RTG: Unified Training in Geometry and Topology

摘要奖：DMS 1148490主要研究者：丹尼尔S. Freed，Dan F.作者：Alan W. Reid德克萨斯大学几何/拓扑研究培训补助金（RTG）是一个垂直整合的计划，以加强本科生，研究生和博士后助理在德克萨斯大学的培训，并通过在网络上发布，超越。 十三名教师将建立在我们以前的RTG赠款的成功，并启动新的活动。 后者包括一个定向阅读计划配对本科生和研究生导师;一个新的期刊研讨会的所有研究生在几何和拓扑学;一个新的系列的第一次削减讲座的各种核心主题，将在网络上传播;一个新的年度本科生数学会议;和一个新的系列研究生冬季学校的研究生和博士后。 所有RTG活动都是围绕几何和拓扑的统一观点设计的。 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、几何表示论和同伦理论的思想的健康组合;通常理论物理的概念和灵感也起着重要的作用。 RTG培训活动旨在向青年数学家推广几何和拓扑学的统一观点，更广泛地说，推广数学和科学的统一观点。 几何学，空间的研究，长期以来一直与真实的世界有联系和应用。 拓扑学非常粗略地研究不依赖于局部结构的空间的整体几何性质。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，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。 在我们的现代世界先进的思想几何基础抽象理论的物理学，也有更直接的实际影响，例如在导航，计算机图形学等大规模结构拓扑学研究出现在生物和物理科学;有新的应用数据分析和信号处理以及。 有效的培训下一代的几何和拓扑是至关重要的继续发展这些想法，无论是在数学和外部应用。 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。 对来自各个方向的思想采取开放的态度是成功应对当今数学和科学所面临的挑战的关键。 通过在奥斯汀的当地活动，并通过网络传播，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.