Department of Mathematics,
University of California San Diego
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Math 211A: Algebra Seminar
Urshita Pal
University of Michigan, Ann Arbor
The generalized Lee--Szczarba conjecture on the cohomology of principal congruence subgroups
Abstract:
I will discuss the rational cohomology of $SL_n(R), Sp_{2n}(R)$, and their principal congruence subgroups for $R$ a number ring. Borel--Serre showed that these groups satisfy a (co)homological duality that lets us study their cohomology groups via certain representations called the `Steinberg modules’, which have a beautiful combinatorial description in terms of Tits buildings. I will describe a conjecture of Lee--Szczarba on the top cohomology of principal congruence subgroups of $SL_n(Z)$, and its resolution due to Miller--Patzt--Putman. I will then discuss forthcoming work on generalizations of this to other Euclidean rings, and also to symplectic groups.
Host: Karthik Ganapathy
February 23, 2026
3:00 PM
APM 7321
Research Areas
Algebra Combinatorics Geometry and Topology Number Theory****************************
