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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.

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.