GWU Mathematics Department Graduate Student Seminar
Fall 2009 - Seminar Presentations

Radmila Sazdanovic

Date: Friday, 20 November 2009   at   11:30 AM

Title: Chromatic Graph Cohomology and Khovanov Link Homology

Abstract: Chromatic graph cohomology was introduced by Helme–Guizon and Rong as a comultiplication free version of the Khovanov cohomology of alternating links, where alternating link diagrams are translated to plane (Tait) graphs. In this talk I will outline both constructions, explain the relations between them and related results about torsion in homology.


Lu Xie, 6 November 2009

An Introduction to and Solutions of Laplace's Equation

In this talk, I will first show an example of the Dirichlet problem for Laplace's equation in the two dimensional case. Then I will derive the radial solutions of Laplace's equation. In the latter part of this talk, I will focus on some basic properties of harmonic functions which we use to study the classical Dirichlet problem of Laplace's equation.


Forest Fisher, 23 October 2009

Monoidal Categories

Monoids are one of the simplest algebraic structures in mathematics, even more elementary than groups. In the introduction to his book, Categories for the Working Mathematician, Saunders MacLane outlines a way to describe monoids in the language of category theory. This talk will be an introduction to monoidal categories, those categories where the appropriate structure exists for defining a monoid. We will discuss several concrete examples and no prior knowledge of category theory will be necessary.


Tyler White, 16 October 2009

Introduction to Substitution Dynamical Systems

In this talk, I will extend on a previous talk by Joseph Herning which covered basic properties of symbolic dynamical systems. I will give examples of different classes of symbolic dynamical systems, including subshifts which are of finite type and those which are sofic. I will then primarily focus on symbolic dynamical systems which arise from substitution maps. I will show that these types of systems satisfy certain properties, such as having zero entropy.


Joseph Herning, 9 October 2009

Introduction to Symbolic Dynamics

Symbolic dynamical systems arise naturally from the consideration of discrete-time dynamical systems but are interesting in their own right as well. I will illustrate this through elementary examples such as irrational rotations, beta-maps, and shift spaces. I will define basic terms in the field such as topological mixing, subshifts, and topological entropy. Focusing mainly on topological properties, I will conclude by proving the Curits-Lydon-Hedlund Theorem which states all semi-conjugacies between subshifts are realized by a simple map called a block code.


Valentina Harizanov, 2 October 2009.

Constructing Computable Linear Orderings with Non-Computable Propreties

We will present an example of a construction of a computable linear ordering with certain non-computable properties. The example will (in a very simple setting) illustrate the main ideas and give the flavor of a computability theoretic technique called the finite injury priority method. This method was invented in the 1950's and represents the first level in the hierarchy of priority methods, which revolutionized computability theory.


Ken Shoda, 25 September 2009.

Introduction to Matroid Theory and Lattices of Cyclic Flats

We will begin with an introduction to Matroid theory and some important concepts including the Tutte Polynomial. This will be followed by some interesting results. Omer Gimenez showed how to construct, for each permutation of [n], a matroid on 4n+5 elements so that all n! resulting matroids are nonisomorphic but have isomorphic lattices of cyclic flats of width 2. We show that these matroids in fact have the same Tutte polynomial. Thus, this gives a super-exponential family of nonisomorphic matroids having isomorphic lattices of cyclic flats and the same Tutte polynomial. Note that this talk utilizes the same material as that given at DCMGSM.


Mike Coleman, 18 September 2009.

MATLAB Software and Programming for Numerical Analysis

After a brief introduction to MATLAB, the program designers and other facts about its history, we will explore some techniques and short cuts that are useful for programming. This seminar will also cover methods for MATLAB that are useful in performing numerical computations. We will investigate some of the ODE solvers in addition to other programming techniques that are necessary for implementing them.


Lowell Abrams, 11 September 2009.

What are Self-dual Graph Embeddings and How can I Construct Them?

Get set for an exciting mix of symmetry, topology and graph theory! This completely introductory talk will explain everything you need to know (from the ground up!) to appreciate recent work of Abrams and Slilaty which answers the second question in the title


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