Georgia Institute of Technology College of Sciences

In 2017 Moseley, Proudfoot, and Young conjectured that the reduced Orlik-Terao algebra of the braid matroid was isomorphic as a symmetric group representation to the cohomology of a certain configuration space. This was proved by Pagaria in 2022. We generalize Pagaria's result from the braid arrangement to arbitrary hyperplane arrangements and recover a new proof in the case of the braid arrangement. Along the way, we give formulas for several other invariants of a hyperplane arrangement. Joint with Nick Proudfoot.

We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms to improve the computational efficiency of matrix cross approximation. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: image compression and the least squares approximation of continuous functions. Our numerical results demonstrate the effective performance of our approach.

We begin with a survey of some Floer-theoretic knot concordance and homology cobordism invariants. Building on these ideas, we describe a new family of homology cobordism invariants and give a new proof that there are no 2-torsion elements with Rokhlin invariant 1. This is joint work in progress with Irving Dai, Matt Stoffregen, and Linh Truong.

 When does a graph admit an orientation with some desired properties? This question has been studied extensively for many years and across many different properties. Specifically, I will talk about properties having to do with degree restrictions, and progress towards a conjecture of Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati dealing with a list-type of degree restriction. This is all joint work with my PhD advisor Jessica McDonald.

In this talk, I will introduce the Fourier-restricted Euler and hypodissipative Navier–Stokes equations. These equations are analogous to the Euler equation and hypodissipative Navier–Stokes equation, respectively, but with the Helmholtz projection replaced by a projection onto a more restrictive constraint space. The nonlinear term arising from the self-advection of velocity is otherwise unchanged. I will prove finite time-blowup when the dissipation is weak enough, by making use of a permutation symmetric Ansatz that allows for a dyadic energy cascade of the type found in the Friedlander-Katz-Pavlović dyadic Euler/Navier–Stokes model equation.

Given a random polynomial f of degree n with integer coefficients each drawn uniformly and independently from an interval [-H, H], what is the probability that the Galois group of the roots of f is NOT the full symmetric group Sₙ? In 1936, van der Waerden conjectured that the answer should be of order 1/H, with the dominant contribution coming from f with a rational root. This conjecture was finally resolved by Bhargava in 2023. In this project (joint w/ Theresa Anderson), we ask the same question for reciprocal (a.k.a. palindromic) polynomials, which arise for instance as the characteristic polynomials of symplectic matrices. Using a suitably modified variant of the Fourier-analytic methods of Bhargava and others, we find that polynomials with non-generic Galois group appear with frequency O(log H/H) and, unlike in van der Waerden's setting, almost all of these polynomials are irreducible.

Under the operation of connected sum, the set of three-manifolds form a monoid. Modulo an equivalence relation called homology cobordism, this monoid (of homology spheres) becomes a group. What is the structure of this group? What families of three-manifolds generate (or don’t generate) this group? We give some answers to these questions using Heegaard Floer homology. This is joint work with (various subsets of) I. Dai, K. Hendricks, M. Stoffregen, L. Truong, and I. Zemke.

We consider the task of estimating a rank-one matrix from noisy observations. Models that fall in this framework include community detection and spiked Wigner models. In this talk, I will discuss pseudo-maximum likelihood theory for such inference problems. We provide a variational formula for the asymptotic maximum pseudo-likelihood and characterize the asymptotic performance of pseudo maximum likelihood estimators. We will also discuss the implications of these findings to least squares estimators. Our approach uses the recent connections between statistical inference and statistical physics, and in particular the connection between the maximum likelihood and the ground state of a modified spin glass.

Based on joint work with Curtis Grant and Aukosh Jagannath.

The Heilbronn triangle problem is a classical problem in discrete geometry with several old and new connections to various topics in extremal and additive combinatorics, graph theory, incidence geometry, harmonic analysis, and projection theory. In this talk, we will give an overview of some of these connections, and discuss some recent developments. Based on joint work with Alex Cohen and Dmitrii Zakharov.

As a society, we are faced with important challenges to combat climate change. Geological Carbon Storage, during which gigatonnes of super-critical CO2 are stored underground, is arguably the only scalable net-negative negative CO2-emission technology that is available. Recent advances in generative AI offer unique opportunities—especially in the context of Digital Twins for subsurface CO2-storage monitoring, decision making, and control—to help scale this technology, optimize its operations, lower its costs, and reduce its risks, so assurances can be made whether storage projects proceed as expected and whether CO2 remains underground.

During this talk, it is shown how techniques from Simulation-Based Inference and Ensemble Bayesian Filtering can be extended to establish probabilistic baselines and assimilate multimodal data for problems challenged by large degrees of freedom, nonlinear multiphysics, and computationally expensive to evaluate simulations. Key concepts that will be reviewed include neural Wave-Based Inference with Amortized Uncertainty Quantification and physics-based Summary Statistics, Ensemble Bayesian Filtering with Conditional Neural Networks, and learned multiphysics inversion with Differentiable Programming.

This is joint work with Rafael Orozco.

Given a digraph $D$, we say that a set of vertices $Q\subseteq V(D)$ is a quasikernel if $Q$ is an independent set and if every vertex of $D$ can be reached from $Q$ by a path of length at most 2.  The Small Quasikernel Conjecture of P.L. Erdős and Székely from 1976 states that every $n$-vertex source-free digraph $D$ contains a quasikernel of size at most $\frac{1}{2}n$.  Despite being posed nearly 50 years ago, very little is known about this conjecture, with the only non-trivial upper bound of $n-\frac{1}{4}\sqrt{n\log n}$ being proven recently by ourself.  We discuss this result together with a number of other related results and open problems around the Small Quasikernel Conjecture.