Georgia Institute of Technology College of Sciences

We will look at various instances of how working with skeins (diagrams) provides a way to describe the existence of various topological quantum invariants that were originally described using representation theory. This provides a very simple description of these invariants. Along the way we will look at how to describe the algebraic data (ribbon categories) topologically and also how one could observe instances of certain dualities that exist between certain categories using these diagrams. 

Consider a sequence of independent and identically distributed SL(2, R) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. I will talk about a recent joint work with Anton Gorodetski and Victor Kleptsyn, where we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, R) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.

This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:

Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)

Complex manifolds:
Stein manifolds
Stein compactifications

Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds 

Depending on how long it takes to discuss these items, I will also attempt to include discussions on:

• Biran-Giroux decompositions of symplectic manifolds • Boothby-Wang bundles and contact plumbings of these • Milnor's fibration theorem for isolated singularities and connections to open book decompositions and Lefschetz fibrations • Open questions and interesting avenues of research

Most of our discussion will, as a side effect, outline the topological structure behind Type IIA String theory (the "topological A-model") which requires a 6-dimensional Calabi-Yau (Kahler) background.

Assuming the Riemann Hypothesis, Montgomery established results concerning the pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results for all level correlations of automorphic $L$-functions. We discover surfaces associated with the zeros of automorphic $L$-functions. In the case of pair correlation, the surface displays Gaussian behavior. For triple correlation, these structures exhibit characteristics of the Laplace and Chi-squared distributions, revealing an unexpected phase transition. This is joint work with Debmalya Basakand Alexandru Zaharescu.