RTG: Geometry and Topology

这个研究培训小组项目的目标是建立一个全面的项目集合，以加强圣母大学几何和拓扑研究小组的本科生、研究生和博士后研究员的研究培训。人人数学系列讲座的目的是在整个圣母大学本科生社区中普及数学思想，从而增加人们对STEM职业的兴趣。通过数学俱乐部交换计划与其他机构的本科生进行更多的互动，通过与研究生定向阅读接触先进材料，以及暑期本科生研讨会，旨在让本科生体验数学研究的创造性方面。另一个项目，加上夏季本科生研讨会，将针对有才华的本科数学专业学生，特别是来自传统上代表性较低的群体，以增加数学研究生课程的多样性。初级研究生将有机会参加暑期项目，以复习必要的背景材料，并与年长的研究生建立关系。拓扑学和几何学系列课程和迷你工作坊将培养学生对拓扑学和几何学的统一认识，使博士后和高级研究人员以及研究生受益。指导活动将确保研究生和博士后助理为他们未来的教学和专业职责做好充分准备。五名研究人员的研究兴趣涵盖从几何和全球分析到同伦理论和量子场论的广泛领域。这个项目的研究活动有可能促进研究人员、他们的博士后同事和学生之间的独特合作，利用该小组广泛的数学专业知识。涉及环空间和场论的研究将为同伦理论中最先进的方法开发新的几何应用。这些主题与对允许正Ricci曲率的流形的研究相吻合，计划中的一项关于卡勒-爱因斯坦和其他特殊度量的研究与物理学有关。对随机莫尔斯函数的研究将发展概率和几何之间的联系。该项目的活动和计划将促进小组成员对几何和拓扑学的统一看法。

The goal of this Research Training Group project is to establish a comprehensive collection of programs to augment the research training of undergraduates, graduate students, and postdoctoral fellows in the geometry and topology research group at the University of Notre Dame. A Math for Everyone lecture series aims to popularize mathematical ideas within the entire Notre Dame undergraduate community, leading to increased interest in STEM careers. Greater interaction with undergraduates at other institutions through the Math Club exchange program, exposure to advanced material through directed reading with graduate students, and summer undergraduate workshops are intended to get undergraduates to experience the creative aspects of mathematical research. Another program, coupled to the summer undergraduate workshops, will target talented undergraduate mathematics majors, especially from traditionally underrepresented groups, to increase diversity in the mathematics graduate program. Beginning graduate students will have access to a summer program to review necessary background material and build relationships with older graduate students. A unified vision of topology and geometry will be cultivated by the topology and geometry survey series and mini workshops, benefiting postdocs and senior researchers as well as graduate students. Mentoring activities will ensure that graduate students and postdoctoral associates are well-prepared for their future teaching and professional duties.The five investigators have research interests covering a broad range of areas from geometric and global analysis to homotopy theory and quantum field theory. The research activities of this project have the potential to foster unique collaborations between the investigators, their postdoctoral associates, and students, taking advantage of the group's wide mathematical expertise. Research involving loop spaces and field theories will develop new geometric applications to state of the art methods in homotopy theory. These topics dovetail with investigations of manifolds that admit positive Ricci curvature, and a planned study of Kahler-Einstein and other special metrics has connections to physics. Research investigating random Morse functions will develop connections between probability and geometry. The project's activities and programs will foster a unified vision of geometry and topology among the group members.