GWU Mathematics Department Graduate Student Seminar
FALL 2007 - Seminar Presentations

Michael Moses, 6 December 2007.

Compactness in Mathematical Logic

Two well known mathematical languages, the classical one of Aristotelian Logic and the everyday one of First Order Logic, are compact in that, for any (infinite) collection of sentences in these languages, if every finite sub-collection 'is satisfiable', then so is the whole collection. It is surprisingly easy to see why these languages are compact. Even more surprising are the ramifications of their compactness, in all areas of mathematics, from classical analysis to modern combinatorics and, most especially, in mathematical logic, where it has some unsettling things to say about the mathematical languages that we employ, and the mathematical proofs that we seek. I will begin this talk with an historical exploration of compactness and its connection with the corresponding topological concept from which it gets its name, follow this with examples of the use of compactness in the non-standard analysis of Abraham Robinson and the probabilistic method of Paul Erdos, and close with a brief discussion of what compactness says about mathematical languages and proofs.


Sarah Pingrey, 29 November 2007.

Computability and the Halting Problem

Computability theory was invented in the 1930's by Turing, Gödel, Church, and Kleene, among others, who gave formal definitions of a computable function. All of these definitions turned out to be equivalent. The Church-Turing Thesis, generally accepted by all computability theorists, says that the formal notion of a computable function captures the intuitive idea of a computable function. I will discuss what it means for a function, set, and relation to be computable, and computably enumerable and then define the Turing degrees. The Turing degree of a set says how uncomputable a set is. Then, we will discuss the halting problem, which is a natural example of a set that is computably enumerable but not computable. This talk is aimed at undergraduates.


Michele Friend, GWU Department of Philosophy. 16 November 2007.

An Introduction to the Realist and Constructivist Philosophies of Mathematics

I shall be outlining the basic positions which fall under "realism" and "constructivism" in logic/mathematics. I shall also be giving a quick set of indicators, so you can tell which one you are talking to! I'll then outline some of the motivations for both positions and examine some contentious formal arguments.


John B. Conway, 1 November 2007.

Matrices and Topology


Jennifer Chubb, 25 October 2007.

An Algorithmic Apporach to Linear Orderings

Starting with first principles, I'll talk about properties of linear orderings and relations on linear orderings in the context of computability theory. After some examples and a brief survey of research in this area, I'll explain some new results. This talk will be accessible to undergraduates.


Larry Chang, 18 October 2007.

An Introduction to de Rahm's Theorem

In 1931, Georges de Rham proved a theorem identifying the de Rham cohomology groups as topological invariants. That is, using de Rham cohomology, one is able to express topological data about smooth manifolds as representations in their cohomology classes. de Rham's theorem therefore states that for a smooth manifold M, the k-dimensional de Rham cohomology groups are isomorphic as real vector spaces with singular cohomology groups with coefficients in R. In this talk, we will develop the necessary background from singular homology and cohomology, then generalize to smooth singular homology, and finally, with the aid of Stokes' Theorem, present an outline for the proof of de Rham's theorem.


Larry Chang, 11 October 2007.

An Introduction to Riemannian Geometry

Given a smooth manifold, a natural question to ask is: How can "distance" be measured? The answer relies on nothing more than understanding how to equip a smooth manifold with an inner product. Thus we aim to define such a structure, examine what it gives us, and what applications are available.


Ken Shoda, 27 September 2007.

Non-isomorphic Matroids having the same Tutte Polynomial and Lattice of Cyclic Flats

The Tutte polynomial is one of the most important invariants of the theory of matroids. Starting with a special lattice of cyclic flats, one can generate super-exponential families of matroids having the same Tutte Polynomial. Also there is an interesting connection to lattice-path matroids.


Kerry Luse, 20 September 2007.

The Transition Polynomial and Chord Diagrams

The transition polynomial for a graph G was defined by F. Jaeger in 1990. In a joint paper with Yongwu Rong, we define a transition polynomial for signed chord diagrams. This polynomial encodes the information necessary to recover the genus of the surface associated to the chord diagram.


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