Given an arbitrary graph, we show that if we are allowed to modify (say) 1% of the edges then it is possible to obtain a much smaller regular partition than in Szemeredi's original proof of the regularity lemma. Moreover, we show that it is impossible to improve upon the bound we obtain.
We shall present a procedure which to any admissible family
of surfaces into an arbitrary closed riemannian manifolds assigns a
smooth, possibly branched, minimal surface
whose area is equal to the width of the corresponding minmax and whose
Morse index is bounded by the
dimension of the familly. We will discuss the question of bounding the
Morse index + Nullity from below as well as possible extensions of
this procedure to more general families.
We prove that the systole (or more generally, any k-th
homology systole) of a minimal surface in an ambient three manifold of
positive Ricci curvature tends to zero as the genus of the minimal
surfaces becomes unbounded. This is joint work with Anna Siffert.
In this talk, I will give an overview on how PCPs, combined with cryptographic tools,
are used to generate succinct and efficiently verifiable proofs for the correctness of computations.
I will focus on constructing (computationally sound) *succinct* proofs that are *non-interactive*
(assuming the existence of public parameters) and are *publicly verifiable*.
In particular, I will focus on a recent result with Omer Paneth and Lisa Yang,
where we show how to construct such proofs for all polynomial time computations,
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 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
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.