##### Department of Mathematics,

University of California San Diego

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### Advancement to Candidacy

## Soumya Ganguly

#### UCSD

## Classifying pseudoconvex domains by properties of Bergman Kernel and Kähler-Einstein Metrics

##### Abstract:

A fundamental problem in geometry is to classify geometric structures. In one complex variable, for example, the Riemann Mapping Theorem asserts that any simply connected region of the plane, other than the plane itself, is biholomorphically equivalent to the unit disk. This is far from true in higher dimensions, where the local CR geometry of the boundary obstructs the existence of biholomorphisms. In this talk, we shall survey some results and open problems characterizing the unit ball and ball quotients, up to biholomorphism, by properties of the Bergman kernel (e.g., the Ramadanov Conjecture and one concerning algebraicity of the kernel) and the Bergman metric (Cheng’s Conjecture). Particular focus will be on generalizing some of the results to algebraic surfaces, weakly pseudoconvex domains and solving Cheng's conjecture for Stein spaces in dimension 2.

Advisors: Peter Ebenfelt, Ming Xiao

### May 16, 2022

### 9:00 AM

AP&M 7218

https://ucsd.zoom.us/j/

Zoom Meeting ID: 979 3649 6792

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