Saturday 11/14/2020 (Eastern Time)
9:00 – 9:50. Van H. Vu (*)
Title: Recent results in combinatorial random matrix theory
Abstract: We are going to survey recent progress on the main combinatorial problems in random matrix theory.
10:00 – 10:50. Jane Pu Gao
Title: The rank of sparse random matrices
Abstract: We determine the rank of a random matrix over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low- density parity check codes. The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations. This work is collaborated with Amin Coja-Oghlan, Alperen Ergur, Samuel Hetterich and Maurice Rolvien.
11:00 – 11:50. Phillip Matchett Wood
Title: A replacement principle for perturbations of non-normal matrices
Abstract: There are certain non-normal matrices whose eigenvalues can change dramaticallywhen a small perturbation is added. However, when that perturbation is an iid random matrix,it appears that the eigenvalues become stable after perturbation and only change slightly whenfurther small perturbations are added. Much of the work is this situation has focused on iidrandom gaussian perturbations. In this talk, we will discuss work on a universality result thatallows for consideration of non-gaussian perturbations, and that shows that all perturbationssatisfying certain conditions will produce the same limiting eigenvalue measure. Interestingly,this even allows for deterministic perturbations to be considered. Joint work with SeanO’Rourke.
12:00 – 1:00 Break
1:00 – 1:50. Steven Evans
Title: The fundamental theorem of arithmetic for metric measure spaces
Abstract: A metric measure space is a complete, separable metric space equipped with a probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov–Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup. Using an explicit family of continuous semicharacters, we establish that each element has a unique factorization into prime elements. This is joint work with Ilya Molchanov (University of Bern).
2:00 – 2:50. Konstantin Tikhomirov
Title: Singularity of random discrete matrices
Abstract: In this talk, I will discuss two recent results, one dealing with singularity of +1/-1 square random matrices, the other (joint work with A.E.Litvak) — singularity of 0/1 matrices with i.i.d. Bernoulli(p) entries.
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Sunday 11/15
10:00 – 10:50. Melanie Matchett Wood
Title: Universality classes for cokernels of random integral matrices
Abstract: We will give an overview of many important universality classes of random integral matrices, based on their limiting cokernel distributions. We will discuss several kinds of random matrix models that are proven to fall into these universality classes, and present many open questions about other kinds of random matrix models.
11:00 – 11:50. András Mészáros
Title: The distribution of sandpile groups of random regular graphs
Abstract: We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the $p$-Sylow subgroup of the sandpile group is a given $p$-group $P$, is proportional to $|Aut(P)|^{-1}$. For finitely many primes, these events get independent in the limit. Similar results hold for undirected random regular graphs, where for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.
This answers an open question of Frieze and Vu whether the adjacency matrix of a random regular graph is invertible with high probability. Note that for directed graphs this was recently proved by Huang.
12:00 – 12:50. Nathan Kaplan
Title: Cokernels of random matrices and distributions of finite abelian p-groups
Abstract: We will discuss distributions on finite abelian p-groups that arise from taking cokernels of families of random p-adic matrices. We will focus on questions about the moments of these distributions, and when individual distributions occur as specializations within larger families.
1:00 – 2:00 Break
2:00 – 2:50. Andrew Newman
Title: Cohen—Lenstra heuristics for torsion in homology of random simplicial complexes
Abstract: This talk will discuss the “torsion burst” in the Linial–Meshulam random simplicial complex model. Considering the Linial–Meshulam model as a stochastic process one finds that immediately before the first nontrivial top-dimensional cycle appears, an enormous torsion group appears in the homology of the random complex. In this talk we discuss results of extensive computational experiments which strongly suggest that the torsion groups that appear follow a natural Cohen–Lenstra distribution. The reasons for this behavior to occur are still unknown leading to several questions and conjectures. This is joint work with Matthew Kahle, Frank H. Lutz, and Kyle Parsons.
3:00 – 3:50. Sam Payne
Title: Random graphs and random curves
Abstract: I will present some probability distributions on graphs of first Betti number g that arise naturally from local coordinates on moduli spaces of curves, focusing on the aspects involving the combinatorics of graphs and their associated Laplacian matrices. Based on this framework, I will suggest some potential models for “the Jacobian of a random totally degenerate curve of genus g” that may be worthy of further investigation using techniques from random integer matrix theory.
(*) The speaker will not be able to present his talk; one of the organizers will go over the speaker’s slides instead.
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Workshop virtual link
https://osu.zoom.us/j/99922820269
Meeting ID: 999 2282 0269
Password: please email either nguyen.1261@osu.edu or elliot.paquette@gmail.com