By a Fourier quasicrystal we mean a pure point measure in R^d,
whose Fourier transform is also a pure point measure. This notion
was inspired by the experimental discovery of quasicrystalline
materials in the middle of 80's.
The classical example of such a measure comes from Poisson's
summation formula. Which other measures of this type may exist?
I will give the relevant background on this problem and present
our recent results obtained in joint work with Alexander Olevskii.

The Continuum Problem is whether there is a set of reals whose cardinality is strictly between the cardinality of the integers and the reals.

I will survey various approximation properties of finitely generated groups and explain how they can be used to prove various longstanding conjectures in the theory of groups and group rings. A large class of groups (no group known to be not in the class) is presented that satisfy the Kervaire-Laudenbach Conjecture about solvability of non-singular equations over groups. Our method is inspired by seminal work of Gerstenhaber-Rothaus, which was the key to prove the Kervaire-Laudenbach Conjecture for residually finite groups.

The inverse Galois problem (over number fields k) is one of the central problems in algebraic number theory. A classical approach to it is via specialization of Galois coverings: Hilbert’s irreducibility theorem guarantees the existence of infinitely many specialization values in k such that the Galois group of the specialization equals the Galois group of the covering. I will consider problems related to the inverse Galois problem which can be attacked using the specialization approach.

One of the mainstream and modern tools in the study of non abelian groups are quasi-morphisms. These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error. Nowadays they are used in many fields of mathematics. For instance, they are related to bounded cohomology, stable commutator length, metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory, orderability, and the study of mapping class groups and of concordance group of knots.

One of the mainstream and modern tools in the study of non abelian groups are quasi-morphisms. These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error. Nowadays they are used in many fields of mathematics. For instance, they are related to bounded cohomology, stable commutator length, metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory, orderability, and the study of mapping class groups and of concordance group of knots.

I will consider classical 2D traveling water waves with vorticity. By
means of local and global bifurcation theory using topological degree,
we can prove that there exist many such waves. They are exact smooth
solutions of the Euler equations with the physical boundary conditions.
Some of the waves are quite tall and steep and some are overhanging.
There are periodic ones and solitary ones. I will exhibit some
numerical computations of such waves. New analytical results will be
presented on waves with favorable vorticity.

Title: “The geometry of eigenvalue extremal problems”
Abstract: When we choose a metric on a manifold we determine the spectrum of
the Laplace operator. Thus an eigenvalue may be considered as a functional
on the space of metrics. For example the first eigenvalue would be the fundamental
vibrational frequency. In some cases the normalized eigenvalues are bounded
independent of the metric. In such cases it makes sense to attempt to find
critical points in the space of metrics. In this talk we will survey two cases in

Abstract - Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory.
We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of those results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.