May 20, 2010:

Nancy Kopell, Boston University

Gamma, beta and cell assemblies: brain rhythms from physiology to function

May 6, 2010:

Michael Brenner, Harvard University

Adventures in self-assembly

Abstract: Self assembly is the idea of creating a system whose component parts spontaneously assemble into a structure of interest. In this talk I will outline our research program aimed at creating self-assembled structures out of very small spheres, that bind to each other on sticking. The talk will focus on (i) some fundamental mathematical questions in finite sphere packings (e.g. how do the number of rigid packings grow with N, the number of spheres); (ii) algorithms for self assembly (e.g. suppose the spheres are not identical, so that every sphere does not stick to every other; how to design the system to promote particular structures); (iii) physical questions (e.g. what is the probability that a given packing with N particles forms for a system of colloidal nanospheres); (iv) comparisons with experiments on colloidal nanospheres. and (v) ways of using microfluidics to enable kinetically driven self assembly.

April 1, 2010:

Athanasios Fokas, Cambridge University

Integrability, imaging, and complexification

Abstract: Ideas and techniques of Integrability have had a significant impact in several areas of science and engineering. In this lecture, two such applications will be reviewed: (a) an analytical approach to certain important medical imaging techniques; (b) a unified approach to analyzing boundary value problems. The latter approach unifies the fundamental contribution to the analytical solution of PDEs of Fourier, Cauchy and Green, and also presents a non-linearization of some of these results.

February 25, 2010:

Leslie Greengard, Courant Institute

A new formalism for electromagnetic scattering in complex geometry

Abstract: We will describe some recent, elementary results in the theory of electromagnetic scattering. There are two classical approaches that we will review - one based on the vector and scalar potential and applicable in arbitrary geometry, and one based on two scalar potentials (due to Lorenz, Debye and Mie), valid only in the exterior of a sphere. In extending the Lorenz-Debye-Mie approach to arbitrary geometry, we have encountered some new mathematical questions involving differential geometry, partial differential equations and numerical analysis. This is joint work with Charlie Epstein.

February 4, 2010:

George Papanicolaou, Stanford University

Imaging with noise

Abstract: It is somewhat surprising at first that it is possible to locate a network of sensors from cross correlations of noise signals that they record. This is assuming that the speed of propagation in the ambient environment is known and that the noise sources are sufficiently diverse. If the sensor locations are known and the propagation speed is not known then it can be estimated from cross correlation information. Although a basic understanding of these possibilities had been available for some time, it is the success of recent applications in seismology that have revealed the great potential of correlation methods, passive sensors and the constructive use of ambient noise in imaging. I will introduce these ideas in an interdisciplinary, mathematical way and show that a great deal can be done with them. Things become more complicated, and a mathematically more interesting, when the ambient medium is also strongly scattering. I will end with a review of what is known so far in this case, and what might be expected.

January 14, 2010:

Charlie Doering, University of Michigan

Heat rises: convection, stability and turbulence

Abstract: Rayleigh-Bénard convection is the buoyancy-driven flow that results when a fluid is heated from below and cooled from above sufficiently to destabilize the conduction state where the fluid is at rest. A key experimental, theoretical, and mathematical challenge is to ascertain the functional dependence of the heat transport on the applied temperature drop and on the material parameters characterizing the fluid. Turbulent convection is of particular interest. In this talk we will review some of the history and scientific applications of Rayleigh-Bénard convection, describe some theory and analysis connecting notions of nonlinear stability to the statistical dynamics of the highly unstable turbulent regime, and compare the results with direct numerical simulations and laboratory experiments.

November 19, 2009:

Jason Fleischer, Princeton University

Optical hydrodynamics

Abstract: It is well-known that the basic equations of nonlinear optics can be mapped to equations from condensed matter physics. For example, the nonlinear Schrödinger description of paraxial beam propagation is identical to the Gross-Pitaevskii treatment of coherent matter waves, e.g. for Bose-Einstein Condensates. In turn, these equations can be mapped to Euler-like fluid dynamics using a polar (Madelung) transformation. Here, we exploit these relations to develop an optical hydrodynamics. For coherent waves, we examine dispersive shock wave formation, hydrodynamic instabilities, and vortex flow. For incoherent waves, we demonstrate all-optical plasma dynamics, including Landau damping, bump-on-tail instabilities, and weak and strong regimes of spatial turbulence. Optical experiments are performed and shown to match very well with theory. The results establish optical systems as an analog simulator for fluid behavior and suggest a variety of fluid solutions to photonic problems, including those of imaging.

November 5, 2009:

Phil Holmes, Princeton University

Neuromechanics of lamprey swimming or, an elastic rod with a mind of its own

Abstract: Locomotion provides superb examples of cooperation among neuromuscular systems, environmental reaction forces, and sensory feedback. As part of a program to understand the neuromechanics of locomotion, here we construct a model of anguilliform (eel-like) swimming in slender fishes. Building on a continuum mechanical representation of the body as an viscoelastic rod, actuated by a traveling wave of preferred curvature and subject to simplified hydrodynamic reaction forces, we incorporate a new version of a calcium release and muscle force model, fitted to data from the lamprey Ichthyomyzon unicuspis, that interactively generates the curvature wave. We use the model to investigate the source of the difference in speeds observed between electromyographic waves of muscle activation and mechanical waves of body curvature, concluding that it is due to a combination of passive viscoelastic and geometric properties of the body and active muscle properties. Moreover, we find that nonlinear force dependence on muscle length and shortening velocity may reduce the work done by the swimming muscles in steady swimming.

October 22, 2009:

Russel Caflisch, UCLA

Multiscale methods for dilute fluids and plasmas

Abstract: Fluids and plasmas exhibit a variety of spatial and temporal scales that lead to interesting physical phenomena and difficult computational challenges. One of the most difficult of these is due to the difference between the continuum (fluid) and the collisional (particle) length scales. For a dilute gas or plasma, the effect of particle collisions is not accurately approximated by a continuum description, so that its evolution requires a particle description such as the Boltzmann equation. Numerical simulation of the Boltzmann equation is usually performed using a Monte Carlo method such as Direct Simulation Monte Carlo (DSMC). Near the fluid limit, however, DSMC becomes computationally intractable, because of the large collision rate. To overcome this problem we have developed a hybrid method that combines a continuum (fluid) description and a particle (DSMC) description. Numerical examples for both fluids and plasmas will be presented to illustrate the performance of the methods.

April 9, 2009:

Marty Golubitsky, Ohio State University

Symmetry-Breaking; Synchrony Breaking

Abstract: A coupled cell system is a network of interacting dynamical Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can We ask: which part of the qualitative dynamics observed in coupled cells is the product of network architecture and which part depends on the specific equations? In our theory, local network symmetries replace symmetry as a way of organizing network dynamics, and synchrony-breaking replaces symmetry-breaking as a basic way in which transitions to complicated dynamics occur. Background on symmetry-breaking and pattern formation will be presented.

Marty Golubitsky  is Distinguished Professor of Mathematics and Physical Sciences at the Ohio State University, where he serves as Director of the Mathematical Biosciences Institute. He received his PhD in Mathematics from M.I.T. in 1970 and has been Professor of Mathematics at Arizona State University (1979-83) and Cullen Distinguished Professor of Mathematics at the University of Houston (1983-2008).Dr. Golubitsky works in the fields of nonlinear dynamics and bifurcation theory studying the role of symmetry in the formation of patterns in physical systems and the role of network architecture in the dynamics of coupled systems. His recent research focuses on some mathematical aspects of biological applications: animal gaits, the visual cortex, the auditory system, and coupled systems. He has co-authored four graduate texts, one undergraduate text, and two nontechnical trade books, (Fearful Symmetry: Is God a Geometer with Ian Stewart and Symmetry in Chaos with Michael Field) and over 100 research papers. Dr. Golubitsky is a Fellow of the American Academy of Arts and Sciences, a Fellow of the American Association for the Advancement of Science (AAAS), the 1997 recipient of the University of Houston Esther Farfel Award, and the 2001 co-recipient of the Ferran Sunyer i Balaguer Prize (for The Symmetry Perspective). He has been elected to the Councils of the Society for Industrial and Applied Mathematics (SIAM), AAAS, and the American Mathematical Society. Dr. Golubitsky was the founding Editor-in-Chief of the SIAM Journal on Applied Dynamical Systems and has served as President of SIAM (2005-06).

Printable poster

May 21, 2009:

Nick Trefethen, Oxford University

Chebfuns: a new kind of numerical computing

 Abstract: Chebfuns represent a new kind of computing  that aims to combine the feel of symbolics with the speed of precision numerics. The idea is to represent functions by piecewise Chebyshev expansions whose length is determined adaptively to maintain an accuracy of close to machine. The software is implemented in object-oriented Matlab, with familiar vector operations such as sum and diff overloaded to analogues for functions such as integration and differentiation, and the chebop extension solves linear ordinary differential equations by typing a backslash. This is joint work with others including Zachary Battles, Folkmar Bornemann, Toby Driscoll, Ricardo Pachon, and Rodrigo Platte.

Nick Trefethen is Professor of Numerical Analysis and head of the Numerical Analysis Group at Oxford University. A Fellow of the Royal Society and a member of the National Academy of Engineering, he is known for books, articles, and software in areas including numerical linear algebra, transition to turbulence, approximation of functions, numerical conformal mapping, and spectral methods for partial differential equations.

Printable poster

January 29, 2009:

Claude Le Bris, Paris

Mathematical challenges in molecular simulation: an overview

 Abstract: Molecular simulation is increasingly important in many engineering sciences and life sciences. The field has only been recently explored by mathematical analysts and numerical analysts, leading to several achievements, but also leaving major challenging issues unsolved, both theoretically and computationally. The talk will present the state of the art and will review major mathematical issues of practical importance and theoretical relevance. It will also relate such issues of molecular simulation with issues in  materials science. It is mostly based on a recent article coauthored with E. Cances and PL. Lions, and published in Nonlinearity, volume 21, T165-T176, 2008.

Claude Le Bris is a Professor of Applied Mathematics at the Ecole Nationale des Ponts et Cahussees, Paris. He is also Civil engineer-in-chief, Associate Professor at the Ecole Polytechnique and scientific director of the MICMAC project (multiscale methods) at INRIA. Professor Le Bris has won numerous awards including the Blaise Pascal Prize 1999 from the French Academy of Sciences, the CS 2002 Prize in Scientific computing from Communications & Systems, and the Giovanni Sacchi-Landriani Prize 2002 from the Lombard Academy of Arts and Sciences.

Printable poster

February 12, 2009:

Thomas Hou, Caltech

Recent progress on dynamic stability and global regularity of 3D incompressible Euler and Navier-Stokes equations

 Abstract: Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the seven Millennium Open Problems posted by the Clay Mathematical Institute. We review some recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. The local geometric regularity of vortex filaments can lead to tremendous cancellation of nonlinear vortex  stretching, thus preventing a finite time singularity. Our studies also reveal a surprising stabilizing effect of convection for the 3D incompressible Euler and Navier-Stokes equations. Finally, we present a new class of solutions for the 3D Euler and Navier-Stokes equations, which exhibit very interesting dynamic growth property by exploiting the special structure of the solution and the cancellation between the convection term and the vortex stretching term, we prove nonlinear stability and the global regularity of this class of solutions.

Thomas Hou is the Charles Lee Powell Professor and Executive Officer of Applied and Computational Mathematics at the California Institute of Technology. Professor Hou has won numerous awards including the Sloan Fellowship (1990-1992), the Feng Kang Prize in Scientific Computing (1997), the APS Francois N. Frenkiel Award (1998), the SIAM James H. Wilkinson Prize in Numerical Analysis and Scientific Computing (2001), the Morningside Gold Medal in Applied Mathematics, International Congress of Chinese Mathematicians (2004), and the Computational and Applied Sciences Award, the United States Association of Computational Mechanics (2005). In addition, he is on numerous editorial boards including as the founding editor of the SIAM Journal on Multiscale Modeling and Simulation.

Printable poster

February 26, 2009:

Carl Bender, Washington University

Quantum Mechanics in the Complex Domain

 Abstract: The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (symmetric under combined matrix transposition and complex conjugation) in order to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H=p^2+ix^3,for example, which is clearly not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a perfectly acceptable quantum mechanics. Evidently, the axiom of Dirac Hermiticity is too restrictive. While the Hamiltonian H=p^2+ix^3 is not Dirac Hermitian, it is PT symmetric; that is, it is symmetric under combined space reflection P and time reversal T. In general, if a Hamiltonian H is not Dirac Hermitian but exhibits an unbroken PT symmetry, there is a procedure for determining the adjoint operation under which H is Hermitian. 

 It is wrong to assume a priori that the adjoint operation that interchanges bra vectors and ket vectors in the Hilbert space of states is the Dirac adjoint. This would be like assuming a priori what the metric g^munu in curved space is before solving Einstein's equations.) In the past a number of interesting quantum theories, such as the Lee model and the Pais-Uhlenbeck model, were abandoned because they were thought to have an incurable disease. The symptom of the disease was the appearance of ghost states (states of negative norm). The cause of the disease is that the Hamiltonians for these models were inappropriately treated as if they were Dirac Hermitian. The disease can be cured because the Hamiltonians for these models are PT symmetric, and one can calculate exactly and in closed form the appropriate adjoint operation under which each Hamiltonian is Hermitian. When this is done, one can see immediately that there are no ghost states and that these models are fully acceptable quantum theories.

Carl Bender is a Professor of Physics at Washington University in St. Louis. Professor Bender has won numerous awards including the Sloan Fellowship (1972-1977), the M.I.T. Graduate Student Council Teaching Award (1976), a Fulbright Fellowship to the United Kingdom (1995-1996), John Simon Guggenheim Memorial Foundation Fellowship (2003-2004), the Ulam Fellowship, Los Alamos National Laboratory, (2006-2007), the Compton Faculty Achievement Award, Washington University (2007), and Wilfred R. and Ann Lee Konneker Distinguished Professor of Physics (2007). In addition, he is a co-author of one of the most influential texts in applied mathematics: Advanced Mathematical Methods for Scientists and Engineers (Bender & Orzag).

Printable poster

October 9, 2008:

David Keyes, Columbia University

Attacking the asymptotic algorithmic bottleneck: scalable solvers for scientific simulation

 Abstract: Many simulations with promise for scientific discovery and technological advance are of multiscale and multiphysics character, and subject to uncertainties of model or data. Asymptotically in mesh resolution, such applications are ultimately bottlenecked by solution algorithms. Simulations based on Eulerian formulations of partial differential equations can be among the first applications to take advantage of petascale capabilities, but not the way most are presently pursued. 

 Weak scaling avoids the fundamental limitation expressed in Amdahl's Law but only optimal implicit formulations can get around another limitation on scaling that is an immediate consequence of Courant-Friedrichs-Lewy stability theory under weak scaling of a PDE. Many PDE-based applications and other lattice-based applications with petascale roadmaps (such as quantum chromodynamics) will likely be forced to adopt optimal implicit solvers. However, even this narrow path to petascale simulation is made treacherous by the imperative of dynamic adaptivity, which drives us to consider algorithms that are less synchronous than those in common use today. After reviewing themes in scalable algorithms at a high level, we focus on some particular advances in the U.S. magnetic fusion energy program and other PDE-based applications enabled by scalable solution algorithms.

David E. Keyes is the Fu Foundation Professor of Applied Mathematics in the Department of Applied Physics and Applied Mathematics at Columbia University, and the Chair-designate of the Mathematical and Computer Sciences and Engineering Division at KAUST. With backgrounds in engineering, applied mathematics, and computer science, Keyes works at the algorithmic interface between parallel computing and the numerical analysis of partial differential equations, across a variety of applications. Newton-Krylov-Schwarz parallel implicit methods, introduced in a 1993 paper, are now widely used throughout computational physics and engineering and scale to many thousands of processors. Keyes is currently the Vice President-at-Large of SIAM, and a member of the advisory committees of the Mathematics and Physical Sciences Directorate and the Office of CyberInfrastructure of the NSF.

Printable poster

October 23, 2008:

John Bush, MIT

Interfacial Biomechanics

Abstract: We examine several biological systems dominated by the influence of surface tension. Particular attention is given to elucidating natural strategies for water-repellency, underwater breathing, fluid transport on a small scale, and walking on water.  Examples are primarily taken from the world of insects, but capillary feeding in shorebirds is also highlighted, and the mechanics of spider capture silk touched upon. A number of Nature's designs are rationalized and serve as inspiration for biomimetic microfluidic devices.

John Bush is on the faculty in the Mathematics Department at MIT and is Director of the department's Fluid Dynamics Laboratory. His research involves an interplay between experimental and theoretical modeling techniques, and is focussed towards identifying and elucidating new fluid dynamical phenomena. He is particularly interested in Surface Tension-Driven Phenomena and Biofluidynamics. Among his distinctions, Professor Bush received the Gallery of Fluid Motion Award of the APS Division of Fluid Mechanics virtually every year since 1999. He received an NSF Career Award in 2002. In 2003, the department faculty selected him to be the initial holder of the Edward F. Kelly Research Award.

Printable poster

November 6, 2008:

Mark Lewis, University of Alberta

Dynamics of emerging wildlife diseases

Abstract: In this talk I will present recent progress in modelling the dynamics of emerging wildlife diseases. I will focus on two examples, one involving interactions between hosts (birds) and disease vectors (mosquitoes) in the outbreak of West Nile virus, and the other involving a "spill over" and "spill back" disease between net pen aquaculture and wild salmon. The focus of the talk will be quantitative assessment of the disease dynamics using dynamical systems, and the resulting interplay between models and data.

 Mark Lewis holds the Canada Research Chair in Mathematical Biology at the University of Alberta. Dr. Lewis' research is in mathematical biology and ecology, including modelling and analysis of nonlinear PDE and integral models in population dynamics and ecology. Applications, made to case studies with detailed data and biology, include: wolf territories, spatial spread and impact of introduced pest species, wildlife diseases, vegetation shift in response to climate.