Georgia Institute of Technology College of Sciences

For a connected graph G, the set of G-parking functions are integer sequences counted by spanning trees that arise in the theory of chip-firing on G.  They can also be defined as the standard monomials of a `G-parking function ideal', whose homological properties have interesting combinatorial interpretations. We extend these constructions to the setting of hypergraphs, where edges can have multiple vertices. We study algebraic and combinatorial aspects of parking functions in this context, employing generalized notions of acyclic orientations and spanning trees. Minimal cellular resolutions of the underlying ideals can be understood in terms of certain generalized permutohedra. This is joint work with Ayah Almousa and Ben Smith, as well as an REU project with Timothy Blanton, Isabelle Hong, Suho Oh, and Zhan Zhan.

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.

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.

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.