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.
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).
In this talk we will survey recent progress on the Beresticky-Caffarelli-Nirenberg Conjecture in Space Forms; that is, let $\Omega$ be an open connected domain of a complete connected Riemannian manifold ($M,g$) and consider the OEP given by
Based on joint works with Xin Li, Noam Solomon and Jiapeng Zhang.