News & Events

The ensemble G(n,d) of random regular graphs, and its spectral properties, has drown a considerable attention as an accessible model for probabilistic expanders, in diverse disciplines of physics and mathematics.

The eigenvectors of an (n,d) graph are believed to contain information about the graph structure and they are being used in various algorithms. However, unlike the spectral properties of the ensemble, not much is currently known about the distribution of the eigenvectors. Motivated by the mixing property of such graphs, and following a conjecture by Michael Berry for chaotic billiards, I will present several recent results and hypotheses concerning the distribution of (n,d) graphs, such as the correlations between different components, a candidate for the components' limiting distribution, the expected number of nodal domains in an eigenvector and some hints for a critical phenomena.

The study of dynamics on networks is becoming increasingly more relevant within biology. An important subclass of biological networks are `pulse-coupled' networks, such as neuronal networks. An important question is: what is the relationship (or map) between a pulse-coupled network's architecture and any given statistical feature of its dynamics? In many circumstances this question cannot be answered easily, and theorists and modelers often resort to simulations in order to probe the properties of this map. I will present a framework which takes a first step towards answering this question. By expressing the desired statistical feature of the network's dynamics in terms of an appropriate integral of the equilibrium distribution of system paths in state-space (i.e., a projection of the system's filtration), one can derive a systematic expansion (in terms of coupling strength) of any desired projection of the network's dynamics. After motivating the derivation, I will present a few examples illustrating the utility of this new method.

I will present a mathematical derivation of equations of motion for the conditional averages of dynamical variables in stochastic multiscale problems. The surprising feature of the resulting formalism is that it yields exact reduction schemes for the dynamics of multiscale problems, and also reduces the problem of evaluating the marginals in large sampling problems to the much easier problem of evaluating conditional expectations. The resulting formulas are of course too complex to be fully evaluated, but having an exact result to start with makes it easier to find good approximations.

I will focus on two applications: the derivation of simplified dynamics for systems with long memory (thus, no separation of scales), with an application to hydrodynamics, and the development of effective chain-free sampling schemes, with an application to glassy systems.

Density Functional Theory (DFT) is one of the most successful electronic structure method used in physics, chemistry and materials science. In the DFT framework, the exact electron density in materials is given by the fixed point of the so called Kohn-Sham self-consistent map. In turn, the electron density determines the ground state energy of the electrons.

While the computational studies and applications of the Kohn-Sham self-consistent map are quite abundant, rigorous analyses on the existence and uniqueness of the fixed points are scarce. In this talk, I will discuss these issues from a mathematical point of view. For a class of approximations to the exchange-correlation potential, that include the Local Density Approximation, I will present results on the existence and uniqueness of the fixed points for the Kohn-Sham self-consistent map at finite temperatures. For zero temperature, I will demonstrate that the Kohn-Sham self-consistent map can have multiple fixed points.

Slow flows of dense granular materials are encountered in a wide variety of industrial processing devices, such as fluidized beds and hoppers. Predictive continuum modeling of such flows is important in designing the devices on a firm numerical basis instead of on empirical trial-and-error approaches. However, reliable constitutive models for the rheological behaviors of dense granular flows that are linked to particle-level properties are not available for general flow conditions. In this presentation, our efforts aimed at constructing such constitutive models using a hierarchical multiscale approach will be presented.

In this multiscale approach, the discrete element method (DEM) is used to simulate the motion of individual particles in sheared assemblies; the stress and kinematics information at the continuum level are then obtained through statistical averaging. The continuum information is then used to establish quantitative closure relations for continuum rheological models.

In this presentation, rheological behaviors of nearly homogeneous assemblies of uniformly sized, spherical particles in periodic domains under steady and unstea

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dy simple shear will be presented. The assembly pressure and stress ratio variations with respect to the particle-level properties and microstructural parameters are examined to establish the closure relations between them. The DEM simulation results will be compared with predictions of continuum constitutive models, supplemented with the closures determined in this study.

Gaussian beam is an efficient way for solving the high frequency wave equations asymptotically with solution being valid at caustics. Our new work is about developing a new Eulerian Gaussian beam method for solving the Schrodinger equation in the semiclassical regime with possible applications in quantum related topics. In the talk I will also briefly include the (standard) Lagrangian Gaussian beam formulation to make the Eulerian methods better understood.

Oscillatory integral transforms and equations arise in many direct and inverse problems pertaining to wave propagation phenomena. Examples abound in fields including seismic migration, acoustic and electromagnetic wave scattering, and radar imaging. However, the rapid evaluation of these transforms is an challenging task due to the oscillatory nature of the kernel.

In this talk, we first review the butterfly algorithm, which was recently developed as a general approach for the rapid evaluation of these oscillatory integrals. However, sometimes the practical efficiency of the butterfly algorithm is limited by its high preprocessing time and high storage requirement. In the second part of this talk, we discuss two applications: (1) sparse Fourier transform and (2) partial Fourier transform, where in each case these constraints can be removed by using tools such as tensor product decomposition and non-standard Chebyshev interpolation.

Suspensions of filaments in the Stokes flow regime have a wide range of applications fromfiber processing to nano-cargo transport systems.  In this talk a slender-body formulation for multiple filaments immersed in Stokes flow will be presented.  Analysis and numerical simulation results will show how the dynamics of semi-flexible filaments could lead to non-trivial filament transport.  Effects of hydrodynamic interactions will be quantified in terms of filament separation, filament flexibility, and slenderness.  Finally a brief summary on recent attempts to efficiently model multi-filament flows will be presented.

This work is an on-going collaboration with Mike Shelley (Courant).