Thursday, September 26, 2019
09/26/2019 - 3:00pm
Eigenvectors of non-Hermitian matrices are typically non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables quantify the stability of the spectrum. They first appeared in the physics literature; well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of the overlaps and their correlations. Some results are based on an explicit solvable P-recursive equation, for which conceptual understanding is lacking. (Joint work with Guillaume Dubach).
09/26/2019 - 4:15pm
Abstract: What is the most powerful topological quantum field theory (TQFT)? And, which 4-manifold invariants can detect the Gluck twist? Guided by questions like these, we will look for new invariants of 3-manifolds and smooth 4-manifolds. Traditionally, a construction of many such invariants and TQFTs involves a choice of certain algebraic structure, so that one can talk about “invariants for SU(2)” or a “TQFT defined by a given Frobenius algebra.” Surprisingly, recent developments lead to an opposite phenomenon, where algebraic structures are labeled by 3-manifolds and 4-manifolds, so that one can speak of VOA-valued invariants of 4-manifolds or MTC-valued invariants of 3-manifolds. Explaining these intriguing connections between topology and algebra will be the main goal of these lectures.