EMSW21-RTG: Training the Research Workforce in Geometry, Topology and Dynamics

这项建议要求继续我们的研究培训计划在密歇根大学的几何，拓扑和动力学。 博士学位研究生和高级研究生将是主要受益者。 UM的数学系拥有全国最大，最具活力的博士后和研究生课程之一，在几何，拓扑和动力学方面培养高质量的研究人员方面有着出色的记录。 我们将加强对这些领域的博士后和研究生的培训，深化和拓宽他们的教育，并为他们提供充分的机会，使他们在研究方面取得卓越成就。 五位资深教师（Richard Canary，John Erik Fornaess，Gopal Prasad，Yongbin Ruan和Ralf Spatzier）将与其他资深教师合作领导该项目。 我们在现有的培训计划中引入了各种创新，并将继续并扩展这些创新：我们将提供密集的探索性研讨会，系列讲座和研讨会。 所有学员将发展演讲技巧，并将受益于密集的辅导。 一些学员将通过前往研究前沿的其他机构来加深他们的科学培训。 最后，我们将通过REU让本科生接触到这些领域的研究。 我们让博士后参与REU活动，从而培训他们监督研究。几何学、拓扑学和动力系统是数学的核心领域。几何学通过曲率等不变量来研究空间的形状。 拓扑学探索在连续变形下保持不变的空间的性质。 一个基本的不变量，即亏格，是曲面中的“洞”的数量。 动力系统关注物理或数学系统随时间的演化。 特别是近年来，几何学和动力学在互利的互动中发展。 这方面的例子是拓扑学，几何学和复杂的动力学在低维，在许多方面相互反映。 这些领域中许多最令人兴奋的发展是真正相互关联的，并且通过想法，类比或实际工具相互受益。 与此同时，与群论、代数几何和数学物理等其他领域的联系也得到了极大的加强。 这些发展的广度和深度令人惊叹，并显示了这些领域的活力。

This proposal calls to continue our research training program in geometry, topology and dynamics at the University of Michigan. Recent Ph.D.'s and advanced graduate students will be the main beneficiaries. The Mathematics Department at UM has one of the largest and most vigorous post-doctoral and graduate programs in the country, with an excellent record of producing high-quality researchers in geometry, topology and dynamics. We will bolster the training of post-docs and graduate students in these areas by deepening and broadening their education and providing the trainees with ample opportunities to excel in their research. Five senior faculty members (Richard Canary, John Erik Fornaess, Gopal Prasad, Yongbin Ruan and Ralf Spatzier) will lead this project in collaboration with other senior faculty. We introduced various innovations into our current training program, and will continue them and expand on them: We will provide intensive exploratory seminars, lecture series and workshops. All trainees will develop lecturing skills, and will benefit from intensive mentoring. Some trainees will deepen their scientific training by traveling to other institutions at the forefront of research. Finally, we will expose undergraduates to research in the these areas via REU's. We involve postdocs in REU activities, thus training them to supervise research. Geometry, topology and dynamical systems are core areas of mathematics. Geometry investigates the shape of spaces, through invariants such as curvature. Topology explores the properties of spaces which remain invariant under continuous deformations. One basic such invariant, the genus, is the number of "holes" in a surface. Dynamical systems concern the evolution of a physical or mathematical system over time. Especially in recent years, geometry and dynamics have developed in mutually beneficial interaction. Case in point are topology, geometry and complex dynamics in low dimension which in many aspects mirror each other. Many of the most exciting developments in these areas are truly interrelated, and benefit from each other either by idea, analogy or actual tool. At the same time, connections with other fields such as group theory, algebraic geometry and mathematical physics have strengthened dramatically. These developments have been amazing in their breadth and depth, and demonstrate the vitality of these areas.