Mar 6

Quantitative Hellytype theorems 
Khaya Keller (BGU) 
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 nonempty 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 bestknown extensions is the AlonKleitman (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 Hellytype 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 AlonKleitman 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 wellstudied packing number, and giving rise to new Ramseytype theorems.
Based on joint works with Shakhar Smorodinsky and Gabor Tardos.

Mar 13

Degeneration’s of Riemann surfaces together with a differential 
Samuel Grushevsky (Stony Brook University) 
The DeligneMumford 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.

Mar 20

Lipschitz geometry of singularities 
Lev Birbrair (Universidade Federal do Ceara) 
“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 ``biLipschitz geometry’’, which
permits alteration of a geometric object by applying limited local
stretching and shrinking. For example, a biLipschitz change to the
geometry of a knife preserves the sharpness of the knife, but may turn
a dinner knife into a butter knife.
Applying biLipschitz 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
biLipschitz 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.

Mar 27

Caustics and Billiards 
Yaron Ostrover (Tel Aviv University) 
Mathematical billiards are a classical and wellstudied 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. ArtsteinAvidan, D. Florentin, and D Rosen.

Apr 10

Entropy and quasimorphisms 
Michal Marcinkowski (University of Regensburg) 
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 realvalued functions on $Diff(S,area)$, called quasimorphisms or quasicharacters, and how to use them to prove that the entropy norm is unbounded.
