Moduli Spaces of Galois Representations

Project: Research project

Project Details

Description

Overview. The PI proposes to conduct research in the p-adic Langlands program, which seeks to relate p-adic Galois representations of p-adic fields and smooth representations of p-adic Lie group. This program has undergone a seismic shift in the last two years, with the formulation of the “categorical p-adic Langlands conjectures”. This has redefined the main problems in the program, from seeking a “pointwise” relationship between individual smooth representations and individual Galois representations to seeking functors between appropriate categories of such objects. This emerging categorical perspective seems very powerful: not only it explains most of the seemingly pathological behaviors observed in the p-adic Langlands program over the last decade, but it also allows the formulation of the most conceptual form of the reciprocity between global Galois representations and automorphic forms manifested in the cohomology of locally symmetric spaces. The categorical p-adic Langlands program is currently in its infancy, and the PI believes the recent developements in the moduli theory of Galois representations (some due to the PI) have provided the necessary tools to explore this new landscape. Intellectual Merit. The PI expects that the categorical view will not just shed new light, but also lead to the right tools to attack major outstanding problems in the p-adic Langlands program. The main goals are to: • Analyze in detail the geometry of the moduli stack of n-dimensional p-adic Galois representations over a p-adic field, together with its collection of substacks cut out by p-adic Hodge theoretic conditions; especially investigating the emerging connection between such objects and objects in geometric representation theory. • Understand how the mod p smooth representation theory of p-adic Lie groups can be explicitly incarnated onto the moduli stack of p-adic Galois representations, and in particular to study how the phenomena in the smooth representation theory can be explained via geometry of the moduli stack on the Galois side. • Use the results from the first two points above to discover and establish structural features of the infinite level mod p cohomology of locally symmetric spaces, such as determination of mod p multiplicities and Gelfand-Kirillov dimension of its Hecke eigenspaces, and deducing consequences for the global Langlands program from these. Broader Impacts. The PI is currently supervising four PhD students. The emerging nature of the categorical p-adic Langlands program makes it fertile ground for exploring new phenomenons, as well as revisiting old observations from a new perspective, leading to many potential thesis topics. Some phenomelogical investigations will likely require substantial experimentation with concrete objects, and can serve as excellent research projects for advanced undergraduate students. The PI has actively engaged with the Vietnamese mathematical community, through the teaching of summer schools to co-organizing sessions for the AMS-VMS Joint Math Meetings. The PI is planning to run a special program on Number Theory in Vietnam during summer 2024, which will in particular include several mini-courses aimed at introducing advanced undergraduate students in Southeast Asia to modern developments in Number Theory. The PI also proposes several concrete steps towards broadening the participation of underrepresented groups and ensuring inclusivity in number theory.
StatusNot started
Effective start/end date7/1/236/30/26

Funding

  • National Science Foundation (DMS-2302619)

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