MPS-Ascend: Eigenvector Statistics of the Adjacency Operator of Graphs

Mckenzie, Theo R, Berkeley CA

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). PI Theo McKenzie 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. McKenzie supports the research project entitled "MPS-Ascend: Eigenvector Statistics of the Adjacency Operator of Graphs", under the mentorship of a sponsoring scientist. The host institution for the fellowship is Harvard University, and the sponsoring scientist is Dr. Hong-Tzer Yau. The project's goal is to study the structure of eigenvectors of operators on graphs. The PI will work in three research directions: deduce eigenvector and eigenvalue statistics such as the number of nodal domains of random and semi-random graphs using the combinatorial structure of the graph to prove eigenvector delocalization; bound the number of eigenvalues near the edge of the spectrum for operators on graphs and manifolds using PI's prior research; and generalize known results in quantum ergodicity to graph families with arbitrary limits to further elucidate the connection between eigenvector statistics of a graph family and those of the infinite limit. PI's broadening participation activities are two-fold. One, the PI will leverage his experience as a lead instructor at the associate degree program at San Quentin State Prison, to teach and introduce a more advanced curriculum in the college programs at prisons in Massachusetts, that have a high population of racial minorities. This project aims to directly help those who are incarcerated learn higher level math and thereby gain agency. Second, using his own experience as a Black mathematician, create a community of postdocs and graduate students from underrepresented backgrounds at Harvard. This group will interact with each other and work on personal development activities thereby creating a welcoming environment for researchers. 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.