Geometry and Billiards
نویسنده
چکیده
This book starts the new collection published jointly by the American Mathematical Society and the MASS (Mathematics Advanced Study Semesters) program as a part of the Student Mathematical Library series. The books in the collection will be based on lecture notes for advanced undergraduate topics courses taught at the MASS and/or Penn State summer REU (Research Experience for Undergraduates). Each book will present a self-contained exposition of a non-standard mathematical topic, often related to current research areas, accessible to undergraduate students familiar with an equivalent of two years of standard college mathematics and suitable as a text for an upper division undergraduate course. Started in 1996, MASS is a semester-long program for advanced undergraduate students from across the USA. The program's curriculum amounts to 16 credit hours. It includes three core courses from the general areas of algebra/number theory, geometry/topology and analysis/dynamical systems, custom designed every year; an interdis-ciplinary seminar; and a special colloquium. In addition, every participant completes three research projects, one for each core course. The participants are fully immersed in mathematics, and this, as well vii viii Foreword: MASS and REU at Penn State University as intensive interaction among the students, usually leads to a dramatic increase in their mathematical enthusiasm and achievement. The program is unique for its kind in the United States. The summer mathematical REU program is formally independent of MASS, but there is a significant interaction between the two: about half of the REU participants stay for the MASS semester in the fall. This makes it possible to offer research projects that require more than 7 weeks (the length of an REU program) for completion. The summer program includes the MASS Fest, a 2–3 day conference at the end of the REU at which the participants present their research and that also serves as a MASS alumni reunion. A non-standard feature of the Penn State REU is that, along with research projects, the participants are taught one or two intense topics courses. Preface Mathematical billiards describe the motion of a mass point in a domain with elastic reflections from the boundary. Billiards is not a single mathematical theory; to quote from [57], it is rather a math-ematician's playground where various methods and approaches are tested and honed. Billiards is indeed a very popular subject: in Jan-uary of 2005, MathSciNet gave more than 1,400 entries for " billiards " anywhere in the database. The …
منابع مشابه
Hyperbolic Billiards and Statistical Physics
Mathematical theory of billiards is a fascinating subject providing a fertile source of new problems as well as conjecture testing in dynamics, geometry, mathematical physics and spectral theory. This survey is devoted to planar hyperbolic billiards with emphasis on their applications in statistical physics, where they provide many physically interesting and mathematically tractable models.
متن کاملQuantal Andreev billiards: Density of states oscillations and the spectrum-geometry relationship
Andreev billiards are finite, arbitrarily shaped, normal-state regions, surrounded by superconductor. At energies below the superconducting energy gap, single-quasiparticle excitations are confined to the normal region and its vicinity, the mechanism for confinement being Andreev reflection. Short-wave quantal properties of these excitations, such as the connection between the density of states...
متن کاملVibrating Quantum Billiards on Riemannian Manifolds
Quantum billiards provide an excellent forum for the analysis of quantum chaos. Toward this end, we consider quantum billiards with time-varying surfaces, which provide an important example of quantum chaos that does not require the semiclassical (~ −→ 0) or high quantum-number limits. We analyze vibrating quantum billiards using the framework of Riemannian geometry. First, we derive a theorem ...
متن کاملSpaces of pseudo-Riemannian geodesics and pseudo-Euclidean billiards
Many classical facts in Riemannian geometry have their pseudoRiemannian analogs. For instance, the spaces of space-like and timelike geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce an...
متن کاملRandom Walks Derived from Billiards
We introduce a class of random dynamical systems derived from billiard maps, which we call random billiards, and study certain random walks on the real line obtained from them. The interplay between the billiard geometry and the stochastic properties of the random billiard is investigated. Our main results are concerned with the description of the spectrum of the random billiard’s Markov operat...
متن کاملSpaces of pseudo - Riemannian geodesics and pseudo - Euclidean billiards Boris
In pseudo-Riemannian geometry the spaces of space-like and timelike geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generaliza...
متن کامل