Georgia Institute of Technology College of Sciences

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.

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.

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)

We study the SIRS process on sparse random graphs with power--law degree distributions.
A large physics literature reports numerical evidence for a positive epidemic threshold for SIRS with waning immunity on scale--free networks, suggesting a transition between short--lived and exponentially long--lived regimes.
In contrast, for the SIS/contact process on power--law graphs with exponent $\tau>3$, it is rigorously known that the critical value is $\lambda_c=0$ and that survival is exponentially long for every $\lambda>0$.
We show that, in a survival--time sense, the true threshold for SIRS on power--law random graphs with $\tau>3$ is also zero. Joint work with Debankur Mukherjee and Souvik Dhara.