In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.

Perspectives from computational algebra and numerical optimization are brought to bear on a scientific application and a data science application. In the first part of the talk, I will discuss cryo-electron microscopy (cryo-EM), an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D projections, recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM presents a particularly rich inverse problem, with unknown orientations, extreme noise, big data and conformational heterogeneity. In particular, this motivates a general framework for statistical estimation under compact group actions, connecting information theory and group invariant theory. In the second part of the talk, I will discuss tensor rank decomposition, a higher-order variant of PCA broadly applicable in data science. A fast algorithm is introduced and analyzed, combining ideas of Sylvester and the power method.

Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes -- the asymptotically compatible schemes -- for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, fast algorithms will be presented to reduce the high computational cost from the numerical implementation of the nonlocal operators. Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here we will try to provide better perspectives of the origin of nonlocality from multiscale modeling and homogenization, which in turn may help the development of more effective numerical methods for homogenization.

We will discuss two issues in the context of applying deep learning methods to multi-scale molecular modelling: 1) how to construct symmetry-preserving neural network models for scalar and tensorial quantities; 2) how to efficiently explore the relevant configuration space and generate a minimal set of training data. We show that by properly addressing these two issues, one can systematically develop deep learning-based models for electronic properties and interatomic and coarse-grained potentials, which greatly boost the ability of ab-initio molecular dynamics; one can also develop enhanced sampling techniques that are capable of using tens or even hundreds of collective variables to drive phase transition and accelerate structure search

Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

An effort at UNC is involved in understanding key mechanisms in the lung related to defense against pathogens. In diseases ranging from Cystic Fibrosis to asthma, these mechanisms are highly compromised, requiring therapeutic strategies that one would like to be able to quantify or even predict in some way. The Virtual Lung Project has focused on one principal component of lung defense: "the mucus escalator" as it is called in physiology texts. My goal in this lecture, with apologies to Tina Turner, is to give a longwinded answer to "what's math got to do with it?", and at the same time to convey how this collaboration is influencing the applied mathematics experience at UNC.

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, they also come with increased difficulty in numerical analysis with nonlocality involved. In the first part of this talk, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results. In the second part of this talk, I will describe our recent effort in computing a nonlocal interface problem arising from segregation of two species with high competition. One species moves according to the classical diffusion and the other adopts a nonlocal strategy. A novel iterative scheme will be presented that constructs a sequence of supersolutions shown to be convergent to the viscosity solution of the interface problem.

Many partial differential equation models arising in applications generate complex patterns evolving with time which are hard to quantify due to the lack of any underlying regular structure. Such models often include some element of stochasticity which leads to variations in the detail structure of the patterns and forces one to concentrate on rougher common geometric features. From a mathematical point of view, algebraic topology suggests itself as a natural quantification tool. In this talk I will present some recent results for both the deterministic and the stochastic Cahn-Hilliard equation, both of which describe phase separation in alloys. In this situation one is interested in the geometry of time-varying sub-level sets of a function. I will present theoretical results on the pattern formation and dynamics, show how computational homology can be used to quantify the geometry of the patterns, and will assess the accuracy of the homology computations using probabilistic methods.

Blood is a concentrated suspension of red cells, white cells, and platelets, each having a unique constitution and serving a different function. Red cells are highly deformable liquid capsules enclosed by a thin incompressible membrane whose resting shape is a biconcave disk. White cells are viscoelastic spherical particles enclosed by a cortical shell. In the unactivated state, platelets are oblate spheroids with an average aspect ratio approximately equal to 0.25. Like red cells, platelets lack a nucleus; like white cells, platelets exhibit a low degree of flow-induced deformation. In this talk, the biomechanics and biofluid-dynamics of these three types of cells will be discussed, recent progress in modeling and simulation methods will be reviewed, and open problems will be outlined.

The Fermi-Pasta-Ulam (FPU) model of coupled anharmonic oscillators has long been of interest in nonlinear science. It is only recently (Friesecke and Wattis 1994, Frieseck and Pego 1999-2003, and Mizumachi (submitted)) that the existence and stability of solitary waves in FPU has been completely understood. In light of the fact that the Korteweg-deVries (KdV) equation may recovered as a long wave limit of FPU and that the theory of soliton interaction is both beautiful and completely understood in KdV, it is of interest to describe the interaction of two colliding solitary waves in the FPU model. We show that the FPU model contains an open set of solutions which remain close to the linear sum of two long wave low amplitude solitions as time goes to either positive or negative infinity.

Coupling between free fluid flow and flow through porous media is important in many industrial applications, such as filtration, underground water flow in hydrology, oil recovery in petroleum engineering, fluid flow through body tissues in biology, to name a few.

Stokes flows appear in many applications where the fluid viscosity is high and/or the velocity and length scales are small. The flow through a porous medium can be described by Darcy's law. A region that contains both requires a careful coupling of these different systems at the interface through appropriate boundary conditions.

Our objective is to develop a method based on the boundary integral formulation for computing the fluid/porous medium problem with higher accuracy using fundamental solutions of Stokes and Darcy's equations. We regularize the kernels to remove the singularity for stability of numerical calculations and eliminate the largest error for higher accuracy.

A new numerical method being developed with Charles Peskin is described which simulates interacting fluid, membrane, and particle systems in which thermal fluctuations play an important role. This method builds on the "Immersed Boundary Method" of Peskin and McQueen, which simplifies the coupling between the fluid and the immersed particles and membranes in such a way as to avoid complex boundary problems. Thermal fluctuations are introduced in the fluid through the theory of statistical hydrodynamics. We discuss some approximate analytical calculations which indicate that the immersed particles should exhibit some physically correct properties of Brownian motion. Our intended use of this numerical method is to simulate microphysiological processes; one advantage this method would have over Langevin particle dynamics approaches is its explicit tracking of the role of the fluid dynamics.

The simplest example of a boundary value problem is the Dirichlet Poisson problem: we seek to recover a function, defined on a smooth domain, its values at the boundary of the domain and the divergence of its gradient for all points inside the domain. This problem has been studied for more than 200 years, and has many applications in science and engineering. Analytic solutions are available only for a limited number of cases. Therefore one has to use a numerical method. The basic goals in designing a numerical method is guaranteed quality of the solution, in reasonable time, in a black-box fashion. Surprisingly, a robust, black-box, algorithmically scalable method for the Poisson problem does not exist. The main difficulties are related to robust mesh generation in complex geometries in three dimensions. I will review different approaches in solving the Poisson problem and present a new method based on classical Fredholm integral equation formulation. The main components of the new method are a kernel-independent fast summation method, manifold surface representations, and superalgebraically accurate quadrature methods. The method directly extends to problems with non-oscillatory known Green's functions. In addition to the Poisson problem I will present results for the Navier, modified Poisson, and Stokes operators.

The upper respiratory tract is the portal of entry for inhaled air and anything we breath in with it. For most of us, the nasal passages do most of the work cleansing, humidifying, and warming inhaled air using a lining of highly vascularized tissue coated with mucus. This tissue is susceptible to damage from inhaled material, can adversely affect life quality if deformed or diseased, and is a potential route of systemic exposure via circulating blood. To understand nasal physiology and the effects of inhalants on nasal tissue, information on airflow, gas uptake and particle deposition patterns is needed for both laboratory animals and humans. This information is often difficult to obtain in vivo but may be estimated with three-dimensional computational fluid dynamics (CFD) models. At CIIT Centers for Health Research (CIIT-CHR), CFD models of nasal airflow and inhaled gas and particle transport have been used to test hypotheses about mechanisms of toxicity, help extrapolate laboratory animal data to people, and make predictions for human health risk assessments, as well as study surgical interventions and nasal drug delivery. In this talk an overview of CIIT-CHR's nasal airflow modeling program will be given with the goal of illustrating how CFD modeling can help researchers clarify, organize, and understand the complex structure, function, physiology, pathobiology, and utility of the nasal airways.

Many problems arising from scientific and engineering applications can be naturally formulated as optimization problems, most of which are nonconvex. For nonconvex problems, obtaining a local minimizer is computationally hard in theory, never mind the global minimizer. In practice, however, simple numerical methods often work surprisingly well in finding high-quality solutions for specific problems at hand.

In this talk, I will describe our recent effort in bridging the mysterious theory-practice gap for nonconvex optimization. I will highlight a family of nonconvex problems that can be solved to global optimality using simple numerical methods, independent of initialization. This family has the characteristic global structure that (1) all local minimizers are global, and (2) all saddle points have directional negative curvatures. Problems lying in this family cover various applications across machine learning, signal processing, scientific imaging, and more. I will focus on two examples we worked out: learning sparsifying bases for massive data and recovery of complex signals from phaseless measurements. In both examples, the benign global structure allows us to derive geometric insights and computational results that are inaccessible from previous methods. In contrast, alternative approaches to solving nonconvex problems often entail either expensive convex relaxation (e.g., solving large-scale semidefinite programs) or delicate problem-specific initializations.

Completing and enriching this framework is an active research endeavor that is being undertaken by several research communities. At the end of the talk, I will discuss open problems to be tackled to move forward.

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.

I will discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and classification, can be posed as minimizing functionals on such graphs. We consider functionals involving graph cuts and graph laplacians and their limits as the number of data points goes to infinity. In particular we establish for what graph constructions the minimizers of discrete problems converge to a minimizer of a functional defined in the continuum setting. The talk is primarily based on joint work with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Moritz Gerlach, Matthias Hein, Thomas Laurent, James von Brecht and Matt Thorpe.

We study limiting behavior of rescaled size distributions in several models of clustering or coagulation dynamics, `solvable' in the sense that the Laplace transform converts them into nonlinear PDE. The scaling analysis that emerges has many connections with the classical limit theorems of probability theory, and a surprising application to the study of shock clustering in the inviscid Burgers equation with random-walk initial data. This is joint work with Govind Menon.

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.

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.

Topological insulators (TIs) are materials characterized by topological invariants. One of their remarkable features is the asymmetric transport observed at the interface between materials in different topological phases. Such transport is itself described by a topological invariant, and therefore ``protected" against random perturbations. This immunity makes TIs extremely promising for many engineering applications and actively researched.

In this talk, we present a PDE model for such TIs, introduce a topology based on indices of Fredholm operators, and analyze the influence of random perturbations. We confirm that topology is an obstruction to Anderson localization, a hallmark of wave propagation in strongly heterogeneous media in the topologically trivial case and to some extent quantify what is or is not protected topologically. For instance, a quantized amount of transmission is protected while back-scattering, a practical nuisance, is not.

In this talk, I will present a kernel-free boundary integral (KFBI) method for the variable coefficient elliptic partial differential equation on complex domains. The KFBI method is a generalization of the standard boundary integral method. But, unlike the standard boundary integral method, the KFBI method does not need to know an analytical expression for the kernel of the boundary integral operator or the Green's function associated with the elliptic PDE. So it is not limited to the constant-coefficient PDEs. The KFBI method solves the discrete integral equations by an iterative method, in which only part of the matrix vector multiplication involves the discretization of the boundary integral. With the KFBI method, the evaluation of the boundary integral is replaced by interpolation from a structured grid based solution to an equivalent interface problem, which is solved quickly by a Fourier transform or geometric multigrid based fast elliptic solver. Numerical examples for Dirichlet and Neumann BVPs, interface problems with different conductivity constants and the Poisson-Boltzmann equations will be presented.

We will discuss recent approximations to boundary value problems to non-linear systems of Boltzmann or Landau (for Coulombic interactions) equations coupled to the Poisson equation. The proposed approximation methods involve hybrid schemes of spectral and Galerkin type, were conservation of flow invariants are achieved by a constrain minimization problem. We will discuss some analytical and computational issues related to these approximations.

A very classical shape optimization problem consists in optimizing the topology and geometry of an elastic structure subjected to fixed boundary loads. One typically aims to minimize a weighted sum of material volume, structure perimeter, and structure compliance (a measure of the inverse structure stiffness). This task is not only of interest for optimal designs in engineering, but e.g. also helps to better understand biological structures. The high nonconvexity of the problem makes it impossible to find the globally optimal design; if in addition the weight of the perimeter is chosen small, very fine material structures are optimal that cannot even be resolved numerically. However, one can prove an energy scaling law that describes how the minimum of the objective functional scales with the model parameters. Part of such a proof involves the construction of a near-optimal design, which typically exhibits fine-scale structure in the form of branching and which gives an idea of how optimal geometries look like. (Joint with Robert Kohn)

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.

Hilbert space frames have traditionally been used in signal/image processing. Recently, there have arisen a variety of new applications to speeding up the internet, producing cell phones which won't fade, quantum information theory, distributed processing and more. We will review the fundamentals of frame theory and then look at the myriad of applications of frames.

We present an overview of experiments, numerical simulations, and mathematical analysis of the breakup of a low viscosity drop in a viscous fluid, and consider the role of surface contaminants, or surfactants, on the dynamics near breakup. As part of our study, we address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the important limit of very large values of bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a narrow transition layer near the drop surface or interface in which the surfactant concentration varies rapidly, and its gradient at the interface must be determined accurately to find the drop’s dynamics. Accurately resolving the layer is a challenge for traditional numerical methods. We present recent work that uses the narrowness of the layer to develop fast and accurate `hybrid’ numerical methods that incorporate a separate analytical reduction of the dynamics within the transition layer into a full numerical solution of the interfacial free boundary problem.

In seismic and SAR imaging, fitting cross-correlations of wavefields rather than the wavefields themselves can result in much improved robustness vis-a-vis model uncertainties. This approach however raises two challenges: (i) new spurious local minima may complicate the inversion, and (ii) one must find a good subset of cross-correlations to make the problem well-posed. I will explain how to address these two problems with lifting, semidefinite relaxation, and expander graphs. This mix of ideas has recently proved to be the right approach in other contexts as well, such as angular synchronization (Singer et al.) and phase retrieval (Candes et al.). Joint work with Vincent Jugnon.

We will discuss some recent results on singularity formation for finite-energy strong solutions to the 3D Euler system based on the analysis of scale-invariant solutions. The work consists of three parts: local well-posedness in critical spaces, the proof of blow-up for scale-invariant solutions, and then a cut-off argument to ensure finite energy. This is joint work with I. Jeong.

Single-cell RNA sequencing (scRNA-Seq) has emerged as a powerful tool to sample the complexity of large populations of cells and observe biological processes at unprecedented molecular resolution. This offers the exciting prospect of understanding the molecular programs that guide cellular differentiation during development. Here, we introduce Waddington-OT: a mathematical framework for understanding the temporal dynamics of development based on snapshots of expression profiles. The central challenge in analyzing these data arises from the fact that scRNA-Seq is destructive, which means that one cannot directly measure the trajectory of any given cell over time. We model the population of developing cells mathematically with a time-varying probability distribution (i.e. stochastic process) on a high-dimensional gene expression space, and we propose to recover the temporal coupling of the process with optimal transport. We demonstrate the power of Waddington-OT by applying the approach to study 315,000 scRNA-seq profiles collected at 40 time points over 16 days during reprogramming of fibroblasts to induced pluripotent stem cells. We construct a high-resolution map of reprogramming that rediscovers known features; uncovers new alternative cell fates including neural- and placental-like cells; predicts the origin and fate of any cell class; and implicates regulatory models in particular trajectories. Of these findings, we highlight Obox6, which we experimentally show enhances reprogramming efficiency. Our approach provides a general framework for investigating cellular differentiation.