Sidney Webster


Sidney Webster
Eckhart 311
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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:

  1. Determine Fefferman's asymptotic expansion of the Bergman and Szego kernels more precisely in terms of these and related invariants.
  2. The holomorphic embedding problem (local existence and regularity) for formally integrable CR structures.
  3. 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.