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。动力系统在这里代表一组抽象的点，它们具有一个演化定律，该定律支配着点随时间移动的方式。结果表明，许多关于用有理数同时逼近真实的数的问题都可以用某些轨道的性质来描述，而与整数方程或不等式理论有关的各种现象，一旦用动力系统语言来表述，就能得到更好的理解。此外，在这种情况下出现的系统是代数性质的，这使得有可能使用各种各样的复杂的工具来进行调查。教育部分包括几个成分：开发新的“创造性思维”课程和监督本科生和研究生的暑期研究项目，并保持一个介绍式的跨学科系统。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.