Georgia Institute of Technology College of Sciences

In 1992 Hitchin discovered distinguished components of the PSL(d,R) character variety for closed surface groups pi_1S and asked for an interpretation of those components in terms of geometric structures. Soon after, Choi-Goldman identified the SL(3,R)-Hitchin component with the space of convex projective structures on S. In 2008, Guichard-Wienhard identified the PSL(4,R)-Hitchin component with foliated projective structures on the unit tangent bundle T^1S. The case d \ge 5 remains open, and compels one to move beyond projective geometry to flag geometry. In joint work with Alex Nolte, we obtain a new description of the SL(3,R)-Hitchin component in terms of concave foliated flag structures on T^1S. 

Young tableaux arise in the enumerative geometry of linear series on curves in formulas for the Chow class and the holomorphic Euler characteristic of Brill--Noether varieties. I will discuss an intriguing tropical generalization of these two facts: the formulas for Chow class and Euler characteristic of Brill--Noether loci on a general curve occur in the first and last terms of the Ehrhart polynomial of the tropical Brill--Noether loci on a chain of loops. I will speculate on some generalizations and algebraic analogs of this calculation.

A basic result of probabilistic combinatorics, originally due to Erdős and Rényi, is the determination of the threshold at which the random graph $G_{n,p}$ contains a triangle with high probability. But one can also ask more refined versions of this question, where we ask not just for one triangle but for many triangles which interact in complicated ways. For example, what is the threshold at which we can no longer partition $G_{n,p}$ into two triangle-free subgraphs?


In this talk, I will discuss the proof of the Kohayakawa–Kreuter conjecture, which gives a general answer to all such questions. Rather surprisingly, a key step of the proof is a purely deterministic graph decomposition statement, closely related to classical results such as Nash-Williams' tree decomposition theorem, whose proof uses techniques from combinatorial optimization and structural graph theory.

Based on joint works with Micha Christoph, Eden Kuperwasser, Anders Martinsson, Wojciech Samotij, and Raphael Steiner.

In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. Such results include delocalization properties of eigenvectors and tail estimates for the smallest singular value. A key ingredient in their proofs is a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace.  Building on work of Livshyts and Livshyts, Tikhomirov and Vershynin, we introduce a new distance theorem for inhomogeneous vectors and subspaces spanned by the columns of an inhomogeneous matrix. Such a result has a number of applications for generalizing results about the quantitative behavior of i.i.d. matrices to matrices without any identical distribution assumptions. To highlight this, we show that the smallest singular value estimate of Rudelson and Vershynin, proven for i.i.d. subgaussian rectangular matrices, holds true for inhomogeneous and heavy-tailed matrices.
This talk is partially based on joint work with Max Dabagia.