Events

Title: The rank of sparse symmetric matrices over arbitrary fields

Abstract: Consider a sequence of sparse Erdös-Rényi random graphs (G_{n,d/n})_n on n vertices with edge probability d/n. Moreover, we equip the edges of G_{n,d/n} with prescribed non-zero edge weights chosen from an arbitrary field F.

Title: Diagonal coefficients of Kirchhoff polynomials of 2k-regular graphs and the proof of the c_2 completion conjecture

Abstract: I have for many years been interested in graph invariants with the same symmetries as the Feynman period. Recently Erik Panzer found a new such invariant coming from a particular coefficient of the Martin polynomial. Together we used this to prove an over 10 year old conjecture on an arithmetic graph invariant known as the c_2 invariant, and came to understand that diagonal coefficients of Kirchhoff polynomials tie together many of the known graph invariants with the symmetries of Feynman periods and unlock previously inaccessible proofs.

Joint work with Erik Panzer.

Title: A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut Covering Problem, Part II

Abstract: A cut in a graph \(G = (V, E)\) is a set of edges which has precisely one endpoint in \(S\), for a given subset \(S\) of \(V\). The fractional cut-covering number is the optimal value of a linear programming relaxation for the problem of covering each edge by a set of cuts. We define a semidefinite programming relaxation of fractional cut covering whose approximate optimal solutions may be rounded into a fractional cut cover via a randomized algorithm.

Title: Restricted Intersections and the Sunflower Problem

Abstract: A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erdos and Rado showed that a family of sets of size $n$ contains a sunflower if there are more than $n!(r-1)^n$ sets in the family. Alweiss et al. and subsequently Rao improved this bound to $(O(r \log(rn))^n$.

Title: Minors of random representable matroid over finite fields

Abstract: Consider a random n by m matrix A over GF(q) where every column has k nonzero elements, and let M[A] be the matroid represented by A. In the case that q=2, Cooper, Frieze and Pegden (RSA 2019) proved that given a fixed binary matroid N, if k is sufficiently large, and m/n is sufficiently large (both depending on N), then whp. M[A] contains N as a minor. We improve their result by determining the sharp threshold (of m/n) for the appearance of a fixed q-nary matroid N as a minor of M[A], for every k\ge 3, and every prime q. This is joint work with Peter Nelson.

Title: Ramsey degrees, big and small

Abstract: Many of the seminal results in finite Ramsey theory can be phrased by saying that a certain class of finite structures has the Ramsey property, such as the ordinary finite Ramsey theorem (the class of finite linear orders), the dual Ramsey theorem (the class of finite lex-ordered Boolean algebras), the Graham-Leeb-Rothschild theorem (the class of lex-ordered, finite-dimensional vector spaces over a fixed finite field), and the Nesetril-Rodl theorem (the class of finite ordered triangle-free graphs, among many others).

Title: Douglas–Rachford Algorithm for Control- and State-constrained Optimal Control Problems

Abstract: The Douglas - Rachford algorithm has been applied to many optimization problems due to its simplicity and efficiency but the application of this algorithm to optimal control is less common. In this talk we utilize this method to solve state- and control-constrained linear-quadratic optimal control problems.

Title: Modular representations of the symmetric group and categorification (part I)

Abstract: I'll give two talks on the representations of the symmetric group over small finite fields, in particular, their block structure, with an emphasis on the perspective from categorical actions of Lie algebras.  No previous background in modular representation theory will be assumed.  

Title: Ramsey Cayley graphs, random graph models, and information theory

Abstract: A graph is Ramsey if its largest clique or independent set is of size logarithmic in the number of vertices. While almost all graphs are Ramsey, there is still no known explicit construction of Ramsey graphs. Alon conjectured that every finite group has a Ramsey Cayley graph.

Title: Budget Feasible Mechanisms

Abstract: In the setting of budget feasible mechanism design, a buyer wants to purchase items from a set of agents. Each agent can supply at item at an incurred cost of c_i to themself, and the buyer wants to optimize over their own valuation for the set of items bought. The cost c_i is private information that the buyer doesn't have access to. The goal is to design a mechanism that is truthful, in the sense that the sellers do not have incentive to deviate from reporting their true costs, and budget feasible, in the sense that the total payments made to the sellers is within some budget B.