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# Recent Progress in Nonabelian Cohen--Lenstra Program

## Connections Workshop: Diophantine Geometry February 02, 2023 - February 03, 2023

February 02, 2023 (11:00 AM PST - 12:00 PM PST)
Speaker(s): Yuan Liu (University of Illinois at Urbana-Champaign)
Location: MSRI: Simons Auditorium, Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video
The nonabelian Cohen—Lenstra program studies the distribution of the Galois group of maximal unramified extensions of a family of global fields. In this talk, we will discuss some new developments for this type of question. We will introduce a cohomological invariant of a Galois extension of $\mathbb{F}_q$. We show that by keeping track of this invariant we can generalize the nonabelian Cohen—Lenstra Heuristics given by Liu, Wood and Zureick-Brown to cover the case when the base field contains extra roots of unity; and moreover, we show that the new conjecture is a nonabelian generalization of the work by Lipnowski, Tsimerman and Sawin. We will prove the conjecture with large $q$ limit, and discuss how to make a similar conjecture for number fields.