Abstract: Both the Higgs bundle moduli space and the moduli space of holomorphic connections have a natural stratification induced by a C* action. In both of these stratifications, each stratum is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and holomorphic connections, this homeomorphism does not preserve the respective strata. The closed strata on the Higgs bundle side is the image of the Hitchin section and the closed strata in the space of flat connections is the space of opers. In this talk we will show how many of the relationships between opers and the Hitchin section extend to general strata. This is based on joint work with Richard Wentworth.
Abstract: We discuss whether it is possible to reconstruct an affine connection, a (pseudo)-Riemannian metric or a Finsler metric
by its unparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for
general relativity. We show how to understand whether all curves from a sufficiently
big family are unparameterized geodesics of a certain affine connection, and how
to reconstruct algorithmically a generic 4-dimensional metric by its unparameterized
geodesics. I will also explain how this theory helped to solve two problems
explicitly formulated by Sophus Lie in 1882. This portion of results is joint with R. Bryant, A. Bolsinov, V. Kiosak, G. Manno, G. Pucacco. At the end of my talk, I will explain that the so-called chains in the CR-geometry are geodesics of a so-called Kropina Finsler metric. I will show that sufficiently many geodesics determine the Kropina Finsler metric, which reproves and generalizes the famouse result of Jih-Hsin Cheng, 1988, that chains dermine the CR structure. This correspondence between chains and Kropina geodesics allows us to use the methods of metric geometry to study chains, we employ it to re-prove the result of H. Jacobowitz, 1985, that locally any two points of a strictly pseudoconvex CR manifolds can be joined by a chain, and generalize it to a global setting. This portion of results is joint with J.-H. Cheng, T. Marugame, R. Montgomery.