• I. Kossovskiy (U. Western Ontario), Mappings of 2-nondegenerate hypersurfaces in dimension 3
  • Let (M,p) and (M',p') be two real hypersurfaces with distinguished points in complex affine n-space and let H(M,p;M',p') be the space of local biholomorphic mappings of the ambient space preserving the hypersurfaces and the distinguished points. How "rich" can the space H(M,p;M',p') be? Poincaré (for n=2) and later Chern and Moser (for arbitrary n) in their famous papers gave the answer to this question for Levi non-degenerate hypersurfaces. Their results generated a big stream of further papers on CR-geometry and led to remarkable theorems in complex analysis. Using a new approach, we avoid the difficulties which occur in the Levi-degenerate case and reproduce the Poincaré-Chern-Moser theory for the case of 2-nondegenerate hypersurfaces in complex 3-space. Joint work with Valery Beloshapka.
• Y. Zhang (UCSD), Chern-Moser-Weyl tensor theory and its applications to the Hopf Lemma for CR maps
• P. Dragnev (IPFW), Convexity of harmonic measures on real line and circle
• A. Coffman (IPFW), Counterexamples to unique continuation for a Beltrami system in C²
• Y. Pan (IPFW), Solvability of nonlinear PDE systems in dimension two
  • In this talk, we present a general existence result (local and global) for a nonlinear partial differential system of any order in dimension two. In particular it implies the local existence of J-holomorphic curves on a almost complex manifold, due to Nijenhuis and Woolf, and also implies the existence of harmonic maps from the unit disk to any Riemannian manifold with prescribed tangent vector, which could be new. As a consequence of the method, we prove that any nonlinear partial differential system with a power m of the Laplace operator as principal part can be always solvable locally for any jet of order 2m-1 at the origin. At the same time, global solutions can be obtained, provided the system vanishes to first order at the origin. These results are almost best possible due to the classical theory of Ahlfors and Osserman.

Zhang's visit as Scholar in Residence supported in part by IPFW Office of Research, Engagement, and Sponsored Programs

To the Department of Mathematical Sciences