We first discuss a general method to derive macroscopic laws from certain microscopic interactions that can be applied to a large class of particle systems. In particular we consider a broad class of systems that are linear in a special algebra, so-called tropical algebra. Some natural connections among the scaling limits of these random systems, the solutions to specific partial differential equations (Hamilton-Jacobi type) and combinatorial optimization problems have been noticed. If time allows, we further discuss a useful application of the variational formula (microscopic version of Hopf-Lax formula) to a well-known interacting particle system, totally asymmetric simple exclusion process.

Given two probability measures and a transportation cost, the optimal transport problem asks to find the most cost efficient way to transport one measure to the other. Since its introduction in 1781 by Gaspard Monge, the optimal transport problem has found applications in logistics, economics, physics, PDEs, and more recently data science. However, despite sustained attention from the numerics community, solving optimal transport problems has been a notoriously difficult task. In this talk I will introduce the back-and-forth method, a new algorithm to efficiently solve the optimal transportation problem for a general class of strictly convex transportation costs. Given two probability measures supported on a discrete grid with n points, the method computes the optimal map in O(n log(n)) operations using O(n) storage space. As a result, the method can compute highly accurate solutions to optimal transportation problems on spatial grids as large as 4096 x 4096 and 384 x 384 x 384 in a matter of minutes. If time permits, I will demonstrate an extension of the algorithm to the simulation of a class of gradient flows. This talk is joint work with Flavien Leger.

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.

The kernel-free boundary integral method is a Cartesian grid based method for solving elliptic partial differential equations (PDEs). It solves elliptic PDEs in the framework of boundary integral equations (BIEs). The method evaluates boundary and volume integrals by solving equivalent simple interface problems on Cartesian grids. It takes advantages of the well-conditioning properties of the BIE formulation, the convenience of grid generation with Cartesian grids and the availability of fast and efficient elliptic solvers for the simple interface problems. In this talk, I will present recent developments of the method for the reaction-diffusion equations in computational cardiology, the nonlinear Poisson-Boltzmann equation in biophysics, the Stokes equation in fluid dynamics as well as some free boundary and moving interface problems.

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. Technically, the heart of the argument are new laws of large numbers at the exponential scale, proved through an explicit combinatorics approach. Our result only requires weak regularity on the interaction kernel in negative Sobolev spaces, thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes. For dissipative gradient flows, we may allow any singularity lower than the Poisson kernel. This talk corresponds to a joint work with Z. Wang and an upcoming work with D. Bresch and Z. Wang.

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.

Some reaction-advection-diffusion equations admit traveling wave solutions; these are simple models of a combustion reaction spreading with constant speed. Even in a random medium, solutions to the initial value problem may develop fronts propagating with a well-defined asymptotic speed. First, I will describe this behavior when the nonlinearity is the Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearity and the randomness comes from a prescribed random drift (a simple model of turbulent combustion). Next, I will describe propagation of fronts when the nonlinearity is a random ignition-type nonlinearity. In the latter case, there exist special solutions that generalize the notion of a traveling wave in the random setting.

We have witnessed a lot of exciting development of data science in recent years. From the perspective of optimization, many modern data-science problems involve some basic ``non’’-properties that lack systematic treatment by the current approaches for the sake of the computation convenience. These non-properties include the coupling of the non-convexity, non-differentiability and non-determinism. In this talk, we present rigorous computational methods for solving two typical non-problems: the piecewise linear regression and the feed-forward deep neural network. The algorithmic framework is an integration of the first order non-convex majorization-minimization method and the second order non-smooth Newton methods. Numerical experiments demonstrate the effectiveness of our proposed approach. Contrary to existing methods for solving non-problems which provide at best very weak guarantees on the computed solutions obtained in practical implementation, our rigorous mathematical treatment aims to understand properties of these computed solutions with reference to both the empirical and the population risk minimizations. This is based on joint work with Jong-Shi Pang, Bodhisattva Sen and Ziyu He.

Fluid-structure interactions exist in many aspects of our daily lives. Some biomedical engineering examples are blood flowing through a blood vessel and blood pumping in the heart. Fluid interacting with moving or deformable structures poses more numerical challenges for its complexity in dealing with transient and simultaneous interactions between the fluid and solid domains. To obtain stable, effective, and accurate solutions is not trivial. Traditional methods that are available in commercial software often generate numerical instabilities.

In this talk, a novel numerical solution technique, Immersed Finite Element Method (IFEM), is introduced for solving complex fluid-structure interaction problems in various engineering fields. The fluid and solid domains are fully coupled, thus yield accurate and stable solutions. The variables in the two domains are interpolated via a delta function that enables the use of non-uniform grids in the fluid domain, which allows the use of arbitrary geometry shapes and boundary conditions. This method extends the capabilities and flexibilities in solving various biomedical, traditional mechanical, and aerospace engineering problems with detailed and realistic mechanics analysis. Verification problems will be shown to validate the accuracy and effectiveness of this numerical approach. Several biomechanical problems will be presented: 1) blood flow in the left atrium and left atrial appendage which is the main source of blood in patients with atrial fibrillation. The function of the appendage is determined through fluid-structure interaction analysis, 2) examine blood cell and cell interactions under different flow shear rates. The formation of the cell aggregates can be predicted when given a physiologic shear rate.

A wide range of viral infections, such as HIV or influenza, can now be treated using antiviral drugs. Since viruses can evolve rapidly, the emergence and spread of drug resistant virus strains is a major concern. We shall describe within and between host models that can help indicate settings in which resistance is more or less likely to be problematic. In particular, we shall discuss the potential for the emergence of resistance in the context of human rhinovirus infection, an acute infection that is responsible for a large fraction of 'common cold' cases.

In the event of a vascular injury, a blood clot will form to prevent bleeding. This response involves two intertwined processes: platelet aggregation and coagulation. Activated platelets are critical to coagulation in that they provide localized reactive surfaces on which many of the coagulation reactions occur. The final product from the coagulation cascade directly couples the coagulation system to platelet aggregation by acting as a strong activator of platelets and cleaving blood-borne fibrinogen into fibrin which then forms a mesh to help stabilize platelet aggregates. Together, the fibrin mesh and the platelet aggregates comprise a blood clot, which in some cases, can grow to occlusive diameters. Transport of coagulation proteins to and from the vicinity of the injury is controlled largely by the dynamics of the blood flow. It is crucial to learn how blood flow affects the growth of clots, and how the growing masses, in turn, feed back and affect the fluid motion. We have developed the first spatial-temporal model of platelet deposition and blood coagulation under flow that includes detailed descriptions of the coagulation biochemistry, chemical activation and deposition of blood platelets, as well as the two-way interaction between the fluid dynamics and the growing platelet mass.

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.

Whenever we wish to determine the characteristics of an object based on data of how it scatters incoming radiation we must solve an inverse scattering problem. This is a frequent situation in many fields, such as geophysical imaging or biomedical imaging. To reconstruct objects from the measured data, we can design optimization problems in which the boundary value problems governing the incident radiation act as constraints. Then we implement descent techniques to approach a global minimum. However, the process may stagnate without converging, either due to lack of convexity or to small gradients. We propose a method to overcome this difficulty combining topological derivative based optimization to generate first approximations with iteratively regularized Gauss-Newton techniques to ensure convergence. Numerical simulations illustrate fast reconstruction of objects formed by multiple non convex components in extreme situations such as holographic microscopy set-ups, in which the only data available are intensity measurements for one incident light beam on a limited screen.

In this talk, we will introduce a dynamic mathematical model that describes the interaction of the immune system with the human immunodeficiency virus (HIV). Using optimal control theory, we will illustrate that optimal dynamic multidrug therapies can produce a drug dosing strategy that exhibits structured treatment interruption, a regimen in which patients are cycled on and off therapy. In addition, sensitivity analysis of the model including both classical sensitivity functions and generalized sensitivity functions will be presented. Finally, we will describe how stochastic estimation can be used to filter and estimate states and parameters from noisy data. In the course of this analysis it will be shown that automatic differentiation can be a powerful tool for this type of study.

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.

In this work, we combine the adaptive mesh refinement (AMR) framework with high order finite difference weighted essentially non-oscillatory (WENO) method in space and TVD Runge-Kutta (RK) method in time (WENO-RK) for hyperbolic conservation laws. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. To maintain high order accuracy, we use high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from the coarse to find grid, and at ghost points. The resulting scheme is high order accuracy, robust and efficient, due to the mesh adaptivity and has high order accuracy in both space and time. We have experimented the third and fifth order AMR-finite difference WENO-RK schemes. The accuracy of the scheme is demonstrated by applying the method to several smooth test problems, and the quality and efficiency are demonstrated by applying the method to the shallow water and Euler equations with different challenging initial conditions. From our numerical experiment, we conclude a significant improvement of the fifth order AMR - WENO scheme over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, which we think is due to the very low numerical diffusion of high order schemes. This work is in collaboration with Dr. Chaopeng Shen and Professor Jing-Mei Qiu.

High performance transducers utilizing piezoceramic, electrostrictive, magnetostrictive or shape memory elements offer novel control capabilities in applications ranging from flow control to precision placement for nanoconstruction. To achieve the full potential of these materials, however, models, numerical methods and control designs which accommodate the constitutive nonlinearities and hysteresis inherent to the compounds must be employed. Furthermore, it is advantageous to consider material characterization, model development, numerical approximation, and control design in concert to fully exploit the novel sensor and actuator capabilities of these materials in coupled systems.

In this presentation, the speaker will discuss recent advances in the development of model-based control strategies for high performance smart material systems. The presentation will focus on the development of unified nonlinear hysteresis models, inverse compensators, reduced-order approximation techniques, and nonlinear control strategies for high precision or high drive regimes. The range for which linear models and control methods are applicable will also be outlined. Examples will be drawn from problems arising in structural acoustics, high speed milling, deformable mirror design, artificial muscle development, tendon design to minimize earthquake damage, and atomic force microscopy.

It is often desirable to derive an effective stochastic model for the physical process from observational and/or numerical data. Various techniques exist for performing estimation of drift and diffusion in stochastic differential equations from discrete datasets. In this talk we discuss the question of sub-sampling of the data when it is desirable to approximate statistical features of a smooth trajectory by a stochastic differential equation. In this case estimation of stochastic differential equations would yield incorrect results if the dataset is too dense in time. Therefore, the dataset has to sub-sampled (i.e. rarefied) to ensure estimators' consistency. Favorable sub-sampling regime is identified from the asymptotic consistency of the estimators. Nevertheless, we show that estimators are biased for any finite sub-sampling time-step and construct new bias-corrected estimators.

Linear boundary value problems occur ubiquitously in many areas of science and engineering, and the cost of computing approximate solutions to such equations is often what determines which problems can, and which cannot, be modelled computationally. Due to advances in the last few decades (multigrid, FFT, fast multipole methods, etc), we today have at our disposal numerical methods for most linear boundary value problems that are "fast" in the sense that their computational cost grows almost linearly with problem size. Most existing "fast" schemes are based on iterative techniques in which a sequence of incrementally more accurate solutions is constructed. In contrast, we propose the use of recently developed methods that are capable of directly inverting large systems of linear equations in almost linear time. Such "fast direct methods" have several advantages over existing iterative methods: (1) Dramatic speed-ups in applications involving the repeated solution of similar problems (e.g. optimal design, molecular dynamics). (2) The ability to solve inherently ill-conditioned problems (such as scattering problems) without the use of custom designed preconditioners. (3) The ability to construct spectral decompositions of differential and integral operators. (4) Improved robustness and stability. In the talk, we will also describe how randomized sampling can be used to rapidly and accurately construct low rank approximations to matrices. The cost of constructing a rank k approximation to an m x n matrix A for which an O(m+n) matrix-vector multiplication scheme is available is O((m+n)*k). This cost is the same as that of Lanczos, but the randomized scheme is significantly more robust. For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)), which should be compared to the O(m*n*k) cost of existing deterministic methods.

We present several versions of non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for (a) multiple scattering configurations, (b) bounded composite scatterers with piecewise constant material properties, and (c) layered media. We show that DDM solvers give rise to important computational savings over other existing solvers, especially in the challenging high-frequency regime.

I will present several results on the boundedness of maximal operators and Hilbert transforms along variable curves and surfaces, in dimension two or higher. Connections to the circular maximal operator, and the polynomial Carleson operator will also be discussed.

A chain of pendula connected by nearest neighbor coupling is a near-- integrable system if the coupling is weak. As a consequence of KAM, for most initial data the energy of each pendulum stays near its initial value for all time. We show that these KAM motions coexist with ``diffusing" motions for which the energy can leak from any pendulum to any other pendulum, and it can do so with a prescribed itinerary. This is joint work with Vadim Kaloshin.

In order to understand the nonlinear stability of many types of time-periodic traveling waves on unbounded domains, one must overcome two main difficulties: the presence of zero eigenvalues that are embedded in the continuous spectrum and the time-periodicity of the associated linear operator. I will outline these issues and show how they can be overcome in the context of time-periodic Lax shocks in systems of viscous conservation laws. The method involves the development of a contour integral representation of the linear evolution, similar to that of a strongly continuous semigroup, and detailed pointwise estimates on the resultant Greens function, which are sufficient for proving nonlinear stability under the necessary assumption of spectral stability.

The nonlinear Schrodinger equation

As time permits the talk will also discuss the energy - critical problem in R^d \ W,

Wasserstein Distance is a way of measuring the distance between two probability distributions (minimizing it is a main problem in Optimal Transport). We will give a gentle Introduction into what it means and then use it to prove (1) a completely elementary but possibly new and quite curious inequality for real-valued functions and (2) a statement along the following lines: linear combinations of eigenfunctions of elliptic operators corresponding to high frequencies oscillate a lot and vanish on a large set of co-dimension 1 (this is already interesting for trigonometric polynomials on the 2-torus, sums of finitely many sines and cosines, whose sum has to vanish on long lines) and (3) some statements in Basic Analytic Number Theory that drop out for free as a byproduct.

Nonsmooth and nonconvex loss functions are often used to model physical phenomena, provide robustness, and improve stability. While convergence guarantees in the smooth, convex settings are well-documented, algorithms for solving large-scale nonsmooth and nonconvex problems remain in their infancy.

I will begin by isolating a class of nonsmooth and nonconvex functions that can be used to model a variety of statistical and signal processing tasks. Standard statistical assumptions on such inverse problems often endow the optimization formulation with an appealing regularity condition: the objective grows sharply away from the solution set. We show that under such regularity, a variety of simple algorithms, subgradient and Gauss Newton like methods, converge rapidly when initialized within constant relative error of the optimal solution. We illustrate the theory and algorithms on the real phase retrieval problem, and survey a number of other applications, including blind deconvolution and covariance matrix estimation.

One of the main advantages of smooth optimization over its nonsmooth counterpart is the potential to use a line search for improved numerical performance. A long-standing open question is to design a line-search procedure in the stochastic setting. In the second part of the talk, I will present a practical line-search method for smooth stochastic optimization that has rigorous convergence guarantees and requires only knowable quantities for implementation. While traditional line-search methods rely on exact computations of the gradient and function values, our method assumes that these values are available up to some dynamically adjusted accuracy that holds with some sufficiently high, but fixed, probability. We show that the expected number of iterations to reach an approximate-stationary point matches the worst-case efficiency of typical first-order methods, while for convex and strongly convex objectives it achieves the rates of deterministic gradient descent.

The nascent technique of 4D printing has the potential to revolutionize manufacturing in fields ranging from organs-on-a-chip to architecture to soft robotics. By expanding the pallet of 3D printable materials to include the use stimuli responsive inks, 4D printing promises precise control over patterned shape transformations. With the goal of creating a new manufacturing technique, we have recently introduced a biomimetic printing platform that enables the direct control of local anisotropy into both the elastic moduli and the swelling response of the ink.

We have drawn inspiration from nastic plant movements to design a phytomimetic ink and printing process that enables patterned dynamic shape change upon exposure to water, and possibly other external stimuli. Our novel fiber-reinforced hydrogel ink enables local control over anisotropies not only in the elastic moduli, but more importantly in the swelling. Upon hydration, the hydrogel changes shape accord- ing the arbitrarily complex microstructure imparted during the printing process.

To use this process as a design tool, we must solve the inverse problem of prescribing the pattern of anisotropies required to generate a given curved target structure. We show how to do this by constructing a theory of anisotropic plates and shells that can respond to local metric changes induced by anisotropic swelling. A series of experiments corroborate our model by producing a range of target shapes inspired by the morphological diversity of flower petals.

It is a common expectation in chemistry that a chemical transformation which takes place in the presence of a catalyst must also take place in its absence, though perhaps at a much slower rate. A reaction network will be called ``saturated'' if it satisfies such an expectation. I propose a mathematical definition for saturated networks and show that the associated dynamical systems have no boundary equilibria in positive stoichiometric classes, and are therefore permanent. This result is independent of the specific rates, and generalizes previous results for complete networks by Gnacadja, atomic event-systems by Adleman et al. and constructive networks by Shinar et al. I require no assumption of complex balance or deficiency restrictions. The question of permanence for weakly-reversible reaction networks remains a long-standing open problem.

The use of PDE models to describe complex systems in the social sciences, such as socio-economic segregation and crime, has been popularized during the past decade. In this talk I will introduce some PDE models which can be seen as basic models for a variety of social phenomena. I will then discuss how these models can be used to explore and gain understanding of the real-world systems they describe. For example, we learn that a populations innate views toward criminal activity can play a significant role in the prevention of crime-wave propagation.

Phylogenetics is a science that aims at reconstructing the history of evolution. Phylogenetic tree models are generalizations of well-known Markov chains. In my talk I will present so-called group-based models and their relations to algebra and combinatorics. To a model of evolution one associates an algebraic variety that is the Zariski closure of points corresponding to probability distributions allowed by the model. Many important varieties arise by this construction, e.g. secant varieties of Segre products of projective spaces. It turns out that group-based models provide toric varieties. In particular, they may be studied using tools from toric geometry relating to combinatorics of lattice polytopes.