The classical Hardy-Littlewood maximal operators (averaging over families of Euclidean balls and cubes) are known to satisfy L^p bounds that are independent of dimension. This talk will extend these results to spherical maximal functions acting on Cartesian products of cyclic groups equipped with the Hamming metric.

The study of various questions about networks have increased dramatically in recent years across a number of areas of application, including communications, sociology, and phylogenetic biology. Important questions about communities and groupings in networks have led to a number of competing techniques for identifying communities, structures and hierarchies. We discuss results about the networks of (1) NCAA Division I-A college football matchups and (2) committee assignments in the U.S. House of Representatives. In college football, the underlying structure of the network strongly influences the computer rankings that contribute to the Bowl Championship Series standings. In Congress, the changes of the hierarchical structure from one Congress to the next can be used to investigate major political events, such as the "Republican Revolution" of 1994 and the introduction of the Select Committee on Homeland Security following September 11th. While many structural elements in each case are seemingly robust, we include attention to variations across identification algorithms as we investigate the roles of such structures.

An efficient method for simulating the propagation of a localized solution of the Schroedinger equation near the semiclassical limit is presented. The method is based on a time dependent transformation closely related to Gaussian wave packets and yields a Schroedinger type equation that is very ammenable to numerical solution in the semi-classical limit. The wavefunction can be reconstructed from the transformed wavefunction whereas expectation values can easily be evaluated directly from the transformed wavefunction. The number of grid points needed per degree of freedom is small enough that computations in dimensions of up to 4 or 5 are feasible without the use of any basis thinning procedures. This is joint work with Giovanni Russo.

The Boltzmann equation governs the dynamics of a dilute gas. In this talk, I will address the L^p-stability problem of the Boltzmann equation near vacuum and a global Maxwellian. In a close-to-vacuum regime, I will explain the nonlinear functional approach motivated by Glimm's theory in hyperbolic conservation laws. This functional approach yields the uniform L^1-stability estimate. In contrast, in a close-to-global maxwellian regime, I will present the L^2-stability theory which establishes the uniform L^2-stability of several classical solutions.

There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been some work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does previous partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a classical argument due to Karatsuba that shows VMVT "asymptotically" and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.

We consider the basis pursuit problem: find the solution of an underdetermined system $Ax=y$ that minimizes the $l_1$-norm. We formulate a min-max principle (that, as we learned, actually goes back to 1970's) based on a Largange multiplier, and propose an iterative shrinkage-thresholding type algorithm that seems to work quite well. We show that the numerical algorithm converges to the exact solution of the basis pursuit problem. We also discuss its application to array imaging in wave propagation. The analysis is based on ODE techniques, regularization and energy methods. This is a joint work with M. Moscoso, A. Novikov and G. Papanicolaou.

Recently there has been a large class of research that utilizes nonlinear mappings into low dimensional spaces in order to organize potentially high dimensional data. Examples include, but are not limited to, locally linear embedding (LLE), ISOMAP, Hessian LLE, Laplacian eigenmaps, and diffusion maps. In this talk we will focus on the latter, and in particular consider how to generalize diffusion maps to the setting in which we are given a data set that evolves over time or changes depending on some set of parameters. Along with describing the current theory, various synthetic and real world examples will be presented to illustrate these ideas in practice.

Developing algorithms for solving high-dimensional partial differential equations (PDEs) and forward-backward stochastic differential equations (FBSDEs) has been an exceedingly difficult task for a long time, due to the notorious difficulty known as the curse of dimensionality. In this talk we introduce a new type of algorithms, called "deep BSDE method", to solve general high-dimensional parabolic PDEs and FBSDEs. Starting from the BSDE formulation, we approximate the unknown Z-component by neural networks and design a least-squares objective function for parameter optimization. Numerical results of a variety of examples demonstrate that the proposed algorithm is quite effective in high-dimensions, in terms of both accuracy and speed. We furthermore provide a theoretical error analysis to illustrate the validity and property of the designed objective function.

I will present work from these two papers: 1) Regulating Greed Over Time. Stefano Traca and Cynthia Rudin. 2015 Finalist for 2015 IBM Service Science Best Student Paper Award 2) Prediction Uncertainty and Optimal Experimental Design for Learning Dynamical Systems. Chaos, 2016. Benjamin Letham, Portia A. Letham, Cynthia Rudin, and Edward Browne.
There is an important aspect of practical recommender systems that we noticed while competing in the ICML Exploration-Exploitation 3 data mining competition. The goal of the competition was to build a better recommender system for Yahoo!'s Front Page, which provides personalized new article recommendations. The main strategy we used was to carefully control the balance between exploiting good articles and exploring new ones in the multi-armed bandit setting. This strategy was based on our observation that there were clear trends over time in the click-through-rates of the articles. At certain times, we should explore new articles more often, and at certain times, we should reduce exploration and just show the best articles available. This led to dramatic performance improvements.
As it turns out, the observation we made in the Yahoo! data is in fact pervasive in settings where recommender systems are currently used. This observation is simply that certain times are more important than others for correct recommendations to be made. This affects the way exploration and exploitation (greed) should change in our algorithms over time. We thus formalize a setting where regulating greed over time can be provably beneficial. This is captured through regret bounds and leads to principled algorithms. The end result is a framework for bandit-style recommender systems in which certain times are more important than others for making a correct decision.
If time permits I will discuss work on measuring uncertainty in parameter estimation for dynamical systems. I will present "prediction deviation," a new metric of uncertainty that determines the extent to which observed data have constrained the model's predictions. This is accomplished by solving an optimization problem that searches for a pair of models that each provide a good fit for the observed data, yet have maximally different predictions. We develop a method for estimating a priori the impact that additional experiments would have on the prediction deviation, allowing the experimenter to design a set of experiments that would most reduce uncertainty.

We consider a droplet spreading on a thin liquid film where both fluids are Newtonian, incompressible, and immiscible. Rather than following the typical asymptotic derivation for a thin film, we formulate the model through a gradient flow approach. The sign of the spreading parameter indicates the spreading behavior (complete or partial spreading) and is a relation between the three interfacial tensions: fluid/air, fluid/drop, and drop/air. We are particularly interested in the case where the spreading parameter is negative. In this case, the drop is expected to spread to a static lens and we find the corresponding equilibrium solution. Finally, we make a comparison between the theoretical model and experimental results.

Transportation networks are ubiquitous as they are possibly the most important building blocks of nature. They cover microscopic and macroscopic length scales and evolve on fast to slow times scales. Examples are networks of blood vessels in mammals, genetic regulatory networks and signaling pathways in biological cells, neural networks in mammalian brains, venation networks in plant leafs and fracture networks in rocks. We present and analyze a PDE (Continuum) framework to model transportation networks in nature, consisting of a reaction-diffusion gradient-flow system for the network conductivity constrained by an elliptic equation for the transported commodity (fluid).

The optimal transportation problem defines a notion of distance in the space of probability measures over a manifold, the *Wasserstein space*. In his 1994 Ph.D. thesis, McCann discovered that this space is a length space: the distance between probability measures is given by the length of minimizing geodesics called *displacement interpolants*. A surprising number of important functionals in physics and geometry turned out to be geodesically convex. In contrast with classical function spaces, the Wasserstein space is not a linear space, but rather an infinite-dimensional analogue of a Riemannian manifold. This analogy has motivated new functional inequalities and new methods for studying evolution equations; however, it has rarely been used in rigorous proofs. I will describe recent work with Benjamin Schachter on differentiating functionals (such as the entropy or the Dirichlet integral) along displacement interpolants. Starting from an Eulerian formulation for the underlying optimal transportation problem, we take advantage of the system of transport equations to compute derivatives of arbitrary order, for probability densities that need not be smooth.

Reaction-diffusion equations are widely used to model spatial propagation, and constant-speed "traveling waves" play a central role in their dynamics. These waves are well understood in "essentially 1D" domains like cylinders, but much less is known about waves with noncompact transverse structure. In this direction, we will consider traveling waves of the Fisher–KPP reaction-diffusion equation in the Dirichlet half-space. We will see that minimal-speed waves are unique (unlike faster waves) and exhibit curious asymptotics. The arguments rest on the theory of conformal maps and a powerful connection with the probabilistic system known as branching Brownian motion.

This is joint work with Julien Berestycki, Yujin H. Kim, and Bastien Mallein.

Continuum models of the flow of granular materials in a hopper admit so-called radial solutions. These describe steady flows that appear realistic, and have been used extensively to design commercial hoppers. However, numerical results demonstrate that these solutions may not be robust to perturbation. Moreover, the time dependent equations are (notoriously) ill-posed. In this talk, I describe preliminary research designed to investigate the extent to which steady solutions may be used to represent granular flow. Using a combination of analysis and numerical experiments, we have explored simple models that are linearly ill posed. While there may be a stable steady state, it is a solution of a discretized continuum model, rather than the original equations. Moreover, the survival time of transients is inversely related to the mesh width, suggesting that the continuum limit is meaningless. While these results are not intended to invalidate the radial solutions, they do raise serious concerns about continuum modeling, and the possibility of designing a robust code that can be used to simulate a variety of granular flows.