Conference on Recent Mathematical Advances in Classical, Quantum and Statistical Mechanics

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

The conference "Recent Mathematical Advances in Classical, Quantum and Statistical Mechanics" will be held at the University of Virginia on March 8-9, 2013. The goal of the conference is to present a survey of some of the major recent results in the area of mathematical analysis which studies the evolution of physical systems with large or infinite number of degrees of freedom in a variety of settings while emphasizing ideas and techniques which are common to these settings. The subject of this conference is one of the most active in contemporary mathematical physics. The conference will bring together leading world-experts in such topics as KAM theory for partial differential equations, rearrangement inequalities in functional analysis, Anderson localization, the study of polaron systems and the non-equilibrium statistical mechanics of coupled anharmonic oscillators. The conference will provide a venue for the dissemination or recent results and methods in the area and will foster further advances in these important topics. A strong component of this conference is educational and aimed at disseminating recent advances in classical, quantum and statistical mechanics. Emphasis will be placed on encouraging interactions between senior and junior researchers. Special efforts will be made to invite and support graduate students, postdocs, as well as mathematicians from underrepresented groups. The study of the evolution of systems with infinite number of degrees of freedom is ubiquitous in modern science. In the situation pertaining to classical mechanics, a typical example is that of the evolution of a fluid where a degree of freedom essentially corresponds to the velocity of the fluid at some point in space. Since there are infinitely many such points one is looking at a complex system with infinitely many degrees of freedom in interaction. These give rise to nonlinear partial differential equations which feature in a huge number of applications such as the models used for weather prediction. In the quantum case, examples of such systems are the polaron systems which describe the motion of particles in an ionic crystal (structures similar to that of ordinary salt). Another example studied under the name of Anderson localization has to do with the motion of a quantum particle in interaction with a random medium. The latter is supposed to model, e.g., the impurities in a semiconductor. Finally, a third regime where systems with infinitely many degrees of freedom are the subject of very active study is that of statistical mechanics, in particular in the non equilibrium situation. An example of mathematical challenge in the area is to understand the conduction of heat or electricity, not in an ad hoc phenomenological description, but rather through a derivation from first principles: the basic laws of classical or quantum mechanics at the microscopic level.