Mathematical Sciences: Ergodic Theory, P-adic Lie Groups andNumber Theory

Ratner的项目将把Raghunathan的猜想扩展到p进李群，2)研究SL（2,Q_p）齐次空间上测大地线和环流的动力学性质，3)将先前的结果应用于单幂流的轨道闭包，以求二次不等式积分解的渐近增长率。4)研究主同余群和Ramanujan-Selberg猜想。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类时与几何联系在一起。

Ratner's project will 10 extend Raghunathan's conjectures to p-adic Lie groups, 2) study the dynamical properties of the geodesics and the horocycle flows on homogeneous spaces of SL(2,Q_p), 3) apply previous results on orbit closures of unipotent flows to find asymptotic growth rate of integral solutions of quadratic inequalities, and 4) study Principal Congruence groups and the Ramanujan-Selberg Conjecture. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.