GWU Mathematics Department Graduate Student Seminar
SPRING 2007 - Seminar Presentations

E.A. Robinson, 4 May 2007.

Thirty-six Views of the Rauzy Fractal

How do you draw a picture of an algebraic number? This talk discusses a method for drawing a fractal set that is robustly associated with a certain type of algebraic integer called a Pisot number. This fractal set is amazingly robust and can be drawn using several of the most familiar fractal drawing algorithms.


Barbara Nimershiem, 27 April 2007.

Hyperbolic Geometry meets Number Theory

Amazingly, number theorists' Markoff spectrum (defined using binary quadratic forms) has connections to closed geodesics on the punctured torus (a beautiful hyperbolic 2-manifold). I will describe these connections and summarize results from three papers: one by Carolyn Series characterizing simple geodesics on the punctured torus, one by David Crisp and William Moran about geodesics with single self-intersections, and one by David Crisp, et al. on doubly self-intersecting geodesics. I will conclude by explaining why I'm particularly interested in the connection between number theory and hyperbolic geometry and how I hope these results will relate to hyperbolic 3-manifolds.


Michael J. Coleman, 20 April 2007.

Introduction to Sobolev Spaces and Embeddings

This presentation will feature an introduction to Sobolev Spaces and its associated notion of weak differentiability. We will approach this topic from some fimilar functional analysis concepts including the Lp, Banach and Hilbert Spaces. Relations and embeddings of the Sobolev Spaces will also be introduced as well as some elementary applications to PDE Theory.


Timothy McNicholl, 30 March 2007.

Computable Aspects of Inner Functions

The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also very useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u, the order of the zero at 0 (if there is any) and the Blaschke sum of u provide the exact amount of information necessary to compute the factorization of u. Along the way, we discuss some uniform computability results for Blaschke products. These results play a key role in the analysis of factorization. We use Type II Effectivity as our foundation.

Slides here.

Sarah Pingrey and Jennifer Chubb, 9 March 2007.

Each presentor gave a half hour talk on joint work with John Chisholm, Valentina Harizanov, Denis Hirschfeldt, Carl Jockusch, and Tim McNicholl.

Complexity of Relations on Computable Structures (Sarah Pingrey)

Let A be a computable structure and R be an additional relation on A. The Turing degree spectrum of R on A is the set of all Turing degrees of the images of R under all isomorphisms from A to computable models. Harizanov found a sufficient and necessary condition for the Turing degree spectrum to contain all Turing degrees, which can be relativized to truth-table reducibility. She also showed that the Turing degree spectrum of the ω-part of a linear ordering of type ω+ω* is all of the Δ 02 degrees. This is not the case for tt- degrees, and it can be shown that there is a computable enumerable set D such that D is not weak truth-table reducible to any initial segment of any computable scattered linear order.

Strong Reducibilities, Scattered Linear Orderings, Ranked Sets, and Kolmogorov Complexity (Jennifer Chubb)

A linear order is scattered if it fails to contain a copy of the rationals. We consider the truth-table and weak truth-table degrees of initial segments of scattered computable linear orders, and show that there is a c.e. set that is not wtt-reducible to any initial segment of any computable scattered linear order. Here, we use algorithmic information theory to establish this result.

Slides for the later talk are here.

Fanny Jasso-Hernandez and Kerry Luse, 23 February 2007.

A Basic Introduction to some Useful LaTeX Packages

Among the many useful things one can do with LaTeX are inserting multiple types of figures and diagrams, from simple to complicated, and producing professional looking presentations. To do this, you can use packages like Beamer and XY-pic. We will explain some basics associated with using these packages. We are far from experts on this subject, but we want to share with you what we have learned so far.

Slides here.

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