Events

Title: The Integrality Gap for the Santa Claus Problem

Abstract: 

In the max-min allocation problem, a set of players are to be allocated disjoint subsets of a set of   indivisible resources, such that the minimum utility among all players is maximized.  In the restricted variant, also known as the Santa Claus Problem,  each resource (``toy'') has an intrinsic positive value, and each player (``child'') covets a subset of the resources. Thus Santa wants to distribute the toys amongst the children, while (to satisfy
jealous parents?) wishing to maximize the minimum total value of toys received by each child. This problem turns out to have a natural reformulation in terms of hypergraph matching.

Title: Jack Derangements

Abstract: For each integer partition $\lambda \vdash n$ we give a simple combinatorial formula for the sum of the Jack character $\theta^\lambda_\alpha$ over the integer partitions of $n$ with no singleton parts. For $\alpha = 1,2$ this gives closed forms for the eigenvalues of the permutation and perfect matching derangement graphs, resolving an open question in algebraic graph theory. Our proofs center around a Jack analogue of a hook product related to Cayley's $\Omega$--process in classical invariant theory, which we call \emph{the principal lower hook product}.

Title: Lineup polytopes and applications in quantum physics

Abstract:  To put it simply, Pauli's exclusion principle is the reason why we can't walk through walls without getting hurt. Pauli won the Nobel Prize in Physics in 1945 for the formulation of this principle. A few years later, this principle received a geometrical formulation that is still overlooked today. This formulation uses the eigenvalues of certain matrices (which represent a system of elementary particles, for example electrons). These eigenvalues form a symmetric geometric object obtained by cutting a hypercube: it is a hypersimplex.

Title: On the Adaptivity Gap of Stochastic Orienteering

Abstract: This talk highlights the stochastic orienteering problem, in which we are given a budget B and a graph G=(V,E) with edge distances d(u,v) and a starting vertex x. Each vertex v represents a job with a deterministic reward and a random processing time, drawn from a known distribution.

Title: Bipartite Matching in Almost-Linear Time and More

Abstract:  This talk will present an algorithm that computes maximum bipartite matchings in m^{1 + o(1)} time, and discuss its connections with optimization, graph algorithms, and data structures.

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.

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: 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 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: Recursions and Proofs in Coxeter-Catalan combinatorics

Abstract:  The collection of parking functions under a natural Sn-action (which has Catalan-many orbits) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups W; the main candidates are known as the W-non-nesting and W-non-crossing parking functions.