What is the probability that a finite m-torsion R-module H arises as the m-torsion of a "random" finite R-module?

For the case where R is a (Noetherian) non-singular domain of dimension 1, a celebrated work of Cohen and Lenstra essentially says that if a random finite module M is distributed with the probability inversely proportional to the size of its automorphism group Aut(M), then the probability in question is asymptotically equal to (1/#Aut(H))(1 - 1/q)(1/q^2)(1 - 1/q^3)..., where q = #(R/m). In this talk, I will survey several different ways to produce a distribution of R-modules when R = F_q[t], the polynomial ring over the finite field of q elements. Strikingly, all these different distributions produce the same answer to the above question. This is joint work with Yifeng Huang and Zhan Jiang.

If time permits, I will explain some enlightening approaches to answering why various distributions follow the same probability measure. This is by answering a version of "Hausdorff moment problem", and this is due to a combination of the works by Clancy-Kaplan-Leake-Payne-Wood, Ellenberg-Venkatesh-Westerland, and Fulman-Kaplan. If more time permits, I will also share some questions for the audience to think about (at home), which, in my best knowledge, are still open.

Any undergraduates are encouraged to attend! Speaker(s): Gilyoung Cheong (University of Michigan)