Abstract: In this talk we show that given any regular cone with entropy less than that of round cylinder, all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class. This is joint work with J. Bernstein.
Based on joint works with Xin Li, Noam Solomon and Jiapeng Zhang.
Abstract: The Allen-Cahn equation behaves as a desingularization of the area functional. This allows for a PDE approach to the construction of minimal hypersurfaces in closed Riemannian manifolds. After presenting and overview of the subject, I will discuss recent results regarding a Weyl Law and its consequences for the density of minimal hypersurfaces in generic metrics. This is joint work with P. Gaspar.
Abstract: The Clifford torus is the simplest nontotally geodesic minimal surface in S^3. It is a product surface, it is helicoidal, and it is a solution obtained by separation of variables. We will show that there are more minimal submanifolds with these properties in S^n and in R^4.
The Unique-Games Conjecture is a central open problem in the field of PCP’s (Probabilistically Checkable Proofs) and hardness of approximation, implying tight inapproximability results for wide class of optimization problems.
We will discuss PCPs, the Unique-Games Conjecture and some recent progress. (no familiarity with PCPs or with last week's talk are needed).