GWU Mathematics Department Graduate Student Seminar
FALL 2006 - Seminar Presentations


Ken Shoda, 8 December 2006.

An Introduction to Matroids

What is a matroid? This is the introduction to matroid theory with many visualizations. Topics include some basic concepts and diagrams, vector matroids, graphic matroids, transversal matroids, affine and projective geometries, duality, cyclic flats and more.



Sarah Pingrey, 1 December 2006.

A Theorem on Strong Degree Spectra

We will take a short tour through computable model theory in order to understand the statement and importance of a theorem in one of my recent papers. Starting with a short introduction to model theory, we will then discuss linear orderings. Next, we will go over some basic computability theory and attempt to combine everything together at the end of the talk. This is joint work with John Chisholm, Jennifer Chubb, Valentina Harizanov, Denis Hirschfeldt, Carl Jockusch, and Timothy McNicholl.

Slides here.

Hillary Einziger, 20 November 2006.

The Incidence Hopf Algebra of Finite Lattices

In this talk, I will define and give examples of the algebra and coalgebra structures on a family of finite lattices. I will then show how these structures define a Hopf algebra structure on the bialgebra. The definition of the coproduct leads to both a recursive and an explicit formula for the antipode in the Hopf algebra, which can then be used to find an expression for the inverse of any multiplicative function in the incidence algebra -- in particular, it can be used to derive P. Hall's formula for the Mobius function, by considering it as the inverse of the Zeta function. I will conclude with several results concerning formulae for the antipode of the incidence Hopf algebra of particular familes of lattices.



Tanya Andress, 10 November 2006.

Invariants for Discrete Dynamical Systems

I will show how a map, called substitution, on a finite alphabet can generate a discrete dynamical system (X,T), where X is a topological space and T is a transformation from the space to itself. In our case X is always homeomorphic to a Cantor set and therefore we study Cantor dynamical systems.

It is well knows that all Cantor spaces are homeomorphic, therefore to classify Cantor dynamical spaces arising from the substitutions, algebraic invariants are used that somehow incorporate the transformation T. For example, the dynamical cohomology group that is a group of all continuous integer valued functions on X modulo coboundary or the functions that are related via transformation f - fT .

I will discuss my recent results that for some substitutions this group is just a finitely generated free abelian group, and also the generators of this group can be linked to the measures of certain sets of X.



Michael J. Coleman, 3 November 2006.

Population Models with Ordinary Differential Equations

Population modeling is a common application of ordinary differential equations and can be studied even the non-linear case. We will investigate some cases of differential equations beyond the separable case and then expand to some basic systems of ordinary differential equations. The phase line and phase plane will be used to assist in plotting the solutions of these systems and consequently, to aide understanding the behavior of a hypothetical environment over time.

Slides here.

Radmila Sazdanovic, 27 October 2006.

An Introduction to Knot Theory

We start with basic introduction to knot theory: definitions, various notation and knot invariants such as colorings and polynomials. In addition we state several interesting open problems and experimental results obtained using computer program LinKnot.



Milena Pabiniak, 20 October 2006.

An Introduction to Symplectic Topology

Part I: Linear symplectic geometry.

The first part of this talk will be devoted to simplectic vector spaces and symplectic maps between them. This will be an elementary, linear algebra talk. Only basic knowledge about matrices is required.

Part II: Symplectic manifolds.

We will apply notions from Part I to tangent spaces of manifolds. We will begin with a bit of calculus on manifolds and introduce tangent spaces, differential forms, and derivatives of a form. Then, using this language, we will define a symplectic manifold (a manifold with non-degenarate, closed 2-form, called a symplectic structure). We will see examples of such manifolds as well as non-trivial examples of a manifold which do not allow symplectic structure.

We will analyze symplectomorphisms (maps between symplectic manifolds which preserve symplectic structure of our basic example of a symplectic manifold. The most famous theorems by Darboux, Eliashberg and Gromov will be given (without proofs:))

Slides here.

Fanny Jasso-Hernandez, 13 October 2006.

How to use Homology Groups to Store Information of Knots and Graphs

This talk aims to introduce some basic notions on knot theory invariants and some algebraic concepts that will help to understand the ideas related with my research (with Yongwu Rong). Even though the motivation of this work comes from knot theory, our work could be classified only in the field of graph theory invariants. I will explain the ideas of our construction of homology groups associated to graphs.

If time allows I will show a nice connection between planar graphs and a certain kind of links and how they relate (only partially, so far) in our framework. This is a consequence of a Theorem by Laure, Jozef and Yongwu.

Slides here.

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