Applied Math Colloquium

Abstract: In the 1600s, Christiaan Hyugens realized that two pendulum clocks (an invention of his!) placed in the same wooden table eventually fall into synchrony. Since then, synchronization of coupled oscillators has been an important subject of study in classical mechanics and nonlinear dynamics. The Kuramoto model, proposed in the 1970s, has become a prototypical model used for rigorous mathematical analysis in this field. A realization of this model consists of a collection of identical oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph.

In this talk we discuss which graphs are globally synchronizing, meaning that all but a measure-zero set of initial conditions converge into the fully synchronized state. We show that large expansion of the underlying graph is a sufficient condition (but far from necessary) and solve a conjecture of Ling, Xu and Bandeira stating that Erdos-Renyi random graphs are globally synchronizing above their connectivity threshold.

Time permitting, we will discuss connections with studying the non-convex landscape of the Burer-Monteiro algorithm for Community Detection in the Stochastic Block Model. Joint work with Pedro Abdalla (ETHZ), Martin Kassabov (Cornell), Victor Souza (Cambridge), Steven H. Strogatz (Cornell), Alex Townsend (Cornell).

Abstract: Flows in the fluidic interior of living cells can serve function, and by their structure shed light on how forces are exerted within the cell. Some of these flows can arise through novel collective instabilities ofthe cytoskeleton, that set of polymers, cross-linkers, and molecular motors that underlie much of the mechanics within and between cells. I'll discuss experiments, mathematical modeling and analysis, and simulations of two such cases. One is understanding the emergence of cell-spanning vortical flows in developing egg cells, while the other arises from studying the nature of force transduction in the dynamics of microtubule arrays inside of synthetic cells. Both show the importance of polymer density in determining dynamics and time-scales, and have required the development of new coarse-grained models and simulation methods.

Bio: Dr. Michael J. Shelley is an applied mathematician who works on the modeling and simulation of complex systems arising in physics and biology. This has included, of late, modeling the dynamics of complex and active fluids, and examining transport and self-organization processes in cellular biophysics. He is co-founder and co-director of the Courant Institute's Applied Mathematics Lab [1] at New York University, and is the Director of the Center for Computational Biology [2] at the Flatiron Institute. He is a Fellow of the American Physical Society, the Society for Industrial and Applied Mathematics, and the American Academy of Arts and Sciences, and is a member of the National Academy of Sciences.

Abstract: I will discuss our solution to a longstanding discrete geometry problem where we determined the maximum number of lines in high dimension pairwise separated by a fixed angle. A key step is in showing that a connected bounded degree graph has sublinear second eigenvalue multiplicity.

These results prompted further investigations into spherical codes and spectral graph theory. I will highlight some ongoing work and open problems.

Abstract: In principle, a random matrix is just a matrix whose entries are random variables. The goal of random matrix theory is to understand the eigenvalues and eigenvectors of such matrices. However, it seems at first sight rather hopeless that we could say anything meaningful at this level of generality. In practice, classical random matrix theory has focused on a small collection of very special models where explicit computations and limit theorems are possible, while matrix concentration inequalities that apply to more general models only give access to very weak information.

Very recently, however, a new approach to this area has breathed new life into the initial question: can we compute the spectrum of a random matrix with a general pattern of entry means and variances, dependencies, and distributions? I will aim to describe the basic ingredients of a theory that is able able to answer this question in a surprisingly general setting. This is made possible by an unexpected phenomenon: in many cases, the spectrum of an arbitrarily structured random matrix is accurately modelled by that of an associated deterministic operator, whose spectral statistics can be explicitly computed or estimated using free probability.

Based on joint works with A. Bandeira, M. Boedihardjo, T. Brailovskaya, G. Cipolloni, and D. Schroeder.

Abstract: In this talk, we will discuss numerical approach to solve high dimensional Hamilton-Jacobi-Bellman (HJB) type partial differential equations (PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled by the actor-critic framework inspired by reinforcement learning, based on neural network parametrization of the value and control functions. Within the actor-critic framework, we employ a policy gradient approach to improve the control, while for the value function, we derive a variance reduced least-squares temporal difference method using stochastic calculus. We will also discuss convergence analysis for the actor-critic method, in particular the policy gradient method for solving stochastic optimal control. Joint work with Jiequn Han (Flatiron Institute) and Mo Zhou (UCLA).

Abstract: With the introduction of the AAA algorithm in 2018 (Nakatsukasa-Sete-T., SISC), the computation of rational approximations changed from a hard problem to an easy one. We've been exploring the implications of this transformation ever since. This talk will review the algorithm and then present about 15 demonstrations of applications in various areas including interpolation of missing data, analytic continuation, analysis of solutions of dynamical systems, Wiener-Hopf and Riemann-Hilbert problems, function extension, model order reduction, Dirichlet-to-Neumann maps, and Laplace, Stokes, and Helmholtz calculations.