G. Khimshiashvili

Geometric aspects of Riemann-Hilbert problems

We discuss some recent generalizations of the Riemann-Hilbelrt transmission problem and their connections with certain geometric objects appearing in the theory of loop groups and infinite-dimensional Grassmanians. In particular, we describe in some detail the geometric model for the totality of elliptic Riemann-Hilbert problems in terms of Fredholm pairs od subspaces suggested by B. Bojarski. A number of fundamental geometric and topological properties of related infinite-dimensional Grassmanians are established in this context, with a special emphasis on relations to the theory of Fredholm structures. Several generalizations of the classical Riemann-Hilbert problem are also discussed. The main attention is given to linear conjugation problems for compact Lie groups, Riemann-Hilbert problems for generalized Cauchy-Riemann sysytems, and nonlinear Riemann-Hilbert problems for solutions of generalized Cauchy-Riemann systems. Some geometric aspects of the Riemann-Hilbert monodromy problem for ordinary differential equations with regular singural points are discussed in brief.

Mathematics Subject Classification: 30E25, 32F35, 32S05, 35F15, 58B05, 58G03, 59G20.

Key words and phrases: Holomorphic function, Riemann-Hilbert transmission problem, Tö plitz operator, Birkhoff factorization, partial indices, Fredholm operator, Fredholm index, compact operator, Shatten ideal, Hilbert-Schmidt operator, compact Lie group, linear representation, loop group, holomorphic vector bundle, Fredholm structure, $C^*$-algebra, Hilbert module, Clifford algebra, Dirac operator, generalized Cauchy-Riemann operator, Shapiro-Lopatinski condition, analytic disc, totally real submanifold, complex point, hyper-holomorphic cell, Riemann-Hilbert monodromy problem, regular point.