The Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi (1982) and Leighton (1983 )states that if a graph of n vertices and e>4n edges is drawn in the plane, then the number of crossings between its edges must be at least constant times e^3/n^2. This statement, which is asymptotically tight, has found many applications in combinatorial geometry and in additive combinatorics. However, most results that were obtained using the Crossing Lemma do not appear to be optimal, and there is a quest for improved versions of the lemma for graphs satisfying certain special properties.

Abstract: We will discuss connections between Gromov's work on isoperimetry of waists and Milman's work on the M-ellipsoid of a convex body. It is proven that any convex body K in an n-dimensional Euclidean space has a linear image K_1 of volume one satisfying the following waist inequality: Any continuous map f from K_1 to R^d has a fiber f^{-1}(t) whose (n-d)-dimensional volume is at least c^{n-d}, where c > 0 is a universal constant. Already in the case where f is linear, this constitutes a slight improvement over known results.

Abstract:
Hamiltonian impact systems are dynamical systems in which there are two main mechanisms which dictate the system’s behavior - Hamilton’s equations which govern the motion inside the impact system domain, and the billiard reflection rule which governs the motion upon reaching the domain boundary. As the dynamics in impact systems are piecewise smooth by nature due to the
collisions with the boundary, many of the traditional tools used in the analysis of Hamiltonian