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
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.
This lecture will describe the historical context and some key properties of o-minimality. It will then describe certain results in functional transcendence, generalizing the classical results on exponentiation due to Ax, and sketch how they can be proved.
This talk is mainly concerned with the investigation of small set expansion on the Grassmann Graphs, a study that was motivated by recent applications to Probabilistically Checkable Proofs and hardness of approximation.
This introductory lecture will describe results about counting rational points on certain non-algebraic sets and sketch how they can be used to attack certain problems in diophantine geometry and functional transcendence.