Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. We present recent results on properties of these equilibria and long-time asymptotics of solutions in the setting where attractive and repulsive forces are in competition. In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.

Blood vessel systems and leaf venations are typical biological transport networks. The energy consumption for such a system to perform its biological functions is determined by the network structure. In the first part of this talk, I will discuss the optimized structure of vessel networks, and show how the blood vessel system adapts itself to an optimized structure. Mathematical models are used to predict pruning vessels in the experiments of zebra fish. In the second part, I will discuss our recent modeling work on the initiation process of transport networks. Simulation results are used to illustrate how a tree-like structure is obtained from a continuum adaptation equation system, and how loops can exist in our model. Possible further application of this model will also be discussed.

Fluid velocities and Brownian effects at nanoscales in the near-wall r egion of microchannels can be experimentally measured in an image plane parallel to the wall, using for example, an evanescent wave illumination technique combi ned with particle image velocimetry [R. Sadr et al., J. Fluid Mech. 506, 357-367 (2004)]. Tracers particles are not only carried by the flow, but they undergo r andom fluctuations, the details of which are affected by the proximity of the wa ll. We study such a system under a particle based stochastic approach (Langevin) . We present the modeling assumptions and pay attention to the details of the si mulation of a coupled system of stochastic differential equations through a Mils tein scheme of strong order of convergence 1. Then we demonstrate that a maximum likelihood algorithm can reconstruct the out-of-plane velocity profile, as spec ified velocities at multiple points, given known mobility dependence and perfect mean measurements. We compare this new method with existing cross-correlation t echniques and illustrate its application for noisy data. Physical parameters are chosen to be as close as possible to the experimental parameters.

It is well-known that the asymptotic complexity of matrix-matrix product and matrix inversion is given by the rank of a 3-tensor, recently shown to be at most O(n^2.3728639) by Le Gall. This approach is attractive as a rank decomposition of that 3-tensor gives an explicit algorithm that is guaranteed to be fastest possible and its tensor nuclear norm quantifies the optimal numerical stability. There is also an alternative approach due to Cohn and Umans that relies on embedding matrices into group algebras. We will see that the tensor decomposition and group algebra approaches, when combined, allow one to systematically discover fast(est) algorithms. We will determine the exact (as opposed to asymptotic) tensor ranks, and correspondingly the fastest algorithms, for products of Circulant, Toeplitz, Hankel, and other structured matrices. This is joint work with Ke Ye (Chicago).

In many applications, ranging from computer animation to biology, one wants to quantify how similar two surfaces are to each other. In the last few years, the Gromov-Haussdorff distance has been applied to this problem; this gives good results, but turns out to be very heavy computationally. This talk proposes a different approach, in which (disk-like) 2-dimensional surfaces (typically embedded in 3-dimensional Euclidean space) are first mapped conformally to the unit disk, and the corresponding conformal densities are then compared via optimal mass transportation,. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global Moebius transformations. The metric we construct also defines meaningful intrinsic distances between pairs of "patches" in the two surfaces, which allows automatic alignment of the surfaces. Numerical experiments on "real-life" surfaces to demonstrate possible applications in natural sciences will be shown as well. This is joint work with Yaron Lipman.

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.

Wavespeed selection refers to the problem of determining the long time asymptotic speed of invasion of an unstable homogeneous state by some other secondary state. This talk will review wavespeed selection mechanisms in the context of reaction-diffusion equations. Particular emphasis will be placed on the qualitative differences between wavespeed selection in systems of reaction-diffusion equations and scalar problems as well as some surprising consequences. The primary example will be a system of coupled Fisher-KPP equations that exhibit anomalous spreading wherein the coupling of two equations leads to faster spreading speeds.

Sparsity plays a central role in recent developments of many fields such as signal and image processing, compressed sensing, statistics, and optimization. In practice, sparsity is promoted through the additional of an L1 norm (or related quantity) as a constraint or penalty in a variational model. Motivated by the success of sparsity-inducing methods in imaging and information sciences, there is a growing interest in exploiting sparsity in dynamical systems and partial differential equations. In this talk, we will investigate the connections between compressed sensing, sparse optimization, and numerical methods for nonlinear differential equations. In particular, we will discuss about sparse modeling as well as the advantage of sparse optimization in solving various differential equations arising from physical and data sciences.

Taylor-Couette flow provides one of the pre-eminent examples of bifurcation in fluid dynamics. This phrase refers to the flow between concentric rotating cylinders. If the rotation speed is sufficiently rapid, the primary rotary flow around the axis becomes unstable, leading to a steady secondary flow in approximately periodic cells. Assuming infinite cylinders and exact periodicity in his theory, Taylor obtained remarkable agreement with experiment for the onset of instability, agreement that remains unsurpassed in fluid mechanics to this day. This talk is concerned with incorporating the effect of finite-length cylinders into the theory, an issue whose importance was emphasized by Benjamin. Numerous experiments and simulations of the Navier Stokes equations all support to the following, seemingly paradoxical, observations: No matter how long the apparatus, finite-length effects greatly perturb many of the bifurcating flows but, provided the cylinders are long, hardly perturb others. We understand this paradox as a result of symmetry breaking. The relevant symmetry, which is only approximate, is a symmetry between two normal-mode flows with large, and nearly equal, numbers of cells.

Mathematical models of complex physical processes often involve large number of degrees of freedom as well as events occurring on different time scales. Therefore, direct simulations based on these models face tremendous challenge. This focus of this talk is on the Mori-Zwanzig (MZ) projection formalism for reducing the dimension of a complex dynamical system. The goal is to mathematically derive a reduced model with much fewer variables, while still able to capture the essential properties of the system. In many cases, this formalism also eliminates fast modes and makes it possible to explore events over longer time scales. The models that are directly derived from the MZ projection are typically too abstract to be practically implemented. We will first discuss cases where the model can be simplified to generalized Langevin equations (GLE). Furthermore, we introduce systematic numerical approximations to the GLE, in which the fluctuation-dissipation theorem (FDT) is automatically satisfied. More importantly, these approximations lead to a hierarchy of reduced models with increasing accuracy, which would also be useful for an adaptive model refinement (AMR). Examples, including the NLS, atomistic models of materials defects, and molecular models of proteins, will be presented to illustrate the potential applications of the methods.

Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics have demonstrated novel supersonic, fluid-like phenomena. Shock waves in these and other systems are modeled by a dispersive regularization of Euler's equations, implemented by use of the Whitham averaging technique. Normal and oblique dispersive shock waves (DSWs) connecting two constant states are constructed. Numerical computations of supersonic, dispersive flow over a corner in the special case of systems modeled by the Nonlinear Schrodinger equation (NLS) exhibit stable pattern formation (oblique DSWs) or instability (turbulent-like behavior) depending on the flow parameters. A combination of analytical and computational approaches are used to demonstrate that this change in behavior can be identified with the transition from convective to absolute instability of dark solitons. The linearized NLS behavior about the dark soliton DSW trailing edge is studied in detail to identify the separatrix between convective and absolute instabilities.

The talk is concerned with a matrix-valued Boltzmann equation derived from the Fermi-Hubbard or Bose-Hubbard model for weak interactions. The quantum analogue of the classical distribution function is the Wigner function, which is matrix-valued to accommodate the spin degree of freedom. Conservation laws and the H-theorem can be proven analytically, and numerical simulations illustrate the time dynamics.

Elliptic boundary value problems are well-understood in the case when the boundary, the data, and the coefficients exhibit smoothness. However, it has been long recognized in physics and engineering that irregularities (non-smooth boundary, abrupt change of media, noise or disorder) can decisively influence the properties of the solutions and give rise to completely new phenomena. 

The analysis of general non-smooth elliptic PDEs gives rise to decisively new challenges: possible failure of maximal principle and positivity, breakdown of boundary regularity, lack of the classical L^2 estimates, to mention just a few. Further progress builds on an involved blend of harmonic analysis, potential theory and geometric measure theory techniques. In this talk we are going to discuss some highlights of the history, conjectures, paradoxes, and recent discoveries such as the higher-order Wiener criterion and maximum principle for higher order PDEs, solvability of rough elliptic boundary problems, harmonic measure, as well as an intriguing phenomenon of localization of eigenfunctions -- within and beyond the limits of the famous Anderson localization.

Many common data sets can be modeled by a mixture of manifolds, e.g., facial images of several human subjects under different angles and illumination conditions. However, effective modeling of such data along with careful theoretical analysis is a challenging mathematical problem. In this talk I will introduce the Spectral Curvature Clustering (SCC) algorithm for solving the problem of hybrid linear modeling, i.e., modeling data using linear manifolds, and discuss possible extensions to general multi-manifold data modeling. Our analysis shows that, given data sampled sufficiently around a collection of well separated affine subspaces, the SCC will succeed with high probability. Numerical techniques as well as an application to motion segmentation are also presented.

We apply piecewise constructions and gluing technics to construct asymptotically flat (hyperbolic) manifolds such that associated Laplacians have dense embedded eigenvalues or singular continuous spectra. The method also allows us to provide various examples of operators with embedded singular spectra, including perturbed periodic operators, periodic Jacobi operators, and Stark operators. We establish the asymptotic behavior (WKB for example) of eigensolutions under small perturbations, which implies certain rules for the absence of singular spectra. As a result, several sharp spectral transitions (even criteria) for a single (finitely many or countably many) embedded eigenvalues, singular continuous spectra and essential supports of spectral measures are obtained. The talk is based on several papers, some joint with Jitomirskaya and Ong.

Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and understanding even more challenging. To overcome these ambiguities, I will first discuss modeling based methods. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data and extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds and demonstrate its potential to conduct geometric deep learning on manifolds. This geometric way of defining convolution provides a natural combination of modeling and learning on manifolds. It enables further applications of comparing, classifying and understanding manifold-structured data by combing with recent advances in deep learning.

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.

Many experimental systems can spend extended periods of time relative to their natural time scale in localized regions of phase space, transiting infrequently between them. This display of metastability can arise in stochastically driven systems due to the presence of large energy barriers, or in deterministic systems due to the presence of narrow passages in phase space. To investigate metastability in these different cases, we take the Langevin equation and determine the effects of small damping, small noise, and dimensionality on the dynamics and mean transition time. In finite dimensions, we show the limit of small noise and small damping do not interchange. In the limit of infinite dimensions, we argue the equivalence of the finitely-damped system and the zero-damped infinite energy Hamiltonian system.

Marine snow, composed of organic and inorganic aggregates, plays a major role in marine carbon cycling. Most of these macroscopic particles are extremely porous, allowing diffusion of salt from the ambient fluid to affect the density and therefore the settling of these particles. In a first approximation, these particles can be modeled as spheres. This talk will present a study of the effect of porosity and salt diffusion in the dynamics of a sphere settling under gravity in a salt-stratfied fluid analytically and semi-analytically (depending on the ambient density gradient) in viscosity dominated regimes. For linear stratification, an explicit solution for the sphere's position in time is derived. For more general ambient fluid stratification, the sphere's position can be solved for numerically, under the asymptotic assumptions about the typical time scales of diffusion and settling. A parametric study of the settling behaviors and preliminary comparisons with experiments will be presented.

I’ll discuss a geometric approach to graph partitioning where the optimality criterion is given by the sum of the first Laplace-Dirichlet eigenvalues of the partition components. This eigenvalue optimization problem can be solved by a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective. This partitioning method compares well to state-of-the-art approaches on a variety of graphs constructed from manifold discretizations, synthetic data, the MNIST handwritten digit dataset, and images. I'll present a consistency result for geometric graphs, stating convergence of graph partitions to an appropriate continuum partition.

Regularity of solutions in the optimal transport problem requires very rigid hypotheses (e.g., convexity of certain sets). When such conditions are not available, one can consider the question of partial regularity, in other words, the in-depth analysis of the structure of singular sets. In this talk, I will discuss the regularity of the set of ``free singularities`` which arise in an optimal transport problem with inner product cost, from a connected set to a disconnected set, along with the stability of such sets under suitable perturbations of the data involved. Some of these results are proven via a non-smooth implicit function theorem for convex functions, which is of independent interest. This talk is based on joint work with Robert McCann.

In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex. This is based on an upcoming joint work with Jonathan Luk at Stanford.