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

EMSW21-RTG：几何、拓扑和动力学方面的研究人员培训

• 批准号: 1045119
• 负责人: Ralf Spatzier
• 金额: $ 250万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1045119&HistoricalAwards=false
• 关键词: EMSW21; RTG; Training; Research; Workforce

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.

该提案要求继续在密歇根大学进行几何、拓扑和动力学方面的研究培训计划。 应届博士生和高级研究生将是主要受益者。 密歇根大学数学系拥有全国规模最大、最具活力的博士后和研究生项目之一，在培养几何、拓扑和动力学方面的高素质研究人员方面有着出色的记录。 我们将通过深化和拓宽他们的教育，并为学员提供充足的机会在研究中取得优异成绩，来加强这些领域的博士后和研究生的培训。 五位高级教员（理查德·卡纳利、约翰·埃里克·福纳斯、戈帕尔·普拉萨德、阮永斌和拉尔夫·斯帕齐尔）将与其他高级教员合作领导该项目。 我们在当前的培训计划中引入了各种创新，并将继续这些创新并加以扩展：我们将提供密集的探索性研讨会、系列讲座和研讨会。 所有学员都将培养讲授技能，并将受益于强化指导。 一些学员将通过前往其他处于研究前沿的机构来加深他们的科学培训。 最后，我们将通过 REU 让本科生接触这些领域的研究。 我们让博士后参与 REU 的活动，从而培训他们监督研究。几何、拓扑和动力系统是数学的核心领域。几何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属性。 一个基本的不变量（即属数）是表面上“孔”的数量。 动力系统涉及物理或数学系统随时间的演变。 特别是近年来，几何学和动力学在互惠互利的相互作用中发展。 典型的例子是低维拓扑、几何和复杂动力学，它们在许多方面相互镜像。 这些领域中许多最令人兴奋的发展都是真正相互关联的，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急剧加强。 这些发展的广度和深度都令人惊叹，并展示了这些领域的活力。