A shot of Pedernales Falls on July 4, 2004.
I am interested in geometric and combinatorial group theory, specifically filling invariants related to the word problem, and small cancellation theory, specifically examining the curvature of presentations. Geometric group theory is the study of groups as geometric objects. There are two ways to approach this: we can either study the geometry of a group based on how it acts on geometric spaces, or we can study the group as a geometric space in its own right, via the Cayley graph. The Wikipedia entry for geometric group theory is a decent introduction to the topic. For a complete introduction (and semi-mastery) of the topic, I recommend "Metric Spaces of Non-Positive Curvature," by Martin Bridson and Andre Haefliger.
In June 2006 I passed my advanced oral examination. My syllabus for the exam is here. At left is a shot of the Taj Mahal...it has nothing to do with geometric group theory, or my advanced exam syllabus.
I attended the University of Arkansas Spring Lecture Series in Geometric Group Theory, with principal speaker Martin Bridson, April 4-8, 2006. The link for the website is http://www.uark.edu/depts/mathinfo/sls-2006/ . I compiled the open problems presented at the conference and they are available here.
The Geometry of the Word Problem This is a survey article by Martin Bridson of the word problem and other topics.
Free and Fragmenting Filling Length This 2005 paper by Martin Bridson and Tim Riley gives their two new diagram measurements (variations of a diagram measurement?).
Filling Functions Tim Riley's lectures at the Centre de Recerca Matematica Barcelona, 2005.
The following links are to papers I have found useful as introductions to the subject.
Geometric Group Theory Website This website is a resource for finding people, organizations, conferences, etc. related to GGT.
The Geometry of 3-Manifolds I should have already finished this paper by Peter Scott.
A short course in geometric group theory These notes by Walter Neumann and Michael Shapiro were prepared for the ANU Workshop in 1996. They can be found at www.math.columbia.edu/~neumann/preprints/canberra.ps .
Metric Spaces of Non-Positive Curvature Working Seminar Spring 2006 I am the organizer of a working seminar, covering the analytic foundations of geometric group theory. More information about the seminar can be found at this link.
