Professor

- Office:
- Eckhart 311

- Office Phone:
- 773-702-7426
- Department Email:
- webster@math.uchicago.edu

## Research

I work on the holomorphic geometry of smooth bounded domains in the complex space \(\mathbb{C}^n\). It is conjectured, and known in many cases, that biholomorphic maps of such extend smoothly to the boundary. In the Levi non-degenerate case, the induced CR structure on the boundary has a complete system of invariants, manifested in a normal form (Chern-Moser theory). Some general problems are:

- Determine Fefferman's asymptotic expansion of the Bergman and Szego kernels more precisely in terms of these and related invariants.
- The holomorphic embedding problem (local existence and regularity) for formally integrable CR structures.
- Geometry of CR singularities, especially for real
\(n\)-manifolds in \(\mathbb{C}^n\), normal forms, hulls of holomorphy, etc.

Recently, my former graduate student, Prof. X. Gong, and I have obtained solutions to the local CR-embedding problem (2), and to the integrability problem for CR vector bundles, which have sharp regularity. This has led to my discovery of new invariants, both local and global, of a fundamental solution to the Cauchy-Riemann equations in several complex variables.