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.

该建议呼吁继续我们密歇根大学的几何，拓扑和动态研究培训计划。 最近的博士学位和高级研究生将成为主要受益者。 UM的数学部门是该国最大，最有活力的博士后和研究生课程之一，拥有出色的几何学，拓扑，拓扑和动态研究人员的良好记录。 我们将通过加深和扩大他们的教育并为受训者提供足够的研究机会来加强这些领域的后培训和研究生的培训。 五位高级教师（理查德·卡里（Richard Canary），约翰·埃里克·福纳斯（John Erik Fornaess），戈帕尔·普拉萨德（Gopal Prasad），扬宾·鲁恩（Yongbin Ruan）和拉尔夫·斯帕齐耶（Ralf Spatzier））将领导该项目与其他高级教师合作。 我们将各种创新引入了当前的培训计划，并将继续进行研究：我们将提供密集的探索性研讨会，讲座系列和讲习班。 所有学员都将发展讲课技巧，并将受益于密集的指导。 一些学员将通过前往研究最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揭露本科生在这些领域的研究。 我们将博士后参与REU活动，从而培训他们以监督研究。几何，拓扑和动态系统是数学的核心领域。几何形状通过曲率等不变性研究了空间的形状。 拓扑探讨了在连续变形下保持不变的空间的性质。 这样一个不变的基本不变属是表面中的“孔”数量。 动态系统涉及物理或数学系统的演变。 尤其是近年来，几何和动力学在互惠互动中发展了。 在低维中，拓扑，几何形状和复杂动力学中有一个很好的例子，这些动态在许多方面相互反映。 这些领域中许多最激动人心的发展都是真正相互关联的，并且通过想法，类比或实际工具彼此受益。 同时，与其他领域（例如群体理论，代数几何形状和数学物理学）的联系已显着加强。 这些发展的广度和深度令人惊讶，并证明了这些领域的活力。