Georgia Institute of Technology College of Sciences

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.

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. 

The stein variational gradient descent (SVGD) algorithm is a deterministic particle method for sampling. However, a mean-field analysis reveals that the gradient flow corresponding to the SVGD algorithm (i.e., the Stein Variational Gradient Flow) only provides a constant-order approximation to the Wasserstein gradient flow corresponding to the KL-divergence minimization. In this work, we propose the Regularized Stein Variational Gradient Flow, which interpolates between the Stein Variational Gradient Flow and the Wasserstein gradient flow. We establish various theoretical properties of the Regularized Stein Variational Gradient Flow (and its time-discretization) including convergence to equilibrium, existence and uniqueness of weak solutions, and stability of the solutions. We provide preliminary numerical evidence of the improved performance offered by the regularization.

 Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number  of a graph G and its complement is at most |G|+1.  The Nordhaus-Gaddum graphs are the class of graphs satisfying this inequality with equality, and are well-understood. In this paper we consider a hereditary generalization: graphs G for which all induced subgraphs H of G satisfy that the sum of the chromatic numbers of H and its complement are at least |H|. We characterize the forbidden induced subgraphs of this class and find its intersection with a number of common classes, including line graphs. We also discuss chi-boundedness and algorithmic results.

I will review some recent results in the theory of differentiation of integrals.

In this talk, I will present a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize. DPALM can produce a (near) eps-KKT point within eps^{-2} outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, I will show overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of eps^{-2.5} to produce an eps-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is eps^{-3} to produce a near eps-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.

The uniformization theorem states that every Riemann surface is a quotient of some subset of the complex projective line by a group of Mobius transformations. However, a number of closely related questions regarding the structure of uniformization maps remain open. For example, it is unclear how one might associate a uniformizing map to a given Riemann surface. In this talk we will discuss an approach to this question due to Gunning by attaching a projective line bundle to a Riemann surface and studying its analytic properties.

 For a positive integer , define  to be the smallest number such that the additive energy of any subset and any  is at most . In this talk, I will survey recent results on bounds for , explore the connections with (variants of) the Hausdorff-Young inequality in analysis and with the Balog-Szemeredi-Gowers theorem in additive combinatorics, and then discuss new results on the asymptotic behavior of  as .

The normal distribution appears in a wide and disparate set of circumstances, and this ubiquity is explained by the central limit phenomenon. This talk will explore several forms of the central limit theorem, as well as different methods of proof. Highlights include a new method of moments proof for entries on a hypersphere sphere and results for traces of large random matrices utilizing the Malliavin-Stein method.

As is well known, many materials freeze at low temperatures. Microscopically,
  this means that their molecules form a phase where there is long range order
  in their positions. Despite their ubiquity, proving that these freezing
  transitions occur in realistic microscopic models has been a significant
  challenge, and it remains an open problem in continuum models at positive
  temperatures. In this talk, I will focus on lattice particle models, in which
  the positions of particles are discrete, and discuss a general criterion
  under which crystallization can be proved to occur. The class of models that
  the criterion applies to are those in which there is *no sliding*, that is,
  particles are largely locked in place when the density is large. The tool
  used in the proof is Pirogov-Sinai theory and cluster expansions. I will
  present the criterion in its general formulation, and discuss some concrete
  examples. This is joint work with Qidong He and Joel L. Lebowitz.

We present a lemma, inspired by dependent random choice and sampling procedures from statistical physics, for finding dense structure in arbitrary $d$-partite $d$-uniform hypergraphs. We will then discuss how this lemma leads to the concept of local rank, a notion of tensor rank which is instrumental in proving a "structure vs. randomness" result for tensors (and by extension, polynomials): namely, a relation between the partition and analytic ranks of tensors over finite fields. This is joint work with Guy Moshkovitz.

Given integers $1 < s < t$, what is the maximum size of a $K_s$-free subgraph that every $n$ vertex $K_t$-free graph is guaranteed to contain? This problem was posed by Hajnal, Erdős and Rogers in the 1960s as a way to generalize classical graph Ramsey numbers (which corresponds to the case $s=2$). We  prove almost optimal results in the case $t=s+1$ using recent constructions in Ramsey theory. We also consider the problem where we replace $K_s$ and $K_t$ by arbitrary graphs $H$ and $G$ and discover several interesting new phenomena.  This is joint work with Jacques Verstraete.