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

Kleinbock(Brandeis)：职业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了一组抽象的点，以及一个控制点随时间移动的进化规律。事实证明，许多与有理数同时逼近实数有关的问题可以用某些轨道的行为来解释，而与整数方程或不等式理论有关的各种现象，一旦用动力系统的语言来表述，就可以更好地理解。此外，在这种情况下产生的系统具有代数性质，这使得使用各种复杂工具进行研究成为可能。教育部分包括几个要素：开发新的“创造性思维”课程，指导本科生和研究生的暑期研究项目，以及举办介绍性的跨学科研讨会。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，并使他们接触到广泛的当前数学研究。

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.