**G. Khimshiashvili**

## Geometric aspects of
Riemann-Hilbert problems

**abstract:**

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.