The speaker will talk about several projects that are taking place in an interdisciplinary endeavor between the researchers in the Mathematics Department at the University of Houston, the Texas Heart Institute, Baylor College of Medicine, the Mathematics Department at the University of Zagreb, and the Mathematics Department of the University of Lyon 1. The projects are related to non-surgical treatment of aortic abdominal aneurysm and coronary artery disease using endovascular prostheses called stents and stent-grafts. Through a collaboration between mathematicians, cardiovascular specialists and engineers we have developed a novel mathematical model to study blood flow in compliant (viscoelastic) arteries treated with stents and stent-grafts. The mathematical tools used in the derivation of the effective, reduced equations utilize asymptotic analysis and homogenization methods for porous media flows. The existence of a unique solution to the resulting fluid-structure interaction model is obtained by using novel techniques to study systems of mixed, hyperbolic-parabolic type. A numerical method, based on the finite element approach, was developed, and numerical solutions were compared with the experimental measurements. Experimental measurements based on ultrasound and Doppler methods were performed at the Cardiovascular Research Laboratory located at the Texas Heart Institute. Excellent agreement between the experiment and the numerical solution was obtained. This year marks a giant step forward in the development of medical devices and in the development of the partnership between mathematics and medicine: the FDA (the United States Food and Drug Administration) is getting ready to, for the first time, require mathematical modeling and numerical simulations to be used in the development of peripheral vascular devices. The speaker acknowledges research support from the NSF, NIH, and Texas Higher Education Board, and donations from Medtronic Inc. and Kent Elastomer Inc.

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.

One of the most fundamental questions in partial differential equations is that of regularity and the possible breakdown of solutions. We will discuss this question for solutions to a canonical example of a geometric wave equation; energy critical wave maps. Break-through works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz and Rapha Ì?el-Rodnianski produced examples of wave maps that develop singularities in finite time. These solutions break down by concentrating energy at a point in space (via bubbling a harmonic map) but have a regular limit, away from the singular point, as time approaches the final time of existence. The regular limit is referred to as the radiation. This mechanism of breakdown occurs in many other PDE including energy critical wave equations, Schro Ì?dinger maps and Yang-Mills equations. A basic question is the following: â?¢ Can we give a precise description of all bubbling singularities for wave maps with the goal of finding the natural unique continuation of such solutions past the singularity? In this talk, we will discuss recent work (joint with J. Jendrej and A. Lawrie) which is the first to directly and explicitly connect the radiative component to the bubbling dynamics by constructing and classifying bubbling solutions with a simple form of prescribed radiation. Our results serve as an important first step in formulating and proving the following Radiative Uniqueness Conjecture for a large class of wave maps: every bubbling solution is uniquely characterized by itâ??s radiation, and thus, every bubbling solution can be uniquely continued past blow-up time while conserving energy.

Obtaining predictive dynamical equations from data lies at the heart of science and engineering modeling, and is the linchpin of our technology. In mathematical modeling one typically progresses from observations of the world (and some serious thinking!) first to selection of variables, then to equations for a model, and finally to the analysis of the model to make predictions. Good mathematical models give good predictions (and inaccurate ones do not) --- but the computational tools for analyzing them are the same: algorithms that are typically operating on closed form equations.
While the skeleton of the process remains the same, today we witness the development of mathematical techniques that operate directly on observations --- data, and appear to circumvent the serious thinking that goes into selecting variables and parameters and deriving accurate equations. The process then may appear to the user a little like making predictions by "looking into a crystal ball". Yet the "serious thinking" is still there and uses the same --- and some new --- mathematics: it goes into building algorithms that "jump directly" from data to the analysis of the model (which is now not available in closed form) so as to make predictions. Our work here presents a couple of efforts that illustrate this "new" path from data to predictions. It really is the same old path, but it is traveled by new means.

Boundary value and interface problems for the Laplace equation are often solved by boundary integral methods due to the reduction of dimensionality and its flexibility in domain geometry. However, there are two well-known computational issues with the boundary integral method: (a) evaluation of boundary integrals at points close to domain boundaries usually has low order accuracy; (b) the method typically yields dense coefficient matrices in the resulting discrete systems, which makes the matrix vector multiplication very expensive when the size of the system is very large. In this talk, I will describe a fast accurate boundary integral method for the Laplace boundary value and interface problems. The algorithm uses the high order accurate method proposed by (Beale and Lai 2001) for evaluation of the boundary integrals and applies the fast multipole method for the dense matrix vector multiplication. Numerical results demonstrating the efficiency and accuracy of the method will be presented.

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.

An efficient method for simulating the propagation of a localized solution of the Schroedinger equation near the semiclassical limit is presented. The method is based on a time dependent transformation closely related to Gaussian wave packets and yields a Schroedinger type equation that is very ammenable to numerical solution in the semi-classical limit. The wavefunction can be reconstructed from the transformed wavefunction whereas expectation values can easily be evaluated directly from the transformed wavefunction. The number of grid points needed per degree of freedom is small enough that computations in dimensions of up to 4 or 5 are feasible without the use of any basis thinning procedures. This is joint work with Giovanni Russo.

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.

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.

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.

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.

The Boltzmann equation governs the dynamics of a dilute gas. In this talk, I will address the L^p-stability problem of the Boltzmann equation near vacuum and a global Maxwellian. In a close-to-vacuum regime, I will explain the nonlinear functional approach motivated by Glimm's theory in hyperbolic conservation laws. This functional approach yields the uniform L^1-stability estimate. In contrast, in a close-to-global maxwellian regime, I will present the L^2-stability theory which establishes the uniform L^2-stability of several classical solutions.

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.

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.

Hirschsprung's Disease is a relatively common human congenital defect where the nervous system supporting our gut (the enteric nervous system) fails to develop properly. During embryonic development, the enteric nervous system forms as a result of neural crest cell invasion. Neural crest cells migrate from the hindbrain to the anal end of the gastrointestinal tract. This is one of the longest known cell migration paths, both spatially and temporally, occurring during vertebrate embryogenesis. Neural crest cell invasion is complicated by the simultaneous expansion of underlying tissues and the influence of several growth factors. This presentation outlines a combined experimental and mathematical approach used to investigate and deduce the mechanisms responsible for successful neural crest cell colonization. This approach enables previously hypothesized mechanisms for neural crest cell colonization of the gut tissues to be refuted and refined. The current experimental and mathematical results are focused on population-scale approaches. Further experimental details of cell-scale properties thought to play an important role will be presented. Preliminary discrete modelling results aiming to realize the cell-scale phenomena will also be discussed and outlined as future work.

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.

In vertebrates, ovulation is triggered by a surge of luteinizing hormone (LH) from the pituitary. The precise mechanism by which rising estradiol (E2) from the ovaries initiates the LH surge in the human menstrual cycle remains a mystery. The mystery is due in part to the bimodal nature of estradiol feedback action on LH secretion, and in part to disagreement over the site of the feedback action.

We will describe a differential equations model in which the mysterious bimodality of estradiol action arises from the electrical connectivity of a network of folliculo-stellate cells in the pituitary. The mathematical model is based as closely as possible on current experimental data, and is being used to design and conduct new experiments. No biological background will be assumed.

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.

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales char- acterized by the magnitude of a dimensionless parameterÂ?the Knudsen number (Kn). In the diffusive scaling (Kn ≪ 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well- posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.

The method of geometric optics for incompressible Euler equations is used to study localized shortwave instabilities in ideal fluids. In linear approximation evolution of the shortwave ansatz can be described by a finite dimensional skew-product flow which determines all the linear instabilities coming from essential spectrum. In this talk we will discuss mathematical description of the method, aspects related to vanishing viscosity limit and application to the problem of inherent instability of ideal fluid flows.

The first reaction-diffusion equation developed and studied is the Fisher-KPP equation. Introduced in 1937, this population-dynamics model accounts for the spatial spreading and growth of a species. Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al. I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions. Afterwards I will describe the new model for the cane toads equations and give new results regarding this model. In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads. The talk is based on a joint work with Bouin and Ryzhik.

The emergence of massive data sets, over the past twenty or so years, has lead to the development of Randomized Numerical Linear Algebra. Fast and accurate randomized matrix algorithms are being designed for applications like machine learning, population genomics, astronomy, nuclear engineering, and optimal experimental design. We give a flavour of randomized algorithms for the solution of least squares/regression problems. Along the way we illustrate important concepts from numerical analysis (conditioning and pre-conditioning), probability (concentration inequalities), and statistics (sampling and leverage scores).

In this talk, we study blow-up mechanism of solutions to an integrable equation with cubic nonlinearities and nonlinear dispersion. We will show that singularities of the solutions can occur only in the form of wave-breaking. Some wave-breaking conditons on the initial data are provided. In addition, this equation is known to admit single and multi-peaked solitons, of a different character than those of the Camassa-Holm equation. We will prove that the shapes of these waves are stable under small perturbations in the energy space.

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.

In this talk we discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations. They may arise from optimal control and differential game problems, and are generally difficult to solve numerically in high dimensions.
A major contribution of our works is to consider an optimization problem over a single vector of the same dimension as the dimension of the HJ PDE instead. To do so, we consider the new approach using Hopf-type formulas. The sub-problems are now independent and they can be implemented in an embarrassingly parallel fashion. That is ideal for perfect scaling in parallel computing.
The algorithm is proposed to overcome the curse of dimensionality when solving high dimensional HJ PDE. Our method is expected to have application in control theory, differential game problems, and elsewhere. This approach can be extended to the computational of a Hamilton-Jacobi equation in the Wasserstein space, and is expected to have applications in mean field control problems, optimal transport and mean field games.

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.

The Patlak-Keller-Segel equations (PKS) are widely applied to model the chemotaxis phenomena in biology. It is well-known that if the total mass of the initial cell density is large enough, the PKS equations exhibit finite time blow-up. In this talk, I present some recent results on applying additional fluid flows to suppress chemotactic blow-up in the PKS equations. These are joint works with Jacob Bedrossian and Eitan Tadmor.

The periodic generalized Korteweg-de Vries equation (gKdV) is a canonical dispersive partial differential equation with numerous applications in physics and engineering. In this talk we present the invariance of the Gibbs measure under the flow of the gauge transformed periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized. As a corollary we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed.

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.