Georgia Institute of Technology College of Sciences

The rank r matrix completion problem studies whether a matrix where some of the entries have been filled in with generic complex numbers can be completed to a matrix of rank at most r. This problem is governed by the bipartite rigidity matroid, which is a matroid studied in combinatorial rigidity theory. We show that the study of the bipartite rigidity matroid is related to the study of tensor codes, a topic in information theory, and use this relation to understand new cases of both problems. Joint work with Joshua Brakensiek, Manik Dhar, Jiyang Gao, and Sivakanth Gopi.

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.

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.

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.