Georgia Institute of Technology College of Sciences

We introduce combinatorial B-matrices over ordered blueprints B, which are combinatorial analogues of matrices and correspond to "matroids with a fixed basis". This provides a unifying framework for the study of bimatroids, linking sets, and their valuated analogues.  We then introduce and study their corresponding moduli spaces and describe their relations to the moduli space of matroids, introduced by Baker and Lorscheid. Inspired by the underlying combinatorics in the classical case, this allows us to define several interesting functors between moduli spaces of matrices and moduli spaces of matroids, and, by extension, between moduli spaces of matroids of different ranks.

Parts of this talk are based on joint work in progress with Martin Ulirsch.

Equation learning has been a holy grail of scientific research for decades. Only recently has the capability of learning equations directly from data become a computationally feasible task, due to the availability of high-resolution data and fast algorithms capable of surpassing the inherent combinatorial complexity of most model classes. Weak form equation learning has arisen as an advantageous framework for efficiently selecting models from data with noise and nonsmoothness, qualities inherent to observed data. By viewing the dynamics through the guise of test functions, the weak form affords a flexible representation of the governing equations that naturally incorporates these data maladies. More generally, the weak form has been shown to reveal alternative dynamical descriptions, such as coarse-grained and reduced-order models, opening the door to hierarchical model discovery. In this talk I will give a broad overview of historical advances in weak form equation learning and parameter inference, from the 1950s to WSINDy and more recent algorithms. I will then give an outlook for future research directions in this field, in light of now-known computational limitations and recently demonstrated successes, both theoretical and applied, with applications to molecular dynamics, plasma physics, cell biology, and weather forecasting.

The aim of this talk is to discuss recent advances around the convergence problem in mean field control theory and the study of associated nonlinear PDEs.

We are interested in optimal control problems involving a large number of interacting particles and subject to independent Brownian noises. As the number of particles tends to infinity, the problem simplifies into a McKean-Vlasov type optimal control problem for a typical particle. I will present recent results concerning the quantitative analysis of this convergence. More precisely, I will discuss an approach based on the analysis of associated value functions. These functions are solutions of Hamilton-Jacobi equations in high dimension and the convergence problem translates into a stability problem for the limit equation which is posed on a space of probability measures.

I will also discuss the well-posedness of this limiting equation, the study of which seems to escape the usual techniques for Hamilton-Jacobi equations in infinite dimension.

We prove that the planar Bochner Riesz mean converges almost everywhere for any L^p function in the optimal range, for 5/3

An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which imitates and generalizes the behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, which shows that every hyperbolic group that acts geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. Mainly, the proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.

Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure.  The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.  Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.

We discuss an approach to estimate aggregation and adaptive estimation based upon (nearly optimal) testing of convex hypotheses. The proposed adaptive routines generalize upon now-classical Goldenshluger-Lepski adaptation schemes, and, in the situation where the observations stem from simple observation schemes (i.e., have Gaussian, discrete and Poisson distribution) and where the set of unknown signals is a finite union of convex and compact sets, attain nearly optimal performance. As an illustration, we consider application of the proposed estimates to the problem of recovery of unknown signal known to belong to a union of ellitopes in Gaussian observation scheme. The corresponding numerical routines can be implemented efficiently when the number of sets in the union is “not very large.” We illustrate “practical performance” of the method in an example of estimation in the single-index model.

In this talk, we will discuss recent progress in the theory of smooth star flows that contain singularities and consider their expansiveness, continuity of the topological pressure, and the existence and uniqueness of equilibrium states. We will prove an ergodic version of the Spectral Decomposition Conjecture: $C^1$ open and densely, every singular star flow has only finitely many ergodic measures of maximal entropy, and only finitely many ergodic equilibrium states for Holder continuous potentials satisfying a mild yet optimal condition. Joint with M.J. Pacifico and J. Yang.