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

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。皮安吉尔·罗曼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的参与相关的研究和活动。这项给罗曼博士的奖学金支持了一位赞助科学家的指导下的研究项目，该项目名为《实还原群的C*-代数Mackey双射》。该奖学金的主办机构是密苏里州圣路易斯市的华盛顿大学，赞助科学家是唐翔博士。这个项目试图通过C*-代数的理论和C*-代数的连续域，更详细地研究实可约群的调和的、不可约的酉表示与其Cartan运动群的酉不可约表示之间的Mackey双射。在PI以前研究的基础上，PI将与合作者合作，利用嵌入结果和Mackey对应更好地理解Harish-Chandra Schwartz代数的循环同调，Harish-Chandra Schwartz代数是非对易几何的研究对象。扩大参与的活动有三个方面。国际和平协会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学生。利用PI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将通过一个关于“非对易几何和表示理论”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一点是，作为一名听力迟钝的数学家，PI为这个社区所面临的问题带来了宝贵的洞察力，为数学中的技术术语提供了转录和说明。PI将与加拉德特大学合作开发更好的资源。加拉德特大学是一所顶尖的聋人和听力障碍大学，设有数学本科课程。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.