connected questions of complexity in strong approximation.

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.