An introduction to Birhkhoff Billiards, with emphases on spectral rigidity and integrability
Student No.:50
Time:Thu/Fri 8:00-9:35, 2017-08-10~ 2017-08-25
Instructor:Guan Huang  [University of Maryland]
Place:Conference Room 1, floor 1, Jin Chun Yuan West Building
Starting Date:2017-8-10
Ending Date:2017-8-25





A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary (which is assumed to have infinite mass). This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where "the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered”. Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Not only is their law of motion very physical and intuitive, but also billiard-type dynamics is ubiquitous. Mathematically, they are models in every subclass of dynamical systems (integrable, regular, chaotic, etc.); more importantly, techniques initially devised for billiards have often been applied and adapted to other systems, becoming standard tools and having ripple effects beyond the field.

In this short course, I will explain some fundamental concepts and results on Birkhoff Billiards, including periodic orbits, length spectrum, caustics, integrability, Aubry-Mather theory of exact twist maps, and the Birkhoff conjecture on integrable billiards, etc.





General undergraduate training in Mathematics, particularly, advanced calculus and functional analysis.