The theme of this talk is that polynomial approximations on the sphere are important for applications, but that successful applications of high-degree polynomials need a good understanding of underlying approximation properties. We illustrate with two case studies.

First, for applications in geodesy, there is good reason to use cubature rules that have a high degree of polynomial accuracy .The stability, and even the computability, of such rules depends critically on the properties of the underlying polynomial interpolants. Second, a recent spectral approach to the scattering of sound by three dimensional objects needs for its analysis good approximation properties of the ‘hyperinterpolation’ polynomial approximation scheme. In the course of this talk the existing state of knowledge for both interpolation and hyperinterpolation will be reviewed.

Granular materials are routinely handled in many industries. However, due to the incomplete understanding of such materials, solids-processing plants typically operate much less efficiently than their liquids-processing counterparts. When dealing with granular materials, engineers use models based on phenomenological relations, such as Coulomb's law of sliding friction. Generalizing such a law to multidimensional and/or dynamic settings has proved to be problematic in several respects (it may lead to ill-posedness, for instance).

In this talk, we will study and review several corresponding phenomena and models involving both static granular assemblies and steady-state flows. The mathematical structure of those problems often exhibit nonstandard features, such as the presence of both differential and algebraic constraints. Concrete examples involving an Eikonal formulation (pile formation), system of hyperbolic conservation laws and elliptic systems (granular flows) will be introduced and analyzed. Numerical methods and results will be discussed.

Humans are remarkably adept at processing the sound they receive at their two ears to perceive the spatial location (and other attributes) of sources in an acoustic scene. We are interested in creating auditory virtual reality over headphones. To create convincing sound at specified spatial locations, the cues in the sound used to localize the source must be reintroduced into the sound that is played. While binaural cues (that arise from the differences in the sound received at the two ears) are important, they are incomplete. Additional cues that arise from the scattering of sound off a persons body (external ears, head, and torso), and off the environment are also essential.

The cues that arise due to scattering off the listener are encoded in the so-called "Head Related Transfer Function" (HRTF), while the cues that arise from room scattering are encoded in a Room Transfer Function (RTF). The HRTF exhibits considerable variation from person-to-person Hitherto, the HRTF had to be measured individually, in a tedious measurement procedure that made it impractical to use them widely. We take an approach to modeling the scattering process, which relies on physical modeling and decomposition of the problem, and the use of fast algorithms for computation.

In this talk I will attempt to describe the mathematical formulations of some problems that arise, and some solutions we have proposed. (Joint work with Dmitry Zotkin & Nail Gumerov. The support of NSF and ONR is gratefully acknowledged)

Quantum Chromodynamics (QCD) is a fundamental theory for physics of strong interactions which hold the mysteries of the origin of mass, the formation of proton, neutron, and atomic nuclei, and the quark-gluon plasma in the early stage of the Universe. At the moment, the only direct approach to solve this strongly-coupled quantum field theory is simulate it on a discrete spacetime lattice. In this talk, I discuss the progress, challenges, and prospects of lattice QCD.

Many physical problems arising from oscillatory waves, dispersive waves or Hamiltonian systems require the computations of multivalued solutions which cannot be described by the viscosity methods. In this talk I will review several recent numerical methods for such problems, including the moment methods, kinetic equations and the new level set method with S. Osher. Applications to the semiclassical Schroedinger equation and Euler-Piosson equations with applications to modulated electron beams in Klystrons will be discussed.

We will discuss three numerical methods for Maxwell equations in inhomogeneous media with applications in photonic devices. First, we will discuss recently developed upwinding embedded boundary methods on a Cartesian grid to handle arbitrary material interface. Secondly, we will discuss several issues in discontinuous Galerkin methods such as construction of numerical fluxes, uniaxial PML boundary conditions. Thirdly, we will discuss a new fast integral solver for scattering in multilayered media. Numerical results will also be presented.

Differences in the eigenvalues of an autocovariance matrix indicate directions at which the local Fourier power spectrum of a function is slowly decreasing. This provides a technique to discriminate edge-like singularities from other features in images.

Models of domain coarsening during dynamic processes such as phase transitions in materials science involve many scales and levels of description. A great challenge is to explain the dynamic scaling laws seen in experiment. The simplest classical theory that yields predictions involves a conservation law for the size distribution of a family of particles.

Analyzing this law, we have recently obtained an improved well-posedness theory for measure-valued size distributions with finite mass, using a physically natural topology given by a Wasserstein distance between size distributions. The analysis establishes convergence of a physically meaningful numerical method for the problem.

We also analyze the long-time behavior of solutions. Rigorous analysis shows that this model does not yield the universal self-similar behavior that was classically predicted. Instead, long-time behavior depends sensitively on the initial distribution of the largest particles. E.g., for a dense set of initial data, convergence to any self-similar solution is impossible.

This is joint work with Barbara Niethammer.

We will survey the recent progresses on shock wave theory. There is a deep theory for system of hyperbolic conservation laws in one spacedimension, and serious efforts for multi-dimensional gas flows. We will report on these and raise some future possibilities.

The plasma-sheath transition is a fundamental problem in plasma physics, where the typical length scale can be predicated by dimensional analysis, but the sheath transition and inner layer are determined by a complex interplay of the internal dynamics. Mathematically it provides a challenge to the applied analyst in that there are multiple scales which must be resolved to obtain an adequate description of the physical process.

In a recent work with M. Slemrod we formulate a new mathematical model to catch the dynamics hidden in the plasma-sheath transition layer and the inner sheath layer for planar motion of a plasma. It is shown that the rescaled potential in the plasma-sheath transition layer and inner layer is governed by a perturbed KdV equation, through which some of the complex interactions and couplings among physical mechanisms acting in the plasmasheath formation process are elucidated.