NEWS in the Department of Mathematics
Mark Blumstein - MS final examination
Date: Thursday, May 14, 2015
Place: Weber, 015
Time: 10:00 a.m.
Title: Dimension and Multiplicity of Graded Rings and Modules
Advisor: Dr. Jeanne Duflot
Dr. James Wilson
Dr. Joel Bacon
Abstract: This talk will summarize the work I have done this year for my masters thesis under the guidance of Dr. Duflot. The purpose of my thesis is to define a multiplicity in the category of graded modules and investigate its properties.
This work is guided largely by Serre's work in the local algebra case and, as in Serre, we take a two sided approach. From the commutative algebra perspective we will examine Samuel polynomials and Poincare series, and from the homological algebra side we consider Koszul complexes and their Euler characteristics. As in the local case, each of these approaches produces the same multiplicity. Much of our work has been dedicated to providing a rigorous account of the carryover from the local setting to the graded setting.
Now armed with a variety of ways to compute multiplicity, we prove our main theorem which considers how grading and multiplicity are linked. In particular, given any choice of homogeneous system of parameters for a graded module, the ratio of the multiplicity to the product of the degrees of the elements in the system of parameters is independent of the choice of homogeneous system of parameters.
Matthew Heine - MS final examination
Date: Friday, May 15, 2015
Place: Weber, 201
Time: 8:00 a.m.
Title: A Constrained Optimization Model for Partitioning Students into Cooperative Learning Groups
Advisor: Dr. Michael Kirby
Dr. Olivier Pinaud
Dr. Kimberly Henry
Abstract: The problem of the constrained partitioning of a set using quantitative relationships amongst the elements is considered. An approach based on constrained integer programming is proposed that permits a group objective function to be optimized subject to group quality constraints. A motivation for this problem is the partitioning of students, e.g., in middle school, into groups that target educational objectives. The method is compared to another grouping algorithm in the literature on a data set collected in the Poudre School District.