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The classical Helly’s theorem, dated 1923, asserts that if F is a family of compact convex sets in R^d such that any d+1 sets of F have a non-empty intersection, then all sets of F can be pierced by a single point. Helly’s theorem is a cornerstone in convexity theory, and the need to generalize and extend it has led mathematicians to deep and fascinating new research directions. One of the best-known extensions is the Alon-Kleitman (p,q) theorem (1992) which asserts that for F as above, if among any p sets of F some q intersect (for q>d), then all sets of F can be pierced by a bounded number c(p,q,d) of points.

In this talk we survey the quest for quantitative Helly-type theorems which aim at finding effective bounds on the piercing numbers in various scenarios. We present new bounds on c(p,q,d) for the Alon-Kleitman theorem, which are almost tight for a wide range of parameters. We also show that for several large classes of families, quantitative (p,2) theorems in the plane can be obtained, providing a strong connection between the piercing number of a family to its well-studied packing number, and giving rise to new Ramsey-type theorems.

Based on joint works with Shakhar Smorodinsky and Gabor Tardos.

The Deligne-Mumford compactification of the moduli space of smooth Riemann surfaces is obtained by allowing nodal Riemann surfaces. In various questions in Teichmuller dynamics and algebraic geometry, it is natural to consider also degenerations of a family of Riemann surfaces, each equipped with a suitable meromorphic differential. The main difficulty in compactifying such families is that in the limit the differential may become identically zero on some components of the nodal curve. We will describe recent joint works with M. Bainbridge, D. Chen, Q. Gendron, M. Moeller, and with I. Krichever and C. Norton, which aim at constructing compactifications preserving more information about degenerations of differentials, and at describing the limits geometrically.

“Singularities’’ are points in a geometric region which are different from most nearby points in the region. Their study uses many mathematical tools. One of these tools is what is called ``bi-Lipschitz geometry’’, which permits alteration of a geometric object by applying limited local stretching and shrinking. For example, a bi-Lipschitz change to the geometry of a knife preserves the sharpness of the knife, but may turn a dinner knife into a butter knife.

Applying bi-Lipschitz geometry to singularities retains their basic structure while making them much easier to classify and therefore easier to work with. Despite this, it is only fairly recently that bi-Lipschitz geometry has been applied much in singularity theory, but its use has grown rapidly in the last decade as an increasing number of researchers are starting to work with it. It is a powerful tool for a variety of mathematical problems.

Mathematical billiards are a classical and well-studied class of dynamical systems, “a mathematician’s playground” as described by A. Katok. Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiard dynamics. In this talk we shall survey some known, and some new results and questions related to caustics in Euclidean and Minkowski billiards. The talk is based on joint work with S. Artstein-Avidan, D. Florentin, and D Rosen.

Let $S$ be a compact oriented surface and let $Diff(S,area)$ be the group of area preserving diffeomorphisms of $S$. On $Diff(S,area)$ we have interesting conjugacy invariant norms coming from symplectic geometry (the autonomous norm) or dynamics (the entropy norm). During the talk I will explain how to construct certain real-valued functions on $Diff(S,area)$, called quasimorphisms or quasicharacters, and how to use them to prove that the entropy norm is unbounded.

Accessibility is an important concept in the study of groups and manifolds as it helps decomposing the object in question into simpler pieces. In my talk I will survey some accessibility results of groups and manifolds, and explain how to relate the two. I will then discuss a joint work with Benjamin Beeker on a higher dimensional version of these ideas using CAT(0) cube complexes.

A major concern in Group Theory is the question of linearity of a given group and, in case the group is linear, the question of determining all its possible Linear Representations. While this question is entirely of algebraic nature, many times one approaches it using transcendental methods. Ergodic Theory, classically involves the study of the evolution of a system through time, is in modern view the study of symmetries of a Random System. Through the last few decades, many profound applications of Ergodic Theoretical techniques to Linear Group Theory were found. In my talk I will survey some of these classical results, as well as some of the more recent ones, and I will try to hint on a mathematical theory which partially explains why is Ergodic Theory so prominent in Linear Group Theory.

Expansivness is a fundamental property of dynamical systems. It is sometimes viewed as an indication to chaos. However, expansiveness also sets limitations on the complexity of a system. Ma\~{n}'{e} proved in the 1970’s that if a compact metric space admits an expansive homeomorphism then it is finite dimensional. In this talk we will discuss a recent extension of Ma\~{n}'{e}’s theorem for actions generated by several commuting homeomorphisms, based on joint work with Masaki Tsukamoto. This extension relies on a notion called “topological mean dimension’’ that was introduced by Gromov and developed by Lindenstrauss and Weiss in the late 1990’s.

The primitive element theorem states that every finite separable field extension is generated by a single element. Another classical result states that every central simple algebra is generated by 2 elements over its center. When the base field is replaced with a (commutative) ring, separable extensions and central simple algebras have natural generalizations called finite etale algebras and Azumaya algebras, respectively. After recalling their definition, I will discuss a joint work with Zinovy Reichstein where we generalize the previous results to such algebras. Specifically, we show that when the base ring is noetherian of Krull dimension d and has no finite epimorphic images, every finite etale algebra is generated by d+1 elements and every Azumaya algebra is generated by d+2 elements. In fact, our result applies to a wider range of algebras, and includes as special cases results of Forster and Swan on the number of generators of finitely generated projective modules over a ring of Krull dimension d.

I will also discuss the question of whether our bounds are optimal and several applications to constructing “highly generic” principal homogeneous bundles.

The talk is aimed to point out structural relationships between various spectral notions in the probability literature. Namely, it will be concerned with the relationship between: (1) Spectral density of a stationary stochastic process, a basic notion in the theory of stochastic processes, which generates covariances and provides information on the dependence structure. (2) The operator spectral measure of normal Markov operator, which is significant in both probability theory and functional analysis; also in spectral calculus. (3) The limiting empirical spectral distribution of eigenvalues of a random matrix based on a stationary sequence, which is important in many applied fields. These three, apparently disparate notions, have in their definition the word “spectral”. The talk will show that this is not a coincidence. As a matter of fact they are interconnected and they shed light one on another.

Symmetry, which essentially means “reduction of information”, is a central concept in the arts and sciences of all cultures. Objects with a high degree of symmetry are not only aesthetically pleasing, but easier to describe and analyze than their asymmetric counterparts. In fact, in many cases symmetries are the only reason why we can analyze complex systems at all. Over the last two centuries mathematicians in particular have made themselves very much comfortable on the small island of perfect symmetry.

For a long time it has been believed, that outside this small island only chaos reigns. However, this is not true. Between the small island of perfect symmetry and the vastness of chaos surrounding it lies the realm of aperiodic order. This talk is an invitation to explore this realm, following pioneers such as Hao Wang, Roger Penrose and Yves Meyer. Along the way we will hear about the fascinating story behind the discovery of quasicrystals, which changed the world of crystallography forever.

Time permitting, we will also mention some more recent developments in the field of aperiodic order. While the work of the pioneers mostly concerned aperiodic structures in the Euclidean plane, many of the underlying ideas can also be developed in the context of more general (e.g. hyperbolic) geometries. We will provide some sample results, focussing on the notion of “approximate lattices” developed in joint work with Michael Björklund (Chalmers).

In the 1960’s Erdos and Renyi introduced the notion of a random graph and made fundamental discoveries in that domain. Presently there are numerous different models of random graphs which play an important role in mathematics as well as in computer science and engineering. The simplest and most natural model remains the classical G(n,p) model - A random graph in this model has n vertices and for every every pair of vertices xy we put the edge xy in the graph independently with probability p. One of the crucial contributions of this model is that it gave us for the first time an idea what very large graphs look like typically. These things are currently widely known and well-understood, but when they were first discovered this was a surprise for many people.

A simplicial complex is a collection of finite sets X such that if A is in X and B is a subset of A, then B is is X as well. If the largest member of X has cardinality d+1 we say that dim(X)=d. Note that according to this definition a graph is synonymous with a 1-dimensional simplicial complex. Over 15 years ago Roy Meshulam and I have started a systematic study of random d-dimensional simplicial complexes. For d=1 our model coincides with the Erdos-Renyi G(n,p) model.

A fundamental property of the G(n,p) model is that all monotone graph properties exhibit a threshold behavior. Thus, for p below log n/n a G(n,p) graph is asymptotically almost surely (=aas) disconnected, whereas for slightly larger p it is aas connected. Likewise for p slightly smaller than 1/n aas the graph is the union of small trees and unicyclic graphs, while for slightly larger p it has a “giant” connected component of size linear in n.