Irina Mitrea, Temple University
Abstract: In its classical form, the Riemann-Hilbert problem asks for determining two holomorphic functions defined on either side of a surface $\Sigma$, satisfying a boundary condition of transmission type along $\Sigma$ involving a symbol function $\Phi$. In this regard, I will report on recent progress with Marius Mitrea and Michael Taylor describing the Fredholm solvability in the most geometric measure theoretic setting in which such a problem is meaningfully formulated. This involves replacing a complex plane by a Riemannian manifold $\mathcal{M}$, the surface $\Sigma$ by a uniformly rectifiable subset of $\mathcal{M}$, and the Cauchy-Riemann operator by a general Dirac operator on $\mathcal{M}$ with low regularity assumptions on its coefficients. This topic interfaces with Index Theory on manifolds, and as an application I will discuss the most general Bojarski index formula known to date.