Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous mathematics to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials (See https://clinicaltrials.gov/ct2/results?cond=&term=VRC07&cntry=&state=&city=&dist= ), one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions (See Figure). We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. Ultimately, this method has parallels to other branches of applied mathematics that balance discrete and continuous representations, including physical geometric algorithms, robotics, computer vision, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.

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.

Mean field games (MFGs) study strategic decision-making in large populations where individual players interact via specific mean-field quantities. They have recently gained enormous popularity as powerful research tools with vast applications. For example, the Nash equilibrium of MFGs forms a pair of PDEs, which connects and extends variational optimal transport problems. This talk will present recent progress in this direction, focusing on computational MFG and engineering applications in robotics path planning, pandemics control, and Bayesian/AI sampling algorithms. This is based on joint work with the MURI team led by Stanley Osher (UCLA).

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.

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.

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.

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.

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.

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.

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.

The Explicit Jump Immersed Interface Method (EJIIM) is a finite difference method for elliptic partial differential equations that, like all Immersed Interface Methods, works on a regular grid in spite of non-grid aligned discontinuities in equation parameters and solution. The specific idea is to introduce jumps in function and its derivatives explicitely as additional variables. We present a finite difference based EJIIM for the stationary Stokes flow in saddle point formulation. Challenges related to staggered grid, fast Stokes solver and non-simply connected domains will be discussed.

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.

We begin with an introduction to time-dependent quantum mechanics and the role of Planck's constant. We then describe some mathematical results about solutions to the Schr\"odinger equation for small values of the Planck constant. Finally, we discuss two new numerical techniques for semiclassical quantum dynamics, including one that is a work in progress.

In computational physics, molecular dynamics refers to the computer simulation of a material at the atomic level. I will consider classical deterministic molecular dynamics in which large Hamiltonian systems of ordinary differential equations are used, though many of the same issues arise with other models. Given its scientific importance there is very little rigorous justification of molecular dynamics. From the viewpoint of numerical analysis it is surprising that it works at all. The problem is that individual trajectories computed by molecular dynamics are accurate for only small time intervals, whereas researchers trust the results over very long time intervals. It has been conjectured that molecular dynamics trajectories are accurate over long time intervals in some weak statistical sense. Another conjecture is that numerical trajectories satisfy the shadowing property: that they are close over long time intervals to exact trajectories with different initial conditions. I will explain how these two views are actually equivalent to each other, after we suitably modify the concept of shadowing.

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.

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.

(joint with Mboyo Esole) String theory considerations lead to a non-trivial identity relating the Euler characteristics of an elliptically fibered Calabi-Yau fourfold and of certain related surfaces. After giving a very sketchy idea of the physics arguments leading to this identity, I will present a Chern class identity which confirms it, generalizing it to arbitrary dimension and to varieties that are not necessarily Calabi-Yaus. The relevant loci are singular, and this plays a key role in the identity.

In 1953, the physicist P.E. Rouse proposed to model polymers in dilute solution by taking the polymer to be a series of beads connected by Gaussian springs. Neglecting inertia, the dynamics are set by a balance between the thermal fluctuations in the fluid and the elastic restoring force of the springs. One year later, B. Zimm noted that a polymer will interact with itself through the fluid in a qualitatively meaningful way. In this talk, we consider a more recent Langevin equation approach to dealing with hydrodynamic self-interaction. This involves coupling the continuum scaling limit of the Rouse model with stochastically forced time-dependent Stokes equations. The resulting pair of parabolic SPDE, with non-linear coupled forcing, presents a number of mathematical challenges. On the way to providing an existence and uniqueness result, we shall take time to develop relevant stochastic tools, and consider the modeling implications of certain technical results.

The complexity of solving the time-dependent wave equation via traditional methods scales faster than linearly in the complexity of the initial data. This behavior is mostly due to the necessity of timestepping at the CFL level, and is hampering the resolution of large-scale inverse scattering problems such as reflection seismology, where massive datasets need to be processed. In this talk I will report on some algorithmic progress toward time upscaling of the wave equation using discrete symbol calculus for pseudodifferential and Fourier integral operators. Joint work with Lexing Ying from UT Austin.

In this work we study the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like 2D Euler for times $t << Re^(1/3)$, and in particular exhibits "inviscid damping" (vorticity mixes and weakly approaches a shear flow). For times $t >> Re^(1/3)$, which is sooner than the natural dissipative time scale $O(Re)$, the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by a mixing-enhanced dissipation effect. Afterwards, the remaining shear flow decays on very long time scales $t >> Re$ back to the Couette flow. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to $Re^(-1)$) $L^2$ function. Joint with Nader Masmoudi and Vlad Vicol.

The Radial Basis Functions (RBF) method is not immune from the disastrous effects of the Gibbs phenomenon. When interpolating or solving PDEs whose solutions are piecewise smooth functions, the RBF method loses its notorious spectral accuracy. In this talk, a new method will be presented, based on the RBF method, which incorporates singularities using Heaviside functions and which keeps track of their location using the level set method. The resulting sharp interface method will be shown to recover the lost spectral accuracy and thus overcome the Gibbs phenomenon altogether.

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.

Historically, catalytic research and many areas of surface science have used phenomenological rate equations to build kinetic models for surface processes. The models treat surfaces as a lattice of sites, track the probability of finding a site in a particular state, and use maximal-entropy/well-mixed assumption to reconstruct spatially correlated information. This well-mixed assumption, however, often fails. This talk will develop a hierarchy of models that are able to take into account short range spatial correlations. The hierarchy is developed in the context of averaging an underlying master equation. The talk will continue with some simple examples, an example in catalysis, and conclude with ideas on several other applications for this framework.

The diffusion or adoption of new products (such as fax machines, skype, facebook, Ipad, etc.) is one of the key problems in Marketing research. In recent years, this problem was often studied numerically, using agent-based models (ABMs). In this talk I will focus on analysis of the aggregate diffusion dynamics in ABMs with a spatial structure. In one-dimensional ABMs, the aggregate diffusion dynamics can be explicitly calculated, without using the mean-field approximation. In multidimensional ABMs, we introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the fully-connected Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with "any" spatial structure. This is joint work with Ro'i Gibori and Eitan Muller

The detailed behavior of solutions to the biharmonic equation on regions with corners has been historically difficult to characterize. It is conjectured by Osher (and proven in certain special cases) that the Green�s function for the biharmonic equation on regions with corners has infinitely many oscillations in the vicinity of each corner. In this talk, we show that, when the biharmonic equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of elementary functions which oscillate with a frequency proportional to the logarithm of the distance from the corner. These representations are used to construct highly accurate and efficient Nyström discretizations, significantly reducing the number of degrees of freedom required for solving the corresponding integral equations. We illustrate the performance of our method with several numerical examples.

Epitaxial growth is a process in which a thin film is grown above a much thicker substrate. In the simplest case, no deposition is considered, and all the interactions are assumed to be purely elastic. However, since the film may potentially have different rigidity constant from the substate, such growth leads to a nonuniform film thickness. The equations governing epitaxial growth are high order (generally fourth order), nonlocal, and highly nonlinear. In this talk I will present some recent results about the regularity of solutions to several equations arising from epitaxial growth. Joint work with I. Fonseca and G.Leoni.