CAREER: Dynamical Systems on Homogeneous Spaces and Applications to Number Theory

DMS 0238463D Kleinbock(Brandeis)：Career该项目的研究部分涉及代数动力系统及其在数论中的应用。这里的动力系统代表一组抽象的点，它带有一个演化规律，支配点随时间移动的方式。结果表明，许多与有理数同时逼近实数有关的问题可以转化为某些轨道的行为，而与整数方程理论或不等式有关的各种现象一旦用动力系统语言来表述就可以更好地理解。此外，在这种情况下出现的系统是代数性质的，这使得可以使用各种复杂的工具进行研究。教育部分由几个部分组成：为本科生和研究生开发新的“创造性思维”课程和指导暑期研究项目，以及维持入门式的跨学科研讨会。目标是让学生参与具有挑战性的研究项目，鼓励非标准的独立思考，并让他们接触到当前广泛的数学研究。

AbstractDMS 0238463D Kleinbock(Brandeis): CareerThe research component of the project deals with algebraicdynamical systems and their applications to number theory.A dynamical system here stands for an abstract set of pointstogether with an evolution law which governs the way pointsmove over time. It turns out that many problems concerningsimultaneous approximation of real numbers by rational numberscan be cast in terms of the behavior of certain orbits, andvarious phenomena related to the theory of integer equationsor inequalities can be better understood once they are phrasedin dynamical systems language. Furthermore, systems thatarise in this context are of algebraic nature, which makesit possible to use a wide variety of sophisticated tools fortheir investigation.The educational component consists of several ingredients:developing new `creative thinking' courses and supervisingsummer research projects for both undergraduate and graduatestudents, and maintaining an introductory-style interdisciplinaryseminar. The goals are to engage students in challenging researchprojects, encourage non-standard independent thinking, and toexpose them to a broad spectrum of current mathematical research.