Mean curvature flow is the negative gradient flow of the
volume functional which decreases the volume of (hyper)surfaces in the
steepest way. Starting from any closed surface, the flow exists
uniquely for a short period of time, but always develops singularities
in finite time. In this talk, we discuss some non-uniqueness problems
of the mean curvature flow passing through singularities. The talk is
mainly prepared for non-specialists of geometric flows.
In this talk I would like to explain how methods from
symplectic geometry can be used to obtain sharp systolic inequalities.
I will focus on two applications. The first is the proof of a
conjecture due to Babenko-Balacheff on the local systolic maximality
of the round 2-sphere. The second is the proof of a perturbative
version of Viterbo's conjecture on the systolic ratio of convex energy
levels. If time permits I will also explain how to show that general
systolic inequalities do not exist in contact geometry. Joint work
What is the largest number of projections onto k coordinates guaranteed in every family of m binary vectors of length n? This fundamental question is intimately connected to important topics and results in combinatorics and computer science (Turan number, Sauer-Shelah Lemma, Kahn-Kalai-Linial Theorem, and more), and is wide open for most settings of the parameters. We essentially settle the question for linear k and sub-exponential m.
Based on joint work with Noga Alon and Noam Solomon.
In their seminal works from the 80's, Lubotzky, Phillips and Sarnak proved the following two results: