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Random matrix theory is a source of non-trivial constructions of stochastic processes, which appear as limiting objects as the matrix size goes to infinity. We shall discuss some of these processes, starting with two classical ones and proceeding to two of the less-studied ones.

I will discuss basic properties of conjugation invariant norms on groups. I will explain why some groups are bounded (a group G is bounded if every conjugation invariant norm on G has finite diameter). The examples will include diffeomorphism groups of manifolds and some lattices in semisimple Lie groups. I will discuss the stronger notion of uniform boundedness. I will also provide examples of unbounded groups, some open problems and an application to group actions on manifolds.

The talk covers two unrelated topics in combinatorial geometry that have recently reached a confluence: incidences between points and curves in the plane, or surfaces in higher dimensions, and elimination of cycles in the depth relation of lines in 3-space. Recent progress on the latter problem, inspired by the new algebraic machinery of Guth and Katz, has yielded a nearly tight bound, of roughly $n^{3/2}$, on the number of cuts needed to eliminate all cycles for a set of n lines, or simply-shaped algebraic curves, in 3-space. This in turn leads to a similar bound on the number of cuts that are needed to turn a collection of n constant-degree algebraic arcs in the plane into a collection of pseudo-segments (i.e., each pair of the new subarcs intersect at most once). This leads, among several other applications, to improved incidence bounds between points and algebraic arcs in the plane, which are better than the older general bound of Pach and Sharir, for any number of “degrees of freedom” of the curves. It also leads to several new bounds for incidences between points and planes or points and spheres in three dimensions.

Based on joint works with Boris Aronov, Noam Solomon, and Joshua Zahl.

A Cayley combinatorial object over a group $H$ is a relational structure on $H$ invariant under the group of right translations $H_R$. Cayley graphs over the group $H$ provide a well-known example of such objects. An isomorphism problem for Cayley graphs will be the central topic of my talk. I shall present Klin-Poeschel approach to this problem based on the method of Schur. In addition, some recent results about isomorphism problem for non-graphical Cayley objects will be discussed. (This talk will be part of Yom iuyn on Algebraic Combinatorics on on the occasion of the retirement of Prof. Mikhail Klin)

We give a brief survey of the interaction between class field theory and the theory of Diophantine equations, starting from the theorem of Hasse-Minkowski to recent work on non-abelian reciprocity laws.

A major difficulty in finding polynomial patterns in primes is the need to understand their distribution properties at short scales. We describe how for some polynomial configurations one can overcome this problem by concatenating short scale behavior in “many directions” to long scale behavior.

In this talk, we will try to illustrate the potential of stochastic calculus as a tool for proving inequalities with a geometric nature. We’ll do so by focusing on the proofs of two new bounds related to the Gaussian Ornstein-Uhlenbeck convolution operator, which heavily rely on the use of Ito calculus. The first bound is a sharp robust estimate for the Gaussian noise stability inequality of C. Borell (which is, in turn, a generalization of the Gaussian isoperimetric inequality). The second bound concerns with the regularization of $L_1$ functions under the convolution operator, and provides an affirmative answer to a 1989 question of Talagrand. If time allows, we will also mention an application of these methods to concentration inequalities for log-concave measures.

The arithmetic of elliptic curves is related in multiple ways to explicit class field theory, notably through the theory of {\em complex multiplication}, one of the crown jewels of number theory. This connection plays a key role in recent progress in the recent theorem that there are positive proportions of elliptic curves of rank zero, and of rank one, for which the Birch and Swinnerton-Dyer conjecture is true, growing out of the work of Gross-Zagier, Kolyvagin, Bhargava-Shankar, Skinner-Urban-Wan, and Wei Zhang. I will discuss various relations that exist between elliptic curves and explicit class field theory, such as those above.

Galois theory investigates the symmetry patterns among roots of polynomials over a field. These symmetry patterns are described by the absolute Galois group of the field, whose structure is in general still a mystery. We will describe what is known about this symmetry group: classical facts, consequences of the epochal work by Veovodsky and Rost, and very recent structural results and conjectures related to higher cohomology operations and intersection theorems.

We describe a striking conjectured relation between ``dimensions’’ in topological dynamics and C-algebras. (No previous knowledge of C-algebras or dimension theory will be assumed.) Let $X$ be a compact metric space, and let $h \colon X \to X$ be a minimal homeomorphism (no nontrivial invariant closed subsets). The mean dimension ${\mathit{mdim}} (h)$ of $h$ is a dynamical invariant, which I will describe in the talk, and which was invented for purposes having nothing to do with C-algebras. One can also form a crossed product C-algebra $C^* ({\mathbb{Z}}, X, h)$. It is simple and unital, and there is an explicit description in terms of operators on Hilbert space, which I will give in the talk. The radius of comparison ${\mathit{rc}} (A)$ of a simple unital C-algebra $A$ is an invariant introduced for reasons having nothing to do with dynamics; I will give the motivation for its definition in the talk (but not the definition itself). It has been conjectured, originally on very thin evidence, that the radius of comparison of $C^({\mathbb{Z}},X,h)$ is equal to half the mean dimension of $h$ for any minimal homeomorphism $h$.

In this talk, I will give elementary introductions to mean dimension, the crossed product construction, and the ideas behind the radius of comparison. I will then describe the motivation for the conjecture and some partial results towards it.

The classical Helly’s theorem states that if in a family of compact convex sets in R^d every $d+1$ members have a non-empty intersection then the whole family has a non-empty intersection.

In an attempt to generalize Helly’s theorem, in 1957 Hadwiger and Debrunner posed a conjecture that was proved more than 30 years later in a celebrated result of Alon and Kleitman: For any p,q (p >= q > d) there exists a constant C=C(p,q,d) such that the following holds: If in a family of compact convex sets, out of every p members some q intersect, then the whole family can be pierced with C points. Hadwiger and Debrunner themselves showed that if q is very close to p, then $C=p-q+1$ suffices.

The proof of Alon and Kleitman yields a huge bound $C=O(p^{d^2+d})$, and providing sharp upper bounds on the minimal possible C remains a wide open problem.

In this talk we show an improvement of the best known bound on C for all pairs $(p,q)$. In particular, for a wide range of values of q, we reduce C all the way to the almost optimal bound p-q+1<=C<=p-q+2. This is the first near tight estimate of C since the 1957 Hadwiger-Debrunner theorem.

Joint work with Chaya Keller and Gabor Tardos.