We answer a 1982 conjecture of Erdős and Simonovits about the growth of number of $k$-walks in a graph, which incidentally was studied earlier by Blakley and Dixon in 1966. We prove this conjecture in a more general setup than the earlier treatment, furthermore, through a refinement and strengthening of this inequality, we resolve two related open questions in complexity theory: the communication complexity of the $k$-Hamming distance is $\Omega(k \log k)$ and that consequently any property tester for k-linearity requires $\Omega(k \log k)$.
I will discuss some geometric inequalities that hold on
Riemannian 2-disks and 2-spheres.
For example, I will prove that on any Riemannian 2-sphere there M exist
at least three simple periodic geodesics of length at most 20d, where d is the diameter of M, (joint with A. Nabutovsky, Y. Liokumovich).
This is a quantitative version of the well-known Lyusternik and Shnirelman theorem.
Abstract: Uryson k-width of a metric space X measures how close X is to being k-dimensional. Several years ago Larry Guth proved that if M is a closed n-dimensional manifold, and the volume of each ball of radius 1 in M does not exceed a certain small
constant e(n), then the Uryson (n-1)-width of M is less than 1. This result is a significant generalization of the famous Gromov's inequality relating the volume and the filling radius that plays a central role in systolic geometry.
Abstract : It is a joint work with G. Courtois, S. Gallot and A.Sambusetti. We prove a compactness theorem for metric spaces with anupper bound on the entropy and other conditions that will be discussed.Several finiteness results will be drawn. It is a work in progress.
Abstract: For surfaces immersed into a compact Riemannian manifold, we consider the curvature functional given by the $L^2$ integral of the second fundamental form. We discuss an an area bound in terms of that functional, with application to the existence of minimizers (joint work with V. Bangert).
Abstract: Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the
direct method fails, as the infimum of the Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the
harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow.