Stochastic Dynamics: Models, Analysis, and Numerics
SPRING 2011
Department of Mathematics, University of Maryland, College Park
Lectures: Wednesdays, 3-4 PM, at the room MATH1308
Organizers:
Maria Cameron (cameron@math.umd.edu)
Sandra Cerrai (cerrai@math.umd.edu)
Leonid Koralov (koralov@math.umd.edu)
Dionisios Margetis (dio@math.umd.edu)
Website:
http://www-users.math.umd.edu/~cameron/research-integration-team-r.html
Credit: Students may participate without signing up. Students who sign up for credit are expected to attend regularly and pick one topic and write a book repot on it (a few pages long) by the end of the semester.
Schedule for Spring 2011:
Speaker: Leonid Koralov
Feb. 16 Problems and Challenges in Stochastic Dynamics
Speaker: Maria Cameron
Feb. 23 Overview of SDE's
Speaker: Leonid Koralov
March 2 Large Deviation Theory. Wentzell-Freidlin Action Functional.
Speaker: Sandra Cerrai
March 9 Large Deviation theory (Part II). Exit times.
Speaker Leonid Koralov
March 16 Gradient systems and Minimum Energy Paths. Continuous-time Markov Chains.
Speaker: Maria Cameron
March 30 Transition Path Theory
Speaker: Maria Cameron
April 6 An overview of Monte-Carlo methods
Speaker: Jonathan Weare (University of Chicago, Dept. of Mathematics)
April 20 Smolukhowski-Kramers Approximation
Speaker: Sandra Cerrai
April 27 Problems in materials science: Interfaces and crystal surfaces
Speaker: Dionisios Margetis
May 4 A taste of mathematical physics: Zwanzig-Mori formalism
Speaker: Dionisios Margetis
Scope and Research Focus:
Stochastic dynamics govern a broad range of physical phenomena. Of particular importance are phenomena observed at small length and time scales. An example is the well-known Brownian motion of atomic particles.
Stochastic particle dynamics are usually described by Stochastic Differential Equations (SDEs). SDEs are widely used in physics and chemistry to model and simulate various small-scale phenomena in random environments.
On the other hand, SDEs have been the subject of interest and scrutiny in mathematics, especially in the context of randomly perturbed dynamical systems.
The goal of this RIT is to explore connections of rigorous mathematical theories for SDEs and stochastic dynamics with models and methods of other disciplines (e.g., chemistry, physics, materials science). The topics to be explored include:
(1) Review of the basic mathematical theory of SDEs and stochastic dynamics (e.g., Brownian motion, Large Deviation Theory, Transition Path Theory).
(2) Review of numerical methods used for SDEs and in stochastic modeling (e.g., stochastic integrators for direct simulations, methods for computing the most likely transition paths between metastable states: path-based methods and Hamilton-Jacobi solvers), as well as their applications.
(3) Discussion of some open related problems in the physical sciences.