Events

Title: Stochastic Probing with Applications

Abstract: We will explore a stochastic probing problem. Given a set of elements which have weights and independent probabilities of being "active," the goal is to construct a subset of active elements of maximum weight. To form such a set, we must "probe" elements sequentially to determine whether they are active.

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: 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: 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.