Georgia Institute of Technology College of Sciences

The chromatic polynomial of a graph, which counts colorings of the graph, has a habit of showing up in unexpected places in geometry, e.g. in the theory of hyperplane arrangements. This sometimes has interesting purely combinatorial consequences, such as Huh's proof of Hoggar/Read's conjecture on coefficients of chromatic polynomials. 

I'll discuss a new incarnation of chromatic polynomials. To a graph G, we can naturally associate a sequence of intersection numbers on moduli spaces of stable curves. Surprisingly, we prove that these recover values of the chromatic polynomial of G at negative integers. 

I'll also discuss how this leads to new algebraic invariants of directed graphs.

(Joint with Bernhard Reinke)

The Khovanov-Rozansky skein lasagna module was introduced by Morrison-Walker-Wedrich as an invariant of 4-manifold with a framed oriented link in the boundary. I will discuss an extension of the skein lasagna theory to 4-manifolds with codimension 2 corners, and its behavior under gluing. I will also talk about a categorical framework for computing skein lasagna modules of closed 4-manifolds via trisection, as well as an extended 4d TQFT based on skein lasagna theory. This is joint work with Sarah Blackwell and Slava Krushkal.

This talk explores how to construct meaningful features from noisy, high-dimensional data by leveraging geometric and invariant structures. First, we introduce a geometric framework for dimension reduction using a power-weighted path metric, which effectively de-noises high-dimensional data while preserving its intrinsic geometric structure. This framework is particularly useful for analyzing single-cell RNA data and for multi-manifold clustering, and we provide theoretical guarantees for the convergence of the associated graph Laplacian operators. We then turn to the problem of constructing features invariant to group actions in the multi-reference alignment (MRA) data model. In this setting one has many noisy observation of a hidden signal corrupted by both a group action(s) and additive noise, and one wants to recover the hidden signal from the noisy data. By formulating MRA in function space, we uncover a new connection to deconvolution: the hidden signal can be recovered from second-order Fourier statistics via an approach analogous to Kotlarski’s identity. We extend this identity to general dimensions, analyze recovery in the presence of vanishing Fourier transforms, and validate the resulting deconvolution framework with both theoretical guarantees and numerical experiments.

One of the biggest discoveries in the theory of dynamical systems was that smooth (deterministic) systems can behave very randomly. Since then a rich theory of chaotic properties of smooth dynamical systems was developed using geometric, topological and probabilistic methods. In the talk we will present ideas, highlight main results and discuss techniques that were developed during the last 70 years. In the second part we plan to discuss more recent advancements and present main open questions in the field. In the last part I will focus on connections between smooth ergodic theory and number theory.

The Guderley problem describes the behavior of a strong self-similar shock wave propagating radially in an ideal gas. A spherical shock converges radially inwards to the spatial origin, strengthening as it collapses. At the collapse point, the shock's strength becomes infinite, leading to the formation of a new outgoing shock wave of finite strength, which then propagates outwards to infinity. 

In this talk, I will present recent work on the rigorous construction of the self-similar converging and diverging shock solutions for $\gamma \in (1,3]$. These solutions are analytic away from the shock interfaces and the blow-up point. The proof relies on continuity arguments, nonlinear invariances, and barrier functions.

I'll present various methods, some old, some new,  leading to estimates on the variance of $f(X_1, X_2, \dots, X_n)$ where  

$X_1, X_2, \dots, X_n$ are independent random variables.  These methods will be illustrated with various examples.

From a 3-body problem (CR3BP)
to modeling periodic trajectories with algebraic curves
to minimal problems (related to liaison navigation)
solved via computer algebra (python, Macaulay2).
 

Sampling and mean-field optimization can be viewed as optimization in the space of probability distributions. Stochastic optimization algorithms like stochastic gradient descent have been immensely successful for optimization over Euclidean spaces. However, the infinite-dimensional space of probability distributions poses unique challenges. In this talk, I will discuss my recent work on the design and analysis of a stochastic algorithm for mean-field optimization with applications to the increasingly studied area of mean-field neural networks.

The study of the electronic properties of twisted bilayer graphene (TBG) has garnered much attention from the condensed matter community recently. TBG is obtained by stacking two graphene monolayers on top of each other, and rotating one of them with respect to the other. Theoretical and experimental analyses have found that the electronic properties of TBG depend very strongly on the angle between the layers. In fact, a handful of “magic” angles have been predicted at which TBG becomes a superconductor, and this has even been verified experimentally. The model commonly used to study TBG is an effective one, and was derived by Bistritzer and MacDonald. In this talk, I will present recent results on developing a framework to study TBG from first principles. To be more exact, we consider a tight-binding model for the electrons, but make no further approximations. Using a renormalization group technique, we construct a perturbative expansion to study TBG that is convergent when the twisting angle satisfies certain diophantine conditions. This is joint work with V. Mastropietro.

We give an overview of Teichmuller theory, the deformation theory of Riemann surfaces. The richness of the subject comes from all the perspectives one can take on Riemann surfaces: complex analytic for sure, but also Riemannian, topological, dynamical and algebraic.  In the past 40 years or so, interest has erupted in an extension of Teichmuller theory, here thought of as a component of the character variety of surface group representations into PSL(2,\R), to the study of the character variety of surface group representations into higher rank Lie groups (e.g. SL(n, \R)). We give a even breezy  discussion of that.

Proving positivity of the top Lyapunov exponent ($\lambda_1$​) and obtaining parameter-dependent lower bounds is an interesting and challenging problem for SDEs (stochastic differential equations). We outline methods to obtain lower bounds and establish positivity of $\lambda_1$​ for certain SDEs, combining the coordinate rescaling framework of Pinsky–Wihstutz (1988) for nilpotent linear It\^{o} systems with Fisher information formulas for Lyapunov exponents introduced by J. Bedrossian, A. Blumenthal, and S. Punshon-Smith (2022). This approach uses hypoellipticity and regularity of 2nd order linear PDEs.

We apply these techniques to a 2-D toy SDE to obtain positive lower bounds and small-noise scaling (in terms of noise parameter $\sigma$) for $\lambda_1$​ as $\sigma \to 0$. These techniques avoid computing the stationary density explicitly, using only qualitative regularity of the limiting stationary density coming from hypoellipticity. We also present how a similar approach yields shear-induced chaos for a stochastically driven limit cycle closely related to the Hopf normal form with additive noise, by proving $\lambda_1 > 0$. Finally, we briefly discuss additional SDEs where we believe variants of these ideas may yield positive lower bounds on $\lambda_1$. This work is part of ongoing joint work with Samuel Punshon-Smith.
 

For positive integers $r > \ell \geq 1$, an $\ell$-cycle in an $r$-uniform hypergraph is a cycle where each edge consists of $r$ vertices and each pair of consecutive edges intersect in $\ell$ vertices. For $\ell \geq 2$, we determine the exact threshold for the appearance of Hamilton $\ell$-cycles in an Erd\H{o}s--R\'enyi random hypergraph, confirming a conjecture of Narayanan and Schacht. The main difficulty is that the second moment is not tight for these structures. I’ll discuss how a variant of small subgraph conditioning and a subsampling procedure overcome this difficulty.