Seminar

Title: Sylvester, Gallai, and their complex relatives

Abstract: Given any finite set of points in the real plane, not all collinear, there is a line in the plane that contains exactly two of them. This pretty result was conjectured by Sylvester in 1893 and proved by Gallai in 1944. We will present an extension of the result to higher dimensional complex spaces and discuss some related conjectures. This is joint work with Matthew Kroeker.

Title: Stochastic Minimum Norm Combinatorial Optimization

Abstract: In this work, we introduce and study stochastic minimum-norm optimization. We have an underlying combinatorial optimization problem where the costs involved are random variables with given distributions; each feasible solution induces a random multidimensional cost vector. The goal is to find a solution that minimizes the expected norm of the induced cost vector, for a given monotone, symmetric norm.

Title: A raising operator formula for Macdonald polynomials

Abstract: In this talk, I will give a brief introduction on Catalanimal, a tool that helps us to prove the shuffle theorem under any line, the extended delta conjecture and the Loehr- Warrington conjecture. I will then focus on its variant "Macanimal" which gives us an explicit raising operator formula for the modified Macdonald polynomials. Our method just as easily yields a formula for an infinite series of $GL_l$ characters which truncates to the modified Macdonald polynomials.

Title: Hadamard’s Maximal Determinant Problem and Generalisations

Abstract:  Any matrix $M$ of order $n$ with entries taken from the complex unit disk satisfies Hadamard’s determinantal inequality $|\det M|\leq n^{n/2}$. Matrices meeting this bound with equality have pairwise orthogonal rows and columns. Such matrices are known as Hadamard matrices, and character tables of finite abelian groups give examples at every order.

Title: The Probabilistic Set-Covering Problem

Abstract: In the classical set-covering problem, we have a set of items and a set S of subsets of the items. The objective is to find a min-cost subset C of S which covers every item, i.e., where every item is contained in at least one of the subsets in C. The probabilistic set-covering problem (PSC) generalizes this to a stochastic setting where the objective is to find a min-cost covering which covers a random subset of the items with probability at least p. We will discuss some structural properties of this problem which lead to a branch-and-bound algorithm for solving it.

Title: The ADMM:  Past, Present, and Future

Abstract: Over the past 15 years, the alternating direction method of multipliers (ADMM) has become a standard optimization method.  This talk will cover the origins of the ADMM, its subsequent development, and what to expect in the future.

Title: An introduction to discrete quantum walks

Abstract: A discrete quantum walk is determined by a unitary matrix representation of a graph. In this talk, I will give an overview of different quantum walks, and show how the spectral information of the unitary matrix representation links properties of the walks to properties of the graphs. Part of this talk will be based on the book, Discrete Quantum Walks on Graphs and Digraphs, by Chris and me.

Title: Quasisymmetric functions,  descent sets, immaculate tableaux, and 0-Hecke modules

Abstract:

The first half of this talk will be expository and devoted to a discussion of (quasi)symmetric functions and tableaux.

We define new families of quasisymmetric functions, in particular the new basis of row-strict dual immaculate functions, with an associated cyclic, indecomposable 0-Hecke algebra module. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution \psi on the ring Qsym of quasisymmetric functions. We uncover the remarkable properties of the immaculate Hecke poset induced by the 0-Hecke action on standard immaculate tableaux, revealing other submodules and quotient modules, often cyclic and indecomposable.

Title: Integer programs with bounded subdeterminants and two nonzeros per row

Abstract: Determining the complexity of integer linear programs with integer coefficient matrices whose subdeterminants are bounded by a constant is currently a very actively discussed question in the field. In this talk, I will present a strongly polynomial-time algorithm for such integer programs with the further requirement that every constraint contains at most two variables. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than k vertex-disjoint odd cycles, where k is any constant. Previously, polynomial-time algorithms were only known for k = 0 (bipartite graphs) and for k = 1.

This is joint work with Samuel Fiorini, Gwenaël Joret, and Yelena Yuditsky, which recently appeared at FOCS this year.

Title: Approximate Counting via Lorentzian Polynomials and Entropy Optimization

Abstract: Over the past 20 years, Lorentzian and real stable polynomials have been used to derive a number of combinatorial theorems, from log-concavity statements to counting and volume bounds. One significant thread of this research lies in the utilization of entropy optimization methods to approximately count certain combinatorial objects, such as the matchings of a bipartite graph, the intersection of the sets of bases of two matroids, and the integer points of various polytopes in general. In this talk, we will discuss various results one can achieve using such methods.