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.

The problem of finding transition paths in the Lennard-Jones cluster of 38 atoms became a benchmark problem in chemical physics due to its beauty and complexity. The two deepest potential minima, the face-centered cubic truncated octahedron and an icosahedral structure with 5-fold rotational symmetry, are far away from each other in the configuration space, which makes problem of finding transition paths between them difficult. D. Wales's group created a network of minima and transition states associated with this cluster. I will present two approaches to analyze this network. The first one, a zero-temperature asymptotic approach, is based on the Large Deviation Theory and Freidlin's cycles. I will show that in the gradient case the construction of the hierarchy of cycles can be simplified dramatically and present a computational algorithm for building a hierarchy of only those Freidlin's cycles associated with the transition process between two given local equilibria. The second approach is the Discrete Transition Path Theory, a finite temperature tool. This approach allows us to establish the range of validity of the zero-temperature asymptotic and describe the transition process at still low but high enough temperatures where the zero-temperature asymptotic approach is no longer valid.

Frames are nowadays a standard methodology in applied mathematics, computer science, and engineering when redundant, yet stable expansions are required. Sparsity is a new paradigm in signal processing, which allows for significantly reduced measurements yet still highly accurate reconstruction. In this talk, we will focus on the main two links between these exciting, rapidly growing areas. Firstly, the redundancy of a frame promotes sparse expansions of signals, thereby strongly supporting sparse recovery methods such as Compressed Sensing. After providing an overview of sparsity methodologies, we will discuss new results on sparse recovery for structured signals, in particular, which are a composition of `distinct' components. Secondly, in very high dimensions, frame decompositions might be intractable in applications with limited computing budget. This problem can be addressed by requiring sparsity of the frame itself, and we will show how to derive optimally sparse frames. Finally, we will discuss how some of the presented results generalize to the novel notion of fusion frames, which was introduced a few years ago for modeling distributed processing applications.

We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1, ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a real-valued function. The eigenvalue distribution of these kernel random matrices is studied in the "large p, large n" regime. It is shown that with suitable normalization the spectral density converges weakly, and we identify the limit. Our analysis applies as long as the rescaled kernel function is generic, and particularly, this includes non-smooth functions, e.g. Heaviside step function. The limiting densities "interpolate" between the Marcenko-Pastur density and the semi-circle density.

We will discuss several results concerning quantitative rectifiability in metric spaces, which generalize Euclidean results. We will spend some time explaining both the metric space results as well as their Euclidean counterparts. An example of such a result is a structure theorem, which characterizes subsets of rectifiable curves (the Analyst's Traveling Salesman theorem). This theory is presented in terms of multi-scale analysis and multi-scale constructions, and uses a language which is analogous to that of wavelets. Some of the results we will present will be dimension free.

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.

Phase retrieval problems appear in many imaging applications in which only the magnitude of (e.g., Fourier) transform coefficients of a given signal can be measured. In such settings one desires to learn the original signal (up to a global phase factor) using only such magnitude information. In this talk we discuss methods to rapidly re-learn such lost phase information by using the magnitudes of well-designed combinations of the original transform coefficients. In particular, we develop a fast phase retrieval method which is near-linear time, making it computationally feasible for large dimensional signals. Both theoretical and experimental results demonstrate the method's speed, accuracy, and robustness. We then use this new phase retrieval method to help establish the first known sublinear-time compressive phase retrieval algorithm capable of recovering a given $s$-sparse signal ${\bf x} \in \mathbbm{C}^d$ (up to an unknown phase factor) in just $\mathcal{O}(s \log^5 s \cdot \log d)$-time using only $\mathcal{O}(s \log^4 s \cdot \log d)$ magnitude measurements. This is joint work with Aditya Viswanathan and Yang Wang.

The motion of primary nodal cilia present in embryonic development resembles that of a precessing rod. Implementing regularized singularities to model this fluid flow numerically simulates a situation for which colleagues have exact mathematical solutions and experimentalists have corresponding laboratory studies on both the micro- and macro-scales. Stokeslets are fundamental solutions to the Stokes equations, which act as external point forces when placed in a fluid. By strategically distributing regularized Stokeslets in a fluid domain to mimic an immersed boundary (e.g., cilium), one can compute the velocity and trajectory of the fluid at any point of interest. The simulation can be adapted to a variety of situations including passive tracers, rigid bodies and numerous rod structures in a fluid flow generated by a rod, either rotating around its center or its tip, near a plane. The exact solution allows for careful error analysis and the experimental studies provide new applications for the numerical model. Spectral deferred correction methods are used to alleviate time stepping restrictions in trajectory calculations. Quantitative and qualitative comparisons to theory and experiment have shown that a numerical simulation of this nature can generate insight into fluid systems that are too complicated to fully understand via experiment or exact numerical solution independently.

Compressed sensing, as a signal acquisition technique, has been shown to be highly effective for dimensionality reduction. On the other hand, reconstruction from compressed sensing measurements is highly non-linear and requires digital computers. Thus, quantizing (i.e., digitizing) compressed sensing measurements is an important, albeit under-addressed topic. In this talk, we show that by using $\Sigma\Delta$ quantization instead of the most commonly assumed approach (uniform quantization), a significant reduction in the reconstruction error is possible. In particular, we prove error decay rates of $\lambda^{-c r}$ where $\lambda$ is the ratio of the number of measurements to the sparsity of the underlying signal, and $r$ is the order of the $\Sigma\Delta$ scheme. In addition to the compressed sensing scenario we also consider the quantization of frame expansions, where one collects more measurements than the ambient dimension. We show state of the art results for certain frames (including random frames) and $\Sigma\Delta$ schemes. In particular, we prove error rates of $e^{-c\sqrt{\lambda}}$, where $\lambda$ is the oversampling ratio.

I will present several extremely fast algorithms for recovering a compressible signal from a few linear measurements. These examples span a variety of orthonormal bases, including one large redundant dictionary. As part of the presentation of these algorithms, I will give an explanation of the crucial role of group testing in each algorithm.

The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic microswimmers.

I review some works on modeling and the mathematical theory for dilute suspensions of rigid rods. Such problems appear in modeling sedimentation of suspensions of particles. Similar in spirit models are also used for modeling swimming micro-organisms. Here, we focus on a class of models introduced by Doi and describing suspensions of rod¨Clike molecules in a solvent fluid. They couple a microscopic Fokker-Planck type equation for the probability distribution of rod orientations to a macroscopic Stokes flow. One objective is to compare such models with traditional models used in macoscopic viscoelasticity as the well known Oldroyd model. In particular: For the problem of sedimenting rods under the influence of gravity we discuss the instability of the quiescent flow and the derivation of effective equations describing the collective response. We derive two such effective theories: (i) One ammounts to a classical diffusive limit and produces a Keller-Segel type of model. (ii) A second approach involves the derivation of a moment closure theory and the approximation of moments via a quasi-dynamic approximation. This produces a model that belongs to the class of flux-limited Keller-Segel systems. The two theories are compared numerically with the kinetic equation. (joint work with Christiane Helzel, Univ. Duesseldorf).

In this talk I will discuss the Keller-Segel equation, which is a nonlocal PDE modeling the collective motion of cells attracted by a self-emitted chemical substance. When this equation is set up in 2D with a degenerate diffusion term, it is known that solutions exist globally in time, but their long-time behavior remain unclear. To answer this question, we investigate a general aggregation equation with degenerate diffusion, and prove that all stationary solutions must be radially symmetric up to a translation. As a consequence, this enables us to obtain a convergence result for solutions to 2D Keller-Segel equation with degenerate diffusion as the time goes to infinity. This is a joint work with J. Carrillo, S. Hittmeir and B. Volzone.

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.

PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.
PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.
PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.

When iterative methods are used to solve a discretized linear system for partial differential equations, the key issue is how to make the convergence fast. For different type of problems convergence mechanism can be quite different. In this talk, I will present an efficient iterative method, the fast sweeping method, for a class of nonlinear hyperbolic partial differential equation, Hamilton-Jacobi equation, which is widely used in optimal control, geometric optics, geophysics, classical mechanics, image processing, etc. We show that the fast sweeping method can converge in a finite number of iterations when monotone upwind scheme,  Gauss-Seidel iterations with causality enforcement and proper orderings are used. We  analyze its convergence, which is very different from that for iterative method for elliptic problems. If time permit I will present a new formulation to compute effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations. Both error estimate and stability analysis will be shown.

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.

In this talk we consider the existence of suitable weak solutions for a quantum energy-transport model for semiconductors. The model is formally derived from the quantum hydrodynamic model by J\"{u}ngel and Mili\v{s}i\'{c} (Nonlinear Anal.: Real World Appl., 12(2011), pp. 1033-1046). It consists of a fourth-order nonlinear parabolic equation for the electron density, an elliptic equation for the electron temperature, and the Poisson equation for the electric potential. Our solution is global in the time variable, while the space variables lie in a bounded Lipschitz domain with a mixed boundary condition. The existence proof is based upon a carefully-constructed approximation scheme which generates a sequence of positive approximate solutions. These solutions are shown to be so regular that they can be used to form a variety of test functions , from which we can derive enough a prior estimates to justify passing to the limit in the approximate problems.

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.

The purpose of this presentation is to communicate some mathematical and computational issues in sediment transport for open channels. The main topics are: (1) mathematical simulation for surface and subsurface fluid flow, (2) mathematical modeling of sediment transport in open channels as a 2D problem, and (3) numerical methods for fluid flow and sediment transport.

The problem of mixing via incompressible flows is classical and rich with connections to several branches of analysis including PDE, ergodic theory, and topological dynamics. In this talk I will discuss some recent developments in the area and then present a construction of universal mixers - incompressible flows that asymptotically mix arbitrarily well general solutions to the corresponding transport equation - in all dimensions. This mixing is in fact exponential in time (i.e., essentially optimal) for any initial condition with at least some degree of regularity, while there exists no uniform mixing rate for all measurable initial conditions.

Porous, deformable membranes are encountered in a wide range of applications including red blood cells, vesicles, porous wave makers, and parachutes. The "immersed boundary method" has already proven to be a versatile and robust approach for simulating the interaction of impermeable, elastic structures with an incompressible fluid flow. We demonstrate how to extend the method to handle porous boundaries by incorporating an explicit porous slip velocity that is determined by Darcy's law. We derive a simple, radially-symmetric exact solution, which is then used to validate numerical simulations of porous membranes in two dimensions.

Enhacement of diffusion and reaction processes by advection is a classical subject that has been extensively studied by both physicists and mathematicians. In this talk I will consider reaction-diffusion on a bounded domain in the presence of a fast incompressible flow. I will also consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. For such systems enhacement of diffusion implies, in particular, that the explosion threshold for power-like nonlinearities tends to infinity in the large Rayleigh number limit.

A dielectric boundary in a biomolecular system is a solute-solvent (e.g., protein-water) interface that defines the dielectric coefficient to be one value in the solute region and another in solvent. The inhomogeneous dielectric medium gives rise to the dielectric boundary force that is crucial to the biomolecular conformation and dynamics. This talk begins with a review of the Poisson-Boltzmann theory commonly used for electrostatic interactions in biomolecular interactions and then focuses on the mathematical description of dielectric boundary force. A precise definition and explicit formula of such force are presented. The motion of a cylindrical dielectric boundary driven by the competition between the dielectric boundary force and the reduction of surface energy is then studied. Implications of the mathematical findings to biomolecular conformational stabilities are discussed.

Contemporary and ancient demographic structure in human populations has shaped the genomic variation observed in modern humans, and severely affected the distribution of functional and disease related genetic variation. Using next-generation sequencing technologies, researchers gather increasing amounts of genomic sequencing data for large samples in many different human population groups. These datasets present unprecedented opportunities to study genomic variation in complex demographic scenarios, and this area has received a lot of attention in recent years. In this talk, I will present a method for the inference of demographic histories from full-genome sequencing data of multiple individuals developed by me and my collaborators. I will apply this method to a genomic dataset of Native American individuals to unravel the ancient demographic events underlying the peopling of the Americas. Moreover, I will discuss a novel method for demographic inference that has the potential to improve inference especially in the recent past, which is of particular importance in the context of complex genetic diseases in humans.