RTG: Unified Training in Geometry and Topology

摘要奖：DMS 1148490主要研究员：丹尼尔·S·弗里德、丹·F·克诺夫、艾伦·W·赖德德克萨斯大学几何/拓扑学研究培训助学金(RTG)是一个垂直整合的计划，旨在加强德克萨斯大学本科生、研究生和博士后助理的培训，并通过在网上发布，超越。这13名教职员工将在我们之前RTG助学金成功的基础上继续开展新的活动。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的期刊研讨会；将在网上传播的关于各种核心主题的新的First Cut系列讲座；新的年度本科生数学会议；以及新的研究生和博士后冬季研究生院系列。所有RTG活动都是围绕几何和拓扑的统一视角设计的。它们包括来自低维拓扑、代数几何、微分几何、全局线性和非线性分析、几何群论、几何表示理论和同伦理论的健康混合的思想；通常来自理论物理的概念和灵感也发挥着重要作用。RTG培训活动旨在向青年数学家推广这种几何和拓扑观以及更广泛的数学和科学观。数学是研究数字和空间的学科，是科学和技术的基础。几何学是对空间的研究，长期以来一直与现实世界联系在一起，并在现实世界中得到应用。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性质。几何学和拓扑学之间有很大的相互作用-它们之间没有明显的边界。在古代，测量被引入以促进商业；更先进的概念，如三角学，被引入以了解星星。在我们的现代世界中，几何学的先进思想是物理抽象理论的基础，也具有更直接的实际意义，如在导航、计算机图形学等方面。拓扑学研究的大规模结构出现在生物和物理科学中，在数据分析和信号处理方面也有新的应用。对下一代地质学家和拓扑学家的有效培训对于继续发展这些思想至关重要，无论是在数学内部还是在外部应用。德克萨斯大学几何/拓扑学研究培训补助金促进了对该领域的统一看法。面对当今数学和科学所面临的挑战，对来自各个方向的想法持开放态度是取得成功的关键。通过在奥斯汀的当地活动和在网上传播，区域培训小组将加强各级培训。

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.