Generalized Riemann-Hilbert Transmission and Boundary Value Problems, Fredholm Pairs and Bordisms

TL;DR

This paper explores generalized Riemann-Hilbert problems within the frameworks of K-theory and bordism theory, emphasizing the role of Fredholm pairs and proposing an abstract bordism concept in Hilbert spaces.

Contribution

It introduces a novel abstract formulation of bordism in Hilbert spaces with splittings, connecting classical problems with modern algebraic topology.

Findings

Fredholm pairs are essential in formulating Riemann-Hilbert problems.

A new abstract notion of bordism in Hilbert spaces is proposed.

The approach unifies classical and generalized boundary value problems.

Abstract

We present classical and generalized Riemann-Hilbert problem in several contexts arising from $K$-theory and bordism theory. The language of Fredholm pairs turns out to be useful and unavoidable. We propose an abstract formulation of a notion of bordism in the context of Hilbert spaces equipped with splittings.