# Loop spaces and Riemann-Hilbert problems

Banach Center Publications (2007)

- Volume: 76, Issue: 1, page 411-424
- ISSN: 0137-6934

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topG. Khimshiashvili. "Loop spaces and Riemann-Hilbert problems." Banach Center Publications 76.1 (2007): 411-424. <http://eudml.org/doc/282305>.

@article{G2007,

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.},

author = {G. Khimshiashvili},

journal = {Banach Center Publications},

keywords = {Fredholm structures},

language = {eng},

number = {1},

pages = {411-424},

title = {Loop spaces and Riemann-Hilbert problems},

url = {http://eudml.org/doc/282305},

volume = {76},

year = {2007},

}

TY - JOUR

AU - G. Khimshiashvili

TI - Loop spaces and Riemann-Hilbert problems

JO - Banach Center Publications

PY - 2007

VL - 76

IS - 1

SP - 411

EP - 424

AB - 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.

LA - eng

KW - Fredholm structures

UR - http://eudml.org/doc/282305

ER -

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