RTG: Geometry and Topology

这个研究训练小组项目的目标是建立一个全面的项目集合，以增加圣母大学几何和拓扑研究小组的本科生、研究生和博士后的研究训练。“人人都能学数学”系列讲座旨在在整个圣母大学本科生群体中普及数学思想，从而提高人们对STEM职业的兴趣。通过数学俱乐部交流项目与其他机构的本科生进行更多的互动，与研究生一起通过定向阅读接触高级材料，以及夏季本科生研讨会，旨在让本科生体验数学研究的创造性方面。另一个项目，结合夏季本科生研讨会，将针对有才华的数学专业本科生，特别是来自传统上代表性不足的群体，以增加数学研究生项目的多样性。初学研究生将有机会参加暑期课程，复习必要的背景材料，并与年长的研究生建立关系。通过拓扑和几何调查系列和小型研讨会，将培养拓扑和几何的统一视野，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为他们未来的教学和专业职责做好充分准备。这五名研究员的研究兴趣涵盖了从几何和全局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者、他们的博士后同事和学生之间独特的合作，利用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的最新方法开发新的几何应用。这些主题与对承认正里奇曲率的流形的研究相吻合，并且对Kahler-Einstein和其他特殊度量的计划研究与物理学有关。对随机莫尔斯函数的研究将发展概率论和几何学之间的联系。该项目的活动和方案将在小组成员中培养对几何和拓扑的统一看法。

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.