Combinatorics and Algebra Seminar-Random Groups in Combinatorics and Number Theory and Cokernels of Random Matrices

Title: Random Groups in Combinatorics and Number Theory and Cokernels of Random Matrices

Speaker: Nathan Kaplan, UC Irvine
Date and time: Friday, April 29, 3–4 pm
Place: Phillips 736

Abstract: How many sublattices of Zn have index at most X? If we choose such a lattice Λ at random, what is the probability that Zn/Λ is cyclic? Now let R be a random subring of Zn. What is the probability that Zn/R is cyclic?

Let G be an Erdős-Rényi random graph with n vertices and edge probability q where 0 < q < 1 is a fixed constant. As n goes to infinity, what is the probability that the number of spanning trees of G is odd?

We will explain how these problems and others like them lead to interesting questions about distributions of cokernels of families of random matrices over the integers and over the p-adic integers.