Weyl Law for the phase transition spectrum and density of limit-interfaces

Marco Mendez Guaraco
Member, School of Mathematics
November 5, 2018

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.

On the NP-hardness of 2-to-2 Games

Dor Minzer
Member, School of Mathematics
October 30, 2018

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).

 

The Zilber-Pink conjecture

Jonathan Pila
University of Oxford
October 26, 2018

The Zilber-Pink conjecture is a far reaching finiteness conjecture in diophantine geometry, unifying and extending Mordell-Lang and Andre-Oort. This lecture will state the conjecture, illustrate its varied faces, and indicate how the point-counting strategy can be applied to parts of it.