GWU Mathematics Department Graduate Student Seminar
SPRING 2008 - Seminar Presentations

Frank Baginski, Magda Musielak, and Yongwu Rong, 2 May 2008.

Annual Employment Search Panel

For the second year in a row, we are hosting a seminar/discussion, which aims to help mathematics graduate students start informing themselves and thinking about the job search process. While the discussion will be most immediately relevant to those who will apply for jobs within a year or two, it is never too early for all graduate students to think, for instance, about what they want to be able to put on their CVs when their turn in the job market comes. Both academic and non-academic job searches will be discussed.

The format will be a very interactive panel discussion which will be directed, in large part, by questions from the audience. One aspect that generally keeps such discussions lively is that the panelists may well disagree on some points. Indeed, there are many valid perspectives on job hunting and few definitive answers-- the important thing is to start thinking about the issues.

More information than could possibly be addressed in one seminar is available at: http://www.gwu.edu/~math/graduate/jobweb.html. All are welcome. Come with your questions!


Robert Allen, George Mason University. 25 April 2008.

Isometries on the Bloch Space

Let X be a Banach space of analytic functions on the unit disk of the complex plane. The problem of characterizing the isometries from X to X, that is the linear operators on X which preserve the norm, is open for most spaces. In this talk, I will discuss what is known of the isometries when X is the Bloch space. Also, I will discuss current research in the characterization of isometries amongst specific types of operators. This is joint work with Flavia Colonna of George Mason University.


Karen Lange, University of Chicago. 17 April 2008.

The Relative Strength of the Atomic and Homogeneous Model Theorems

Reverse mathematics and computable model theory both attempt to measure the strength of classical mathematical principles, the former from a proof-theoretic perspective and the latter from a computable one. I will present computability results on the Homogeneous Model Theorem (HMT), an existence theorem for general model-theoretic structures found throughout mathematics. I will discuss how these results provide insight into the reverse mathematical strength of HMT. In particular, I will discuss how HMT compares with the Atomic Model Theorem studied by Hirschfeldt, Shore, and Slaman.


Cindy Merrick, 11 April 2008.

An Introduction To Lie groups, Tangent Spaces, and Lie Algebras

By using the common example of the special unitary matrices SU(n), I will walk through the definitions and constructions that show the general result that every Lie Group gives rise to a Lie Algebra via a tangent space. Bring only your basic knowledge of algebra and matrices.


Hillary Einziger, 4 April 2008.

A Forest Formula for the Antipode in Incidence Hopf Algebras

In 2005, Figueroa described a Zimmerman-type forest formula for the antipode in the incidence Hopf algebra of distributive lattices. In this talk, I will describe a related definition of the forests of a lattice to arrive at a formula for the antipode in any incidence Hopf algebra of lattices. This definition can also be extended to a formula for the antipode in any incidence Hopf algebra of posets.


Diane Holcomb, 14 March 2008.

A Complex Simplification of Differential Geometry

There are close connections between geometric and analytic approaches to exploring the complex plane. Understanding these connections can lead to a greater understanding of both approaches. After a brief review of important results in Complex Analysis and Differential Geometry we will explore connections between the two topics. This will include discussion of the Riemannian metrics, geodesics, and curvature as well as two interpretations of Cauchy's Theorem.


Sara Miller Quinn, Notre Dame. 7 March 2008.

A Tour of Computable Structure Theory

Computable structure theory is a branch of mathematical logic in which the algorithmic complexity of algebraic structures is examined. In other words, we determine how complicated it is to answer certain questions about algebraic structures, such as: "Are these two structures isomorphic?" There is a hierarchy we use to rank the complexity of these questions. In this talk I will give all of the necessary definitions, and then describe the kinds of questions that we ask in computable structure theory.


Tyler White, 29 February 2008.

Inverse Limits of One-dimensional Tiling Spaces and their Cech Cohomology

Anderson and Putnam showed that the cohomology of a 1-dimensional aperiodic tiling space relates to the direct limit of the transition matrix for the corresponding substitutions that "forces its border". Recent work by Barge and Diamond extended this by using a modified complex that will produce the cohomology of a more general substitution tiling space. We will discuss the process of computing the cohomology of 1-dimensional tiling spaces and provide examples demonstrating this method.


Shari Wiley, Howard University. 22 February 2008.

Impact of Harvesting in a Discrete Time Predator-prey Model

A discrete-time model is used to study the impact of fishing exploitations in a predator-prey system. This work was motivated by Basson and Fogarty, who used an aggregate production model with Ricker-type recruitment functions to account for interspecific interactions and exploitation. In this research, we generalize their model to study recruitment mechanisms that exhibit both compensatory (equilibrium) and over-compensatory (oscillatory) dynamics. We explore the implications of these different dynamics on the long-term survival of the exploited predator and prey species. This is joint work with Dr. Abdul-Aziz Yakubu and Dr. Michael Fogarty.


Lakeshia Legette, Howard University. 8 February 2008.

Maximal Groups in the Stone-Cech Compactification of a Discrete Semigroup

Much is known about the size of the Maximal Groups of the Stone-Cech Compactification of N, the natural numbers. It has been shown that they each contain a copy of the free group on 2c generators. However, they may also be as small as a copy of Z, the integers. I will show that when S is a free semigroup, the Stone-Cech Compactification may be a single point.


Joseph E. Bonin, 1 February 2008.

An Introduction to Matroid Theory through Lattice Paths

This first (and longer) part of the talk, which is based on joint work with Anna de Mier and Marc Noy or Universitat Politecnica de Catalunya, will introduce you to matroid theory through very concrete examples that arise from lattice paths. Fix lattice paths P and Q that go from (0,0) to (m,r), with P never going above Q, and consider the lattice paths from (0,0) to (m,r) that stay in the region that P and Q bound. We show that these paths can be identified with the bases of a special type of matroid-- a lattice path matroid. Many important invariants (such as the Tutte polynomial) that are #P-hard to compute for arbitrary matroids have natural interpretations for lattice path matroids, and these interpretations yield polynomial-time algorithms for computing the invariants.

The second part of this talk uses the ideas discussed in the first part to give a bird's eye view of some of the major research areas in the field: representability over fields, well-quasi-ordering, Tutte polynomials, extremal matroid theory, matroid constructions, and more.

Click here for a PDF version of this abstract.

Yongwu Rong, 25 January 2008.

Some Applications of Pure Mathematics

Mathematics is beautiful! Mathematical is useful! These are two kinds of talks that I have wanted to give. This talk will focus on the usefulness of mathematics, especially pure mathematics. It is also an advertisement for our new seminar on mathematical applications.

I will briefly mention some successful stories of pure mathematicians in this direction, including Fields Medalists Michael Freedman (topology to quantum computation), David Mumford (algebraic geometry to image processing), and Terry Tao (harmonic analysis to "compressive sampling"). Then I will explain more details on some specific topics, chosen from foldings in biology, digital topology, combinatorial optimization, and network complexity.


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