# Colloquium abstracts

**Ved Datar***

IISC, Bangalore

July 16, 2020

**Non-linear partial differential equations and positivity conditions in algebraic geometry**: In the early 2000s Demailly and Paun proved that a (1,1) cohomology class on a K"ahler manifold is positive if and only if certain intersection numbers are positive. This is a transcendental analog of the classical Nakai criteria for ampleness from algebraic geometry. Somewhat surprisingly, the proof of Demailly and Paun's theorem relies on Yau's work on the complex Monge-Ampere equation, and his solution to the Calabi conjecture. In 2019, Gao Chen extended the method of Demailly and Paun to prove that another important PDE in Kahler geometry, namely the J-equation, has a solution if and only if certain (twisted) intersection numbers are uniformly positive, thereby settling a uniform version of a conjecture of Lejmi and Szekelyhidi. In my talk, I will describe this circle of ideas. If time permits, I will conclude by describing a recent joint work with my colleague Vamsi Pingali on extending Gao Chen's main theorem to other so-called inverse Hessian equations, albeit on projective mani