Departmental Colloquium

Three is harder than two, and not just when dealing with human beings and celestial bodies. The complexity increases dramatically when we pass from line segments (two-gons) to triangles, from circles (bivalent graphs) to trivalent graphs and from quadratic forms to cubic forms on a complex vector space. While I want to focus on the last of these, pointing out the relationships between cubic forms and lattices, Gorenstein rings and "K3-categories", there is also rich mathematics in trivalent graphs and polygons that I might lead with in a standard colloquium talk. Instead I want to "Benjamin Button" this colloquium, starting with the more advanced mathematics first and working backward to an amusing ending/beginning.

Data assimilation is concerned with estimating the state of a dynamical system from partial observations. In applications such as numerical weather prediction where the state is high dimensional and the dynamics are expensive to simulate, ensemble Kalman filters are often the method of choice. In this talk, I will present new results on structured covariance operator estimation that help explain why these algorithms can be effective even when deployed with a small ensemble size. Our theory also explains the importance of using covariance localization in ensemble Kalman methods for global data assimilation.

Accelerated training algorithms, such as adaptive learning rates and various normalization methods, are widely used in deep learning but not fully understood. When regularization is introduced, standard optimizers like adaptive learning rates may not perform effectively. This raises the need for alternative regularization approaches and the question of how to properly combine regularization with preconditioning. In this talk, we present preconditioning as a unified mathematical framework for understanding various acceleration techniques and deriving appropriate regularization schemes. We will explain how preconditioning with AdaGrad, RMSProp, and Adam accelerates training; discuss the interaction between regularization and preconditioning, and demonstrate how normalization methods accelerate training and how this perspective can lead to new preconditioning training algorithms.

In this talk, we discuss some recent development of Lagrangian and game theoretic (i.e. stochastic and two-player control generalizations of the method of characteristics) approaches for multi-scale and multi-dimensional reaction-diffusion-advection equations. Through two case studies, we show how stochastic interacting particle methods (IPM) work out as a mesh-free and self-adaptive computational tool. The first case, dated back to Kolmogorov 1937, is concerned with entropy production of reverse-time diffusion processes, and the resulting principal eigenvalue problem of a non-self-adjoint advection-diffusion operator. At a linear complexity rate, the IPM, derived from the Feynman-Kac formula with a genetic interpretation, computes the eigenfuction as a concentrated invariant measure of particle population evolution up to dimension 16. In the second case study of a haptotaxis advection-diffusion system modeling cancer cell spreading, an IPM with a field coupling captures cell merging and expanding dynamics in 3 space dimensions. The third study aims to address a fundamental problem in turbulent combustion by analyzing a curvature dependent level set Hamilton-Jacobi equation (a.k.a. curvature G-equation), and proving the existence of effective front speeds in a cellular flow. To overcome non-coercivity and non-convexity of the Hamiltonian, we combine a one-sided reachability estimate based on the Kohn-Serfaty deterministic two player game characterization, the streamline structure of the flow and a minimum value principle.

In May 2024, a team of seven mathematics students, led by Jody Reimer and Ken Golden, embarked on a 10-day journey to Utqiaġvik, Alaska. This expedition, a cornerstone of the Applied Math NSF RTG program, offered these young researchers a unique opportunity to witness firsthand the dynamic and complex frozen world of sea ice and the ecosystem it supports. This presentation will highlight the mathematical and scientific questions that motivate the research of Golden, Reimer, and their students. We will also share stories and visuals from our trip, and hear from the students about the impact this experience had on them.

I will give a historically motivated account of the connection between modular forms and Galois theory. Ramanujan's 1916 paper "On some arithmetical functions" led to a series of developments that led to the proof of Fermat's Last Theorem: one can draw a line from Ramanujan's paper, to the formulation of Serre's conjecture in the 1970's and 1980's, its connection to Fermat's Last Theorem, and Wiles's proof of Fermat in 1994. I will also indicate recent developments in the subject linking modular forms and Galois representations, for instance the proof of modularity of elliptic curves over Q(i) by Ana Cariani and James Newton, which relies on particular cases of an analog of Serre's conjecture over Q(i) proved in joint work with Patrick Allen and Jack Thorne.