RTG: Unified Training in Geometry and Topology

RTG：几何和拓扑的统一训练

• 批准号: 1148490
• 负责人: Daniel Freed
• 金额: $ 249.91万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1148490&HistoricalAwards=false
• 关键词: RTG; Unified; Training; Geometry; Topology

AbstractAward: DMS 1148490 Principal Investigators: Daniel S. Freed, Dan F. Knopf, Alan W. ReidThe University of Texas Geometry/Topology Research Training Grant (RTG) is a vertically integrated program to enhance the training of undergraduates, graduate students, and postdoctoral associates at the University of Texas and, through publication on the web, beyond. The thirteen faculty members will build on the success of our previous RTG grants and also initiate new activities. The latter include a Directed Reading Program pairing undergraduate students and graduate mentors; a new Journal Seminar for all graduate students in geometry and topology; a new series of First Cut Lectures on a variety of core topics, to be disseminated on the web; a new yearly Undergraduate Mathematics Conference; and a new series of Graduate Winter Schools for graduate students and postdocs. All RTG activities are designed around a unified perspective on geometry and topology. They include a healthy mix of ideas from low dimensional topology, algebraic geometry, differential geometry, global linear and nonlinear analysis, geometric group theory, geometric representation theory, and homotopy theory; often concepts and inspiration from theoretical physics play an important role as well. The RTG training activities are designed to promote this unified view of geometry and topology and, more broadly, of mathematics and science to young mathematicians.Mathematics, the study of Number and Space, lies at the foundation of science and technology. Geometry, the study of Space, has long had connections with and applications to the real world. Topology investigates, very roughly, global geometric properties of spaces which do not rely on local structure. There is great interplay between Geometry and Topology-there is not a sharp boundary between them. In the ancient world measurement was introduced to facilitate commerce; more advanced concepts, such as trigonometry, were introduced to understand the stars. In our modern world advanced ideas in Geometry underlie abstract theories in physics and also have more immediate practical implications, for example in navigation, computer graphics, etc. The large scale structures studied in topology appear in biological and physical science; there are new applications to data analysis and signal processing as well. Effective training of the next generation of Geometers and Topologists is vital for the continued development of these ideas, both within mathematics and for external applications. The University of Texas Geometry/Topology Research Training Grant promotes a unified view of the field. An open attitude towards ideas from all directions is essential for success with the challenges facing mathematics and science today. Through local activities in Austin, and via dissemination on the web, the RTG will enhance training at all levels.

摘要奖：DMS 1148490 首席研究员：Daniel S. Freed、Dan F. Knopf、Alan W. Reid德克萨斯大学几何/拓扑研究培训补助金 (RTG) 是一项垂直整合的计划，旨在加强德克萨斯大学本科生、研究生和博士后的培训，并通过网络发布来加强对其他学生的培训。 这 13 名教职员工将在我们之前 RTG 资助的成功基础上再接再厉，并发起新的活动。 后者包括将本科生和研究生导师配对的定向阅读计划；为所有几何和拓扑学研究生举办的新期刊研讨会；关于各种核心主题的新系列 First Cut 讲座将在网络上传播；新一年一度的本科生数学会议；以及针对研究生和博士后的一系列新的研究生冬季学校。 所有 RTG 活动都是围绕几何和拓扑的统一视角进行设计的。 它们包括低维拓扑、代数几何、微分几何、全局线性和非线性分析、几何群论、几何表示论和同伦论的思想的健康组合；通常，理论物理学的概念和灵感也发挥着重要作用。 RTG培训活动旨在向年轻数学家推广几何和拓扑的统一观点，更广泛地说，是数学和科学的统一观点。数学是对数字和空间的研究，是科学和技术的基础。 几何学是对空间的研究，长期以来与现实世界有着联系和应用。 拓扑学非常粗略地研究不依赖于局部结构的空间的全局几何特性。几何学和拓扑学之间存在着巨大的相互作用——它们之间没有明显的界限。 在古代世界，引入测量是为了促进商业。更先进的概念，例如三角学，被引入以了解恒星。 在我们的现代世界中，几何学中的先进思想是物理学中抽象理论的基础，并且也具有更直接的实际意义，例如在导航、计算机图形学等领域。拓扑学中研究的大规模结构出现在生物和物理科学中；数据分析和信号处理也有新的应用。 对下一代几何学家和拓扑学家的有效培训对于这些思想的持续发展至关重要，无论是在数学内部还是外部应用。 德克萨斯大学几何/拓扑研究培训补助金促进了该领域的统一观点。 对来自各个方向的想法采取开放的态度对于应对当今数学和科学面临的挑战取得成功至关重要。 通过在奥斯汀的当地活动以及网络传播，RTG 将加强各级培训。