MPS-Ascend: The C*-Algebraic Mackey Bijection for Real Reductive Groups

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。PI Angel Roman被授予国家科学基金会数学和物理科学提升博士后研究奖学金（NSF MPS-Ascend），以开展一项与扩大STEM中代表性不足的群体参与相关的研究和活动计划。罗曼博士获得的这一奖学金支持他在一名赞助科学家的指导下进行的名为“真实还原群的C*-代数麦基投影”的研究项目。该奖学金的主办机构是密苏里州圣路易斯的华盛顿大学，赞助科学家是唐翔博士。本项目旨在通过C*-代数理论和C*-代数的连续域更详细地研究实约群的缓和的、不可约的酉表示和其Cartan运动群的酉不可约表示之间的麦基双射。在PI先前研究的基础上，PI将与合作者合作，利用嵌入结果和麦基对应，更好地理解Harish-Chandra Schwartz代数的循环同调，这是非交换几何中的一个研究对象。扩大参与的活动有三个方面。PI将与波多黎各的研究人员合作，在波多黎各组织面向学生的会议，并帮助指导拉丁美洲的学生。利用在宾夕法尼亚州立大学担任跆拳道教练时所使用的领导力发展技能，PI将通过“非交换几何和表示理论”的工作研讨会参与研究生和博士后学者的专业发展。最后但并非最不重要的是，作为一名听力不佳的数学家，PI为这个社区所面临的问题带来了宝贵的见解，包括数学技术术语的转录和说明。PI将与加劳德特大学（Gallaudet University）合作开发更好的资源。加劳德特大学是一所面向聋人和听力障碍者的一流大学，设有数学本科课程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). PI Angel Roman is awarded a National Science Foundation Mathematical and Physical Sciences Ascending Postdoctoral Research Fellowship (NSF MPS-Ascend) to conduct a program of research and activities related to broadening participation by groups underrepresented in STEM. This fellowship to Dr. Roman supports the research project entitled "The C*-Algebraic Mackey Bijection for Real Reductive Groups", under the mentorship of a sponsoring scientist. The host institution for the fellowship is Washington University in St. Louis, Missouri, and the sponsoring scientist is Dr. Xiang Tang. This project seeks to examine the Mackey bijection between tempered, irreducible unitary representations of a real reductive group and the unitary irreducible representations of its Cartan motion groups in greater detail through the theory of C*-algebras and through continuous fields of C*-algebras. Building on PI’s prior research, the PI will work with collaborators to better understand the cyclic homology of Harish-Chandra Schwartz algebras, an object of study in noncommutative geometry, using the embedding result and Mackey correspondence. The broadening participation activities are three-fold. The PI will collaborate with researchers in Puerto Rico to organize student-oriented conferences in Puerto Rico and help mentor Latin American students. Using the leadership development skills the PI used as a Taekwondo coach at Penn State, the PI will engage in graduate student and postdoctoral scholar's professional development through a working seminar on “Noncommutative geometry and Representation Theory.” Last but not the least, as a Hard-of-Hearing mathematician, the PI brings valuable insight to the issues faced by this community with transcription and captions for technical terms in mathematics. The PI will work with Gallaudet University, a premier university for the Deaf and Hard-of-Hearing with an undergraduate program in mathematics, in developing better resources.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.