We will present an overview of the program of statistical description of long time, large scale dynamics of nonlinear waves and highlight some results we obtained: from vanishing mutual information measures in the spatiotemporal chaos induced by hyperbolic structures of PDEs, to confirmation of weak-turbulence spectra, and role of coherent structures in controlling energy transfer in turbulent cycles described by multiple cascade spectra, to effective stochastic dynamics. We will address the issue of how to obtain invariant measures for these systems. Finally, we will report on our study of statistical properties of the focusing nonlinear Schrödinger equation, in the limit of a large number of solitons, corresponding to the semi-classical limit in a periodic domain. Our results demonstrate that the dynamics is described solitonic turbulence and there is a power law for the energy spectrum in the regime. We will discuss the connection between the wave turbulence and solitonic turbulence.

In joint work, Suncica Canic, Eun Heui Kim and I have recently proved the existence of a local solution to the regular reflection problem in the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. There are two kinds of regular reflection, weak and strong, which are distinguished by whether the state immediately behind the reflected shock is subsonic (strong) or supersonic and constant, becoming subsonic further downstream (weak). In the more complicated case of weak regular reflection, the equation, in self-similar coordinates, is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation.

We combine techniques which have been developed for solving degenerate elliptic equations arising in self-similar reductions of hyperbolic conservation laws with an approach to solving free boundary problems of the type that arise from Rankine-Hugoniot relations. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection.

Motion by mean curvature for a network of surfaces arises in many applications. An important example is the evolution of microstructure in a polycrystalline material under heat treatment. Most metals and ceramics are of this type: They consist of many small single-crystal pieces of differing orientation, called grains, that are stuck together. A famous model proposed by Mullins in the 60s describes the dynamics of the network of surfaces that separate neighboring grains from one another in such a material as gradient descent for a weighted sum of the (possibly anisotropic) areas of the surfaces. The resulting dynamics is motion by weighted mean curvature for the surfaces in the network, together with certain conditions that need to be satisfied at junctions along which three or more surfaces may intersect. Typically, many topological changes occur during the evolution, as grains shrink and disappear, pinch off, or junctions collide. A very elegant algorithm -- known as threshold dynamics -- for the motion by mean curvature of a surface was given by Merriman, Bence, and Osher: It generates the whole evolution simply by alternating two very simple operations: convolution with a Gaussian kernel, and thresholding. It also works for networks, provided that all surfaces in the network have isotropic surface energies with equal weights. Its correct extension to the more general setting of unequal weights and possibly anisotropic (normal dependent) surface energies remained elusive, despite keen interest in this setting from materials scientists. In joint work with Felix Otto, we give a variational formulation of the original threshold dynamics algorithm by identifying a Lyapunov functional for it. In turn, the variational formulation shows how to extend the algorithm correctly to the more general settings that are of interest for materials scientists (joint work with Felix Otto and Matt Elsey). Examples of how to use the new algorithms to investigate unsettled questions about grain size distribution and its evolution will also be given.

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 composite media with a broad range of scales, whose effective properties are important in materials science, biophysics, and climate modeling. Examples include random resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The analytic continuation method provides Stieltjes integral representations for the bulk transport coefficients of such systems, involving spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices or discretizations of continuum systems, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. In this lecture we will discuss various implications and applications of these integral representations. We will also discuss computations of the spectral measures of the operators, as well as statistical measures of their eigenvalues. For example, the effective behavior of composite materials often exhibits large changes associated with transitions in the connectedness or percolation properties of a particular phase. We demonstrate that an onset of connectedness gives rise to striking transitional behavior in the short and long range correlations in the eigenvalues of the associated random matrix. This, in turn, gives rise to transitional behavior in the spectral measures, leading to observed critical behavior in the effective transport properties of the media.

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.

Analyzing and inferring the underlying global intrinsic structures of data from its local information are critical in many fields. In practice, coherent structures of data allow us to model data as low dimensional manifolds, represented as point clouds, in a possible high dimensional space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data sets are far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and understanding even more challenging. To overcome these ambiguities, I will first discuss modeling based methods. This approach uses geometric PDEs to adapt the intrinsic manifolds structure of data and extracts various invariant descriptors to characterize and understand data through solutions of differential equations on manifolds. Inspired by recent developments of deep learning, I will also discuss our recent work of a new way of defining convolution on manifolds and demonstrate its potential to conduct geometric deep learning on manifolds. This geometric way of defining convolution provides a natural combination of modeling and learning on manifolds. It enables further applications of comparing, classifying and understanding manifold-structured data by combing with recent advances in deep learning.

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.

Sampling and reconstruction of functions is a central tool in science. A key result is given by the classical sampling theorem for bandlimited functions. We describe the recently developed sampling theory for operators. We call operators bandlimited if their Kohn-Nirenberg symbols are band limited. The addresses engineers and mathematicians and should be accessible for those who have some education in linear algebra and calculus. The talk reviews sampling of functions and introduces some terminology from the theory of pseudodifferential operators. We will also discuss sampling theorems for stochastic operators.

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.

The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional Euler equation remains open. In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of the solution has been double exponential in time. I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp. This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions of 3D Euler equation. The scenario is axi-symmetric. The geometry of the scenario is related to the geometry of 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.

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.

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

Arnold diffusion is the phenomenon of loss of stability of a completely integrable Hamiltonian system: an arbitrarily small perturbation can cause action to change along some orbit by a finite amount. Arnold produced the first example of diffusion and gave an outline of the proof. After a brief overview of related results I will describe the simplest example of Arnold diffusion which we found recently with Vadim Kaloshin. We consider geodesics on the 3-torus, or equivalently rays in a periodic optical medium in $ {\mathbb R} ^3 $ (or equivalently a point mass in a periodic potential in $ {\mathbb R} ^3 $.) Arnold diffusion has a transparent intuitive explanation and a simple proof. Resonances and the so-called ``whiskered tori" acquire a clear geometrical interpretation as well. I will conclude with a sketch of a different but related manifestation of Arnold diffusion as acceleration of a particle by a pulsating potential. This is joint work with Vadim Kaloshin.

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 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.

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

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 will consider some inverse coefficient problems to system of linear and semilinear diffusion equations where the aim is to recover unknown parameters in the equations from partial information on the solutions to the systems. We present some recent theoretical and numerical results, and point out possible applications of such problems in imaging.

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.

We present several high accuracy numerical methods for fluid flow problems and turbulence modeling.

First we consider a stabilized finite element method for the Navier-Stokes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step.

We proceed by introducing a family of defect correction methods for the time dependent Navier-Stokes equations, aiming at higher Reynolds' number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought.

Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost.

We then turn to the turbulence modeling in coupled Navier-Stokes systems - namely, MagnetoHydroDynamics. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, we show that the model preserves the properties of the 3D MHD equations.

Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests.

In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds.

We discuss the geometry of Laplacian eigenfunctions on compact manifolds and combinatorial graphs. The `dual' geometry of Laplacian eigenfunctions is well understood on the torus and euclidean space, and is of tremendous importance in various fields of pure and applied mathematics. The purpose of this talk is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of `similarity' between eigenfunctions is given by a global average of local correlations, and its relationship to pointwise products. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result. This talk will also focus on the applications of discovering such a dual geometry, namely in constructing anisotropic graph wavelet packets and anisotropic graph cuts.