Lie groups and transmission problems on Riemann surfaces

Global Geometric Aspects of Riemann–Hilbert Problems

“…As was shown in [22], the restricted Grassmanian has also a remarkable structure of a cellular complex (CW-complex) which is closely related to the so-called partial indices [1] and gives a visual interpretation of certain phenomena discussed in [1], [2]. Moreover, Fredholm Grassmanians can be turned into differentiable manifolds, which enables one to construct an analogue of the Morse theory and recover in this way the cellular structure obtained from the partial indices [22], [17]. We describe here a simple explicit way of introducing differentiable manifold structures on Fredholm Grassmanians Gr s .…”

“…In order to obtain a proper framework for discussing more subtle geometric properties of Fredholm Grassmanians we proceed by describing some connections with the Fredholm structures theory [10]. As was observed in [16], [17], certain dense subsets of these Grassmanians can be endowed with Fredholm structures. This fact seems to be quite remarkable since a Fredholm structure on an infinite-dimensional manifold enables one to introduce non-trivial global geometric and topological invariants of this manifold.…”

“…An important result due to J.Elworthy and A.Tromba states that a Fredholm structure on M can be constructed from a Fredholm mapping M → H with zero index and also from certain smooth families of zero index Fredholm operators parametrized by the points of M [11]. These facts were used in [15], [17] to construct Fredholm structures on loop groups. The following statement follows from the results of [15], [17].…”

“…These facts were used in [15], [17] to construct Fredholm structures on loop groups. The following statement follows from the results of [15], [17].…”

“…In particular as was suggested in [2] (see also [3]) the totality of elliptic Riemann-Hilbert problems permits a visual geometric description in terms of Fredholm pairs of subspaces of an appropriate functional space. This interpretation enables one to study various global aspects of the Riemann-Hilbert problem in an abstract setting, which eventually led to some conceptual developments [4], [5], [15], [16] and non-trivial geometric results about the so-called Fredholm Grassmanian [2], [4], [17], [18], [24]. Closely related concepts and constructions appeared useful in the geometric theory of loop groups of compact Lie groups [22], [16].…”