Moduli of Galois representations and applications

Project: Research project

Project Details

Description

OVERVIEW:
The PI proposes to conduct research on the theory of p-adic representations of Galois groups of p-adic fields and its connection with Modular representation theory of finite groups and Geometric representation theory. The main goals are:
-Provide a general method to compute many potentially crystalline Galois deformation rings for GLn via the study of new moduli spaces related to Geometric representation theory.
-Establish many cases of Serre weight conjectures for unitary groups and beyond, discover and prove the correct version of the conjecture in the wildly ramified situation.
-Investigate the local nature of various structures on mod p cohomology of locally symmetric manifolds at infinite level, thus leading to new understanding of a sought-after mod p Langlands correspondence.
-Use insights from Galois representations to discover new combinatorial and numerical patterns on affine Springer fibers.
INTELLECTUAL MERIT:
The proposed research creates new links between two well-developed fields of mathematics, via the discovery and study of new objects. As a result, one can apply techniques and insights from one field to solve previously intractable problems of the other field, and transfer conjectures and intuition across to discover new phenomena. This rich interaction will be of great interest to researchers in both fields.
BROADER IMPACTS:
The proposed project leads to a stream of interesting questions that are suitable for a wide range of students: from computational experiments on Macaulay 2 and Sage that require minimal background, to research problems suitable for PhD students. The PI plans to recruit several undergraduate students for summer research, and taking PhD students at Northwestern University.
At University of Chicago, the PI supported graduate students by helping them running study groups, supervised undergraduate math majors in their independent studies, and co-organized the Number Theory Seminar at University of Chicago. The PI expects to continue similar activities at Northwestern University.
This project, if funded, will also support the PI to visit various international collaborators in Canada and Europe, thus promoting international collaboration.
The PI has been giving various mini-courses during summer schools and seminars organized by the Vietnames Institute for Advanced Studies in Mathematics (VIASM) over several years, to popularize modern mathematics in Vietnam. Some of the lecture notes have been published by VIASM, and all have been made available online. The PI also has experience in advising high school students in pursuing further study in mathematics, and is interested in doing this more systematically through outreach programs.
StatusActive
Effective start/end date8/1/187/31/21

Funding

  • National Science Foundation (DMS-1802037-002)

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