Beginning with Yau's solution of the Calabi conjecture, the complex Monge-Ampere equation has a central role in Kahler geometry. Of particular importance in the study of the complex Monge-Ampere equation is the question of $L^\infty$ estimates, and major breakthrough was obtained by Kolodziej in the late 90s using the method of pluripotential theory. A longstanding question has been whether such estimates can be obtained using PDE methods, and potentially be applicable to other nonlinear equations. We will present a new PDE based method for obtaining such $L^\infty$ estimates for a wide class of equations in complex geometry, which also applies in situations where the geometry degenerates. This is based on joint work with B. Guo and D.H. Phong.

Bio: Freid Tong received his undergraduate degree at the University of Toronto. He did graduate studies at Columbia University and received his PhD in 2021 under the supervision of D.H. Phong. He is currently a postdoctoral fellow at Harvard University.