Joint AWM-MOCA Speaker Series 2018-19

Lewis & Clark College

Date: October 9, 2018

Website: https://sites.google.com/a/lclark.edu/ncameron/

Inversion generating functions for signed pattern avoiding permutations

Abstract: We consider the classical Mahonian statistics on the set B_n(\Sigma) of signed permutations in the hyperoctahedral group B_n which avoid all patterns in \Sigma, where \Sigma is a set of patterns of length two. In 2000, Simion gave the cardinality of B_n(\Sigma) in the cases where \Sigma contains either one or two patterns of length two and showed that |B_n(\Sigma)| is constant whenever |\Sigma|=1, whereas in most but not all instances where |\Sigma|=2, |B_n(\Sigma)|=(n+1)!. We answer an open question of Simion by providing bijections from B_n(\Sigma) to S_{n+1} in these cases where |B_n(\Sigma)|=(n+1)!. In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on B_n(21, \bar{2}\bar{1}) and by giving the major index on D_n(\Sigma) for \Sigma ={21, \bar{2}\bar{1}} and \Sigma={12,21}. The main result of this paper is to give the inversion generating functions for B_n(\Sigma) for almost all sets \Sigma with |\Sigma|\leq 2.

Sandia National Laboratories

Date: November 6, 2018

Website: https://sites.google.com/site/martadeliawebsite/

Nonlocal models in computational science and engineering: challenges and applications

Abstract: Nonlocal continuum theories such as peridynamics and nonlocal elasticity can capture strong nonlocal effects due to long-range forces at the mesoscale or microscale. For problems where these effects cannot be neglected, nonlocal models are more accurate than classical Partial Differential Equations (PDEs) that only consider interactions due to contact. However, the improved accuracy of nonlocal models comes at the price of a computational cost that is significantly higher than that of PDEs. In this talk I will present nonlocal models and the Nonlocal Vector Calculus, a theory that allows one to treat nonlocal diffusion problems in almost the same way as PDEs.

Furthermore, I will present current open challenges related to the numerical solution of nonlocal problems and show how we are currently addressing them. Specifically I will describe an optimization-based local-nonlocal coupling strategy and briefly introduce a technique to improve the performance of Finite Element (FE) approximations.

The goal of local-nonlocal coupling methods is to combine the computational efficiency of PDEs with the accuracy of nonlocal models. These couplings are imperative when the size of the computational domain or the extent of the nonlocal interactions are such that the nonlocal solution becomes prohibitively expensive to compute, yet the nonlocal model is required to accurately resolve small scale features. Our approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. I will present consistency and convergence studies and, using three-dimensional geometries, I will also show that our approach can be successfully applied to challenging, realistic, problems.

Finally I will introduce a new concept of nonlocal neighborhood that helps improving the performance of FE methods and show how our approach allows for fast assembling in two- and three-dimensional computations.

Cornell University

Date: April 9, 2019

Website: https://people.orie.cornell.edu/melewis/

Constrained Optimization for Scheduling in Multi-Class Queueing

Abstract: We consider the problem of scheduling a single-server when there are multiple parallel stations to serve. A classic result in scheduling says to create a station dependent index consisting of the product of the holding cost (per customer, per unit time) times the rate at which the service can be completed at that station. The scheduler then prioritizes work in the order of the indices from highest to lowest. Preferences are captured by the various holding costs.

A more natural method for modeling preferences is to assign constraints to the highest priority customers (guaranteeing a fixed quality of service level) and to provide best effort service for the other classes. We consider this formulation, present conditions for optimality and show how to construct an optimal control. We initially focus on the two station model, then explain where the results can be extended. Applications to patient flow in health care are discussed.

Bates College

Date: May 3, 2019

Website: https://sites.google.com/view/asalerno/

Hypergeometric decomposition of symmetric K3 quartic pencils

Abstract: In this talk, we will show the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives. This is joint work with Charles F Doran, Tyler L Kelly, Steven Sperber, John Voight, and Ursula Whitcher.