Pranav Pandit "Buildings, spectral networks, and the Riemann-Hilbert correspondence at infinity"

Pranav Pandit (Vienna)
Buildings, spectral networks, and the Riemann-Hilbert correspondence at infinity

Affine buildings are certain non-positively curved metric spaces that can be viewed as higher dimensional analogues of trees. During the talk, I will introduce buildings and indicate how the notion of a harmonic map between manifolds can be generalized to the setting where the target space is a building.

We will work on a fixed Riemann surface. The map that associates to a connection its monodromy representation defines a complex analytic isomorphism from the moduli space of holomorphic vector bundles with connection to the moduli space of representations of the fundamental group. The goal of this lecture is to explain the sense in which the asymptotic behavior of this map (at infinity) is governed by certain harmonic maps from the universal cover of the Riemann surface to affine buildings. This is based on joint work with Ludmil Katzarkov, Alexander Noll and Carlos Simpson.