# 2008-2009 Colloquium

Dr. Mark Fels, Mathematics & Statistics, Utah State University
Friday, August 14, 2009 11:00 - 12:00 p.m. in RB-2024

Title:    Group Actions and Differential Equations

A symmetry group of a differential equation is a group which acts on the set of solutions to the differential equation.  Therefore I will begin by reviewing some aspects from the theory of group actions on sets (orbits, stabilizers, etc.) using some basic examples. The notions developed for group actions on sets will then be considered in the context of symmetry groups of differential equations.

I will then address how group actions can be used to possibly simplify finding solutions to the equation. For example the notion of a fixed point of a group action leads to the so-called invariant or equivariant solutions.
A number of examples will again be given.

Utilizing symmetry to study the solution space to a differential equation also leads to a number of interesting geometric problems such as the principle of symmetric criticality, and the inverse problem of quotients.
I will explain what some of these problems are, and demonstrate them with some examples.

A second orbit type are those where the action is free. This leads naturally to the theory of quotients for differential equations. If time permits, I will introduce the theory of quotients and how this is related to the idea of a non-linear superposition principle.

Dr. Robert L. Taylor, Mathematical Sciences, Clemson University
Friday, June 19, 2009,  11:00 a.m. - 12:00 p.m. in RB-2024

Title:  Consistency and Validity of Dependent Nonparametric Bootstrap Estimators

The traditional bootstrap resamples with replacement from the original sample observations to form arrays of rowwise independent and identically distributed bootstrap random variables.  There are situations, for example, when sampling from finite populations, where resampling without replacement provides a more realistic bootstrap procedure and produces dependent bootstrap random variables.  The desired properties of consistency and asymptotic validity are shown to hold for certain nonparametric dependent bootstrap estimators.   In addition, it is shown that the smaller variation in dependent bootstrap estimators can be used to increase precision in some of the estimates even in the traditional i.i.d. setting.

Tuesday, March 31, 2009  1:30 p.m. in RB-2023

Title:    Two-Stage Flexible Open-Shop Scheduling with Preemptions

We study a flexible version of two-stage open-shop scheduling problem where the second stage has m identical machines in the parallel setting. Jobs are to be processed once on the first stage, and once on any machine in the second stage. Preemptions are allowed if there is any benefit. The objective is to minimize the makespan so that all jobs are to be finished as soon as possible.
This is a generalization of classical two-machine open-shop problem and m parallel machine problem with preemptions, which are both polynomially solvable. In this study, we first present a lower bound for the optimal makespan of the problem, and then we design an algorithm that achieves this bound. The algorithm is of complexity O(mn), hence the problem is shown to be polynomially solvable.
Finally, a cost analysis is performed to give some guidelines in choosing the number of parallel machines in the second stage.
This work, as part of the master project, has been accepted for presentation at the 39th International Conference on Computers & Industrial Engineering (CIE39) in France, July 2009.
Coffee and refreshments to follow.

Dr. Marshall Hampton, University of Minnesota, Duluth
November 20, 2008, 1:30 - 2:30 in RB-1047

Three years ago William Stein released the first version of Sage, a computational platform based on the popular scripting language Python which aims to create "a viable free open-source alternative to Magma, Maple, Mathematica and Matlab".  Sage unifies a huge collection of mathematical software projects into a coherent and powerful system for mathematics, statistics, and scientific computation.  In this talk I will give an overview of its present capabilities and future directions.

Dr. Walter Whitely, Professor of Mathematics, York University
October 31, 2008, 1:30 - 2:30 in RB-2044

Title:    Geometry and Combinatorics for Protein motions and rigidity.

Walter Whiteley, Professor of Mathematics, at York University is a Member of the Graduate Programs in Mathematics, in Education, Computer Science and Interdisciplinary Studies.

We will outline why protein flexibility and rigidity matters in biochemistry: for function, disease and treatment.  We will then describe a mathematical approach based on rigidity theory which uses geometry and combinatorics to analyze a single 'snap shot' structure (Protein Data Bank model) to produce fast redictions what parts are rigid and flexible. The approach depends on a critical 'molecular conjecture' made by Tay and Whiteley in 1981 which we will describe.  Under an NIH grant, this approach has been implemented in an on-line public program FIRST at Arizona State University.  We will show some sample predictions, and some comparison with alternative (slower - sometimes 1 million times slower) - alternatives. This work is an example of a wider theme of relevance to mathematics education, and mathematics programs: the essential role of geometry and related combinatorics for a wide array of applications, and the importance of interdisciplinary collaborations to address these problems.