Patterns in the Cohomology of Linear Groups

Groups are the mathematician's way of talking about symmetries. The general linear groups encapsulate the symmetries of linear spaces. Group cohomology encodes important information about these symmetries and the linear spaces themselves using a conceptual version of "counting holes". There are deep mathematical connections to many areas of mathematics and physics, including number theory, geometry, K-theory, and quantum field theory. The research will reveal patterns emerging in the cohomology of linear groups. The grant will also support conferences organized by the PI that bring together researchers of various backgrounds as well as outreach programs in Oklahoma exposing students to engaging mathematics. The group cohomology of linear groups has multiple parameters: The cohomological dimension, the size of the matrices, the precise group and ring. Different patterns emerge when varying one or multiple of theses parameters. For example, cohomological stability for the general linear groups means that for fixed cohomological dimension, the cohomology groups stay the same for large sizes of matrices. Principal congruence subgroups defined by a congruence condition on the matrices satisfy a more complicated pattern called representation stability. For fixed cohomological codimension, Church-Farb-Putman conjecture that the cohomology groups of the special linear groups of the integers vanish for large matrix sizes. For congruence subgroups in fixed cohomological codimension, one can ask about stability patterns akin to representation stability. The PI proposes to study a variety of those patterns with specific applications in mind: The Conrey–Farmer–Keating–Rubinstein–Snaith Conjecture on the asymptotics of moments of quadratic L-functions for function fields follows from uniform twisted homological stability for braid groups, which in turn are closely related to homological stability for certain congruence subgroups of odd-dimensional symplectic groups. Part of the Kummer-Vandiver Conjecture is to proof that twelfth K-group of the integers is zero. This would follow from Church-Farb-Putman Conjecture for all codimensions up to 4. This project is jointly funded by the Topology and Geometric Analysis program (TGA) and the Established Program to Stimulate Competitive Research (EPSCoR). 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.