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.

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.

Arithmetic Kleinian groups generated by elements of finite order Abstract: We show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. The proof is based on a generalised Gromov-Guth inequality and bounds for the hyperbolic and tube volumes of the quotient orbifolds. To estimate the hyperbolic volume we take advantage of known results towards Lehmer's problem. The tube volume estimate requires study of triangulations of lens spaces which may be of independent interest.

Abstract:
We will present an elementary problem and a conjecture regarding percolation on planar graphs suggested by assuming quasi invariance of percolation crossing probabilities under coarse conformal uniformization.

Let S be a finite set (the sample space), and 
f_i: S -> R functions, for 1 ≤ i ≤ k. Given a k-tuple (v_1,...,v_k) in R^k
it is natural to ask: 
What is the distribution P on S that maximizes the entropy
      -Σ P(x) log(P(x))
subject to the constraint that the expectation of f_i be v_i?
In this talk I'll discuss a closed formula for the solution P
in terms of a sum over cumulant trees. This is based on a general calculus
for solving perturbative optimization problems due to Feynman, which may be
of interest in its own right. 

Abstract: This talk will introduce the notion of Gaussian and almost Gaussian log-correlated fields. These are a class of random (or almost random) functions many of whose statistics are predicted to coincide in a large system-size limit. Examples of these objects include:
(1) the logarithm of the Riemann zeta function on the critical line (conjecturally)
(2) the log-characteristic polynomial of Haar distributed unitary random matrices (and others),
(3) the deviations of Birkhoff sums of substitution dynamical systems (conjecturally)

In the paper "Mean dimension, small entropy factors and an
embedding theorem, Inst. Hautes Études Sci. Publ. Math 89 (1999)
227-262", Lindenstrauss showed that minimal systems of mean dimension
less than $cN$ for $c=1/36$ embed equivariantly into the Hilbert cubical
shift $([0,1]^N)^{\mathbb{Z}}$, and asked what is the optimal value
for $c$. We solve this problem by proving that $c=1/2$. The method of
proof is surprising and uses signal analysis sampling theory. Joint
work with Masaki Tsukamoto.

Let $A$, $B$ be two rational functions of degree at least two on the Riemann sphere.
The function $B$ is said to be semiconjugate to the function $A$ if there exists a non-constant rational function $X$ such that the equality (*) A\circ X=X\circ B holds.