Education is not the filling of a pail, it is the lighting of a fire. - W. B. Yeats

 

NEWS in the Department of Mathematics
Graduate Announcements

 

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

Committee:
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


Committee:
Dr. Olivier Pinaud
Dr. Kimberly Henry

 

Melody Dodd PhD preliminary examination:

Date: Tuesday, May 19, 2015
Place: Weber, 201
Time: 9:00 a.m.


Title: Computational Advancements in the D-bar Reconstruction Method for 2-D Electrical Impedance Tomography

Advisor: Dr. Jennifer Mueller


Committee:
Dr. Olivier Pinaud
Dr. Margaret Cheney
Dr. Branislav Notaros


Abstract:
In this preliminary exam, I describe two projects dealing with algorithmic and computational improvements to the numerical D-bar method for 2-D electrical impedance tomography. The first is an implementation of the D-bar algorithm that has been optimized to run in real time. The second involves the introduction of \emph{a priori} information into the D-bar algorithm to provide improved spatial resolution in EIT images, which has so far been applied to simulated data. I conclude with an outline of future work to extend the \emph{a priori} algorithm to clinical data.


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