John Peca-Medlin




Thursday, January 24, 2019 - 12:00pm to 1:00pm



RH 340N
Viewed as a linear map on the integers, random integral matrices can be used to model many objects of interest outside probability, including class groups of number fields. Determining when a matrix is injective has been of interest for over half a century. Results by Komlos in the 1960s and later, followed then by more recent results by Tao and Vu and then Bourgain, Vu and P. M. Wood, established square Bernoulli matrices are injective with high probability. The next natural focus is to determine whether such a map is surjective. Recent results by M. M. Wood established the probability a square $n \times n$ matrix is surjective is asymptotically zero. So, what about the surjectivity of rectangular $n \times (n + u)$ matrices? Nguyen and Paquette show surjectivity holds almost surely when $u$ is sufficiently large for a large class of random integral models. So, what if $u$ is small? Nguyen and Wood then give the more general result that the cokernel of a rectangular matrix is isomorphic to a given finite abelian group, or is cyclic, using precise formulas with zeta functions, in agreement with distributions defined by Cohen and Lenstra. In particular, they show for $u = 1$, a rectangular matrix is surjective with positive probability $\prod_{k \ge 2} \zeta(k)^{-1} \approx 0.4358$. I will review the method and tools of Nguyen with Paquette and Wood, along with filling in some details from related questions.