Stochastic Calculus for Applications:
Theory and Numerics joint with MITACS

August 9 – 20, 2004,

Lecturer: E. Vanden-Eijnden (Courant)

The aim is to cover the essentials of stochastic calculus, including fundamental concepts such as Markov chains, Wiener processes, stochastic differential equations, as well as more elaborate ideas such as the Girsanov transformation and path integrals. The material will be presented at a semi-rigorous level by relying only on the standard tools of basic probability, linear algebra, and advanced calculus. Both theoretical and numerical aspects will be covered and illustrated via examples.

Aisenstadt Chair Lecture Series

A.J. Majda (Courant) will be the holder of the Aisenstadt Chair in the Fall 2004. He is well known for both his theoretical contributions to the study of partial differential equations and his applied contributions to diverse areas, most recently climate modeling and prediction.

T.Y. Hou (Caltech) will be the Chair holder during the Spring 2005 semester. He is one of the leading experts in the modeling of free boundary and multiscale problems.

Workshops

Multiscale Rheological Models for Fluids

November 14 – 17, 2004

Organizer: C. Le Bris (Cermics, ENPC, Paris)

This workshop will address the multiscale modeling of non-Newtonian fluids. Models for such systems could be built via constitutive laws, but those are typically very difficult to obtain and validate rigorously. An alternative strategy is to use kinetic theory to simulate the micro-structures that govern the fluid behaviour. (For polymeric fluids, those micro-structures would be polymer chains; for other fluids, it could be particles in suspension.) One then couples this microscopic description with macroscopic continuum equations for the fluid. This workshop will be an opportunity to discuss recent developments along these lines.

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Front Propagation and Nonlinear Stochastic PDEs for Combustion and other Applications

January 26 – 30, 2005

Organizers: A. Bourlioux (Montréal) and P. Souganidis (Texas)

The development of efficient large-scale models for the numerical simulation of turbulent premixed flames requires a good understanding of the mathematical principles governing the dynamics of self-propagating fronts. One of the most challenging issues is the analysis of the complex interactions, at small scales, between advection, reaction and diffusion, including stochastic effects due to the media or the advective flow randomness.

This workshop will be an opportunity for interaction between mathematicians at the forefront of this area and scientists involved in the design of models and numerical methods for various applications, in particular, turbulent combustion.

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Representing Unresolved Degrees of Freedom
for the Atmosphere and Ocean

March 2 – 4, 2005

Organizer : A.J. Majda (Courant)

A central problem in attempts to understand and predict the evolution of atmospheric or oceanic flows is how best to represent the unresolved scales in these flows. In the jargon of dynamic meteorology or physical oceanography this is called the parameterization problem, while in the jargon of turbulence it is called the closure problem. The most pertinent areas of analysis and applied mathematics are homogenization theory, probability and non-linear stochastic PDEs. The purpose of this workshop is to explore two complementary issues that arise in the context of the parameterization problem:

1. the extent to which modern techniques in applied mathematics can be brought to bear on its formulation and partial solution; and
2. the extent to which problems in the representation of atmospheric and oceanic flows create fertile new areas of mathematical inquiry.
   
   

Extracting Macroscopic Information from Molecular Dynamics

April 7 – 9, 2005

Organizers : P.F. Tupper (McGill), A.M. Stuart (Warwick)

Models used in molecular dynamics are high-dimensional dynamical systems (or stochastic dynamical systems) with multiple time-scales. A major challenge for computational mathematics is the extraction of accurate macroscopic information at minimal cost. This workshop will concentrate on two topics:

1. the analysis and development of standard time-stepping algorithms in the context of molecular dynamics, with the purpose of the indirect calculation of macroscopic information; and
2. the design of new algorithms aimed at extracting macroscopic information directly.
   
   

Multiscale Modeling in Solids

April 28 – 30, 2005

Organizers: Weinan E (Princeton), E. Vanden-Eijnden (Courant)

This workshop will focus on energetic and kinetic issues associated with defects, cross-slip, grain boundary migration, and phase boundary dynamics in solids. The objective is to develop mathematical models for complex multiscale phenomena such as crystal plasticity, nucleation and reconstruction of stepped surfaces, and the behaviour of nano-materials in general.

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Integrative Multiscale Modeling and Simulation
in Materials Science, Fluids and Environmental Science

May 11 – 13, 2005

Organizer: T.Y. Hou (Caltech)

Multiscale modeling and simulation have already impacted many scientific and engineering disciplines. Numerous developments have been scattered in various disciplines and there is a great need to integrate isolated efforts. This workshop will recapitulate previous activities, focus on the interdisciplinary interaction among these related fields, and try to develop new tools that combine mathematical analysis, multiscale modeling and computational analysis in an integrative way.

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Stochastic Modeling in Financial Mathematics
joint with SAMSI

June 1 – 5, 2005

Organizers : R. Sircar (Princeton), J.-P. Fouque (North Carolina State)

The theme of this workshop is emerging directions in financial mathematics, with emphasis on stochastic modeling of market uncertainties, theoretical and numerical approximations to pricing, hedging and portfolio optimization control problems, and data estimation issues. The goal is to bring together researchers in a variety of disciplines (mathematics, engineering, operations research and economics, for example) to emphasize different techniques and approaches.

Advanced Course

Fall 2004

Multiscale Systems: Modeling and Simulation

Lecturer: C. Le Bris (Cermics, ENPC, Paris)