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Abstract

We present a survey of recent results concerned with generalizations of the classical Riemann-Hilbert transmission problem in the context of loop spaces. Specifically, we present a general formulation of a Riemann-Hilbert problem with values in an almost complex manifold and illustrate it by discussing two particular cases in more detail. First, using the generalized Birkhoff factorization theorem of A. Pressley and G. Segal we give a criterion of solvability for generalized Riemann-Hilbert problems with coefficients in the loop group of a compact Lie group. Next, we present a visual example of solution to a Riemann-Hilbert problem with values in the immersed loop space of three-dimensional sphere. Finally, we describe a geometric construction of Fredholm structures on loop groups and relate them to the canonical Fredholm structures on Kato Grassmannians.