Peter Patzt
University of Oklahoma, OK·Mathematics & Statistics
Groups are mathematicians' way of describing symmetry. The general linear groups capture the symmetries of linear spaces and can be thought of as groups of invertible matrices.
- Fields
Mathematics & Statistics
- Topics
group cohomology
general linear groups
homological stability
- Grants
- 1 known grant, through May 2027
What they work on
- Study cohomological and representation stability for general linear groups and their principal congruence subgroups, which are matrix groups defined by a divisibility condition. Describe how the cohomology behaves as matrix size grows.
- Pursue the Church–Farb–Putman conjecture that the cohomology of special linear groups over the integers vanishes in fixed codimension for large matrices. Proving it up to codimension 4 would show the twelfth K-group of the integers is zero.
- Establish homological stability for congruence subgroups of odd-dimensional symplectic groups, which is tied to twisted stability for braid groups. This would bear on the Conrey–Farmer–Keating–Rubinstein–Snaith conjecture about quadratic L-functions.
More about the research
Group cohomology is a tool that records information about these symmetries, a bit like a refined way of 'counting holes.' It connects to number theory, geometry, K-theory and quantum field theory. This project looks for patterns in the cohomology of linear groups as several parameters change. These parameters are the cohomological degree, the size of the matrices, and the specific group and ring of numbers used.
Why it matters
This is pure mathematics. Finding stability patterns in the cohomology of linear groups adds to the understanding of symmetry in algebra and topology. Progress could help settle open problems in number theory, including a conjecture on averages of L-functions and part of the Kummer–Vandiver conjecture. It could also inform related areas such as K-theory and geometry.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Jeremy Miller
Purdue University, IN·Mathematics & Statistics
Prove homological stability and its variants, meaning that the hole-counting homology settles down as spaces grow, for classifying spaces of arithmetic groups, using tools such as locally symmetric spaces and homological algebra.
Aaron Landesman
Harvard University, MA·Mathematics & Statistics
Make progress on the Cohen-Lenstra conjectures about class groups of quadratic fields over function fields by computing relevant homology groups with new topological tools.
Andrew Putman
University of Notre Dame, IN·Mathematics & Statistics
Extend results on stable homology, the algebraic hole-counting invariants that settle down as groups grow, from mapping class groups and related groups to their finite-index subgroups.
The school
University of Oklahoma: 11 labs on PI Pages, $474M a year on research, 225 research doctorates a year.
Reach out
A start for your email. Fill in the brackets.