Colloquium

Research seminars

VIGRE Working Groups

Clifford lectures

Mac Hyman,  Tulane University

"Mathematical Models Provide New Insights into Stopping Epidemics"

Abstract:

The pace that new diseases threaten the world is increasing.  The current H1N1 flu pandemic follows on the heels of recent epidemics for Herpes-2, Hepatitis C, HIV/AIDS, SARS, and avian flu.  Public health workers are reaching out to use all available tools to anticipate the spread of new diseases and evaluate the effectiveness of different approaches for bringing epidemics under control.   I will describe how mathematical models, based on the underlying transmission mechanisms, have advanced to help guide these efforts.  Today, mathematical scientists are joining with biological, epidemiological, behavioral, and social scientists to fight these emerging epidemics.  

The talk will provide an overview, for general audiences, of what type of insights these models can provide.  I will also touch briefly on the underlying mathematical advances and theory needed for the next generation of models, and share my personal experiences in controlling the spread of HIV/AIDS, SARS, malaria, foot and mouth disease, and the novel H1N1 (swine) flu.

Giles Auchmuty,  University of Houston

"Spectral Representation of Solutions of Linear Elliptic Boundary Value Problems"

Abstract:

This talk will describe some results about the representation, and approximation, of solutions of Dirichlet  problems for second order  linear elliptic equations using Steklov eigenfunction expansions.

It is well-known that standard eigenfunction expansions provide representations of solutions for Robin and Neumann boundary value problems with non-zero boundary data. This is not the case for  Dirichlet problems with non-zero boundary data. Here we shall describe spectral representations of the solutions of such problems. We first describe Steklov eigenproblems for an elliptic operator and show that the Steklov eigenfunctions  provide  bases for the space of $H^1-$solutions of the homogeneous elliptic equation. This leads to  general representation theorems for the solutions of the Dirichlet problem. It also provides an intrinsic, and constructive, description of the Sobolev trace spaces $H^{1/2}(\partial \Omega)$ and  $H^{s}(\partial \Omega)$.  Important  Hilbert spaces of real harmonic functions on general regions in $R^n$ will be characterized as Reproducing Kernel Hilbert spaces using a kernel defined in terms of the Steklov eigenvalues and eigenfunctions.

Sanjeevi Krishnan,  IHES

"Directed Cohomolgy"

Abstract:

I will present a cohomology theory for "directed spaces," spaces equipped with temporal structure.  Examples include spacetime manifolds and classifying spaces.  On such spaces, ordinary cohomology groups reveal properties invariant under continuous deformations, while directed cohomology monoids detect finer properties invariant under deformations respecting the temporal structure and thus tease out the "qualitative" structure of time.  Directed cohomology extends several well-known properties of its classical analogue: our new invariants admit chain-theoretic constructions, equivalent homotopical descriptions, axiomatic characterizations, and multiplicative structure.  After presenting the basic theory of directed homotopy and cohomology, I will sketch real and potential applications to concurrent engineering, string rewriting, and informatics.  This talk, aimed at a general audience, assumes no prior experience with directed spaces or cohomology theories.

Fall Break

Speaker,   Institution

"Title"

Abstract:

TBA.

Esteban Tabak,  Courant Institute

"The diurnal cycle and the meridional extent of the tropics"

Abstract:

This talk proposes a mathematical theory explaining the sharp transition between tropics and extra-tropics in terms of the diurnal cycle of thermal forcing by the sun. This transition, at a latitude of 30 degrees, coincides with the outer edge of the Hadley cells, and is marked by a steep jump in the height of the troposphere, from fifteen kilometers in the tropics to nine in the mid and high latitudes. The tropics, equatorwards of 30 degrees, are characterized by easterly surface winds -the Trades- and a strong diurnal signal in the wind, pressure and temperature, often marked by regular daily storms in the rainy season.

Polewards of 30 degrees, the winds are westerly, and the weather systems have longer spacio-temporal scales.

This change of behavior can be explained in terms of diurnal waves, created by thermal forcing and trapped equatorwards of 30 degrees by the Coriolis effect. This can be illustrated in simple mathematical models, ranging from forced linear oscillators to nonlinear conservation laws with entraining shock waves, acounting for the entrainment into the troposphere of air from the surface boundary layer.

Jian-Guo Liu ,  Duke University

"The Challenge of Simulating  Fluid Flow Accurately in the Presence of Boundaries"

Abstract:

The physical world has a rich diversity of fluid dynamics, ranging from the micron scale to the galactic scale, and varying from high Mach number compressible flows to low Mach number incompressible flows. Large variations in scales, flow properties, and surrounding environments pose many challenges for computations. These issues are particularly important in domains with boundaries. Much of the scientific and technological impact of the Navier-Stokes equations derives from the effect of no-slip boundary conditions in creating physical phenomena such as lift, drag, boundary-layer separation and vortex shedding, for which the behavior of the pressure near boundaries is of great significance. In this talk, I will present new equivalent formulations of Navier-Stokes equations (NSE) better suited for numerical computations. The emphasis will be on the enforcement of imcompressibility and the discovery of intrinsic stability properties that lead to accurate, efficient and practical computations of three dimensional problems. I will also present some efficient methods for more completed flows such as low-mach flow, MHD, kinetic equations with different scales.

Speaker,   Institution

"Title"

Abstract:

TBA.

Yimin Xiao,  Department of Statistics and Probability, Michigan State University

"Self-Similar Gaussian Random Fields and Their Fractal Properties"

Abstract:

Self-Similar Gaussian random fields are useful as stochastic models in many applied areas and their sample functions are often random fractals. In this talk we present some results on construction of Gaussian random fields and on their geometric and asymptotic properties.

Thanksgiving Holiday.

Shiferaw Berhanu,  Temple University

"Analyticity of solutions of first-order fully nonlinear pdes"

Abstract: