tbranzov

These lectures will present a board survey of recent work on new q-series invariants of 3-manifolds labeled by Spin-C structures. While the original motivation for studying these invariants is rooted in topology, they exhibit a number of unexpected properties and connections to other areas of mathematics, e.g. turn out to be characters of logarithmic vertex algebras. The integer coefficients of these q-series invariants can be understood as the answer to a certain enumerative problem, and when q tends to special values these invariants relate to other invariants of 3-manifolds labeled by Spin and Spin-C structures.

We will start these series by reviewing the general framework of geometric Langlands correspondence, and state the main conjectures in the de Rham and Betti settings. We will also recall V. Lafforgue's theorem about the spectral decomposition in the classical Langlands over function fields. We will then proceed to the formulation of "restricted" Langlands correspondence, which unifies the different contexts. We will state the restricted version of the geometric Langlands conjecture, and explain its relation with the classical Langlands conjecture via the operation of categorical trace.

One of the key developments in combinatorics and algebra of recent years has been the discovery of Lefschetz principles beyond Hodge structures, resolving several long-standing conjectures. I will provide an overview of recent developments, and discuss joint work with Johanna Steinmeyer, Stavros Papadakis and Vasiliki Petrotou.