Research Questions Papers and Preprints For Math 180

Mark O'Brien Universiy of IL at Chicago Chicago, IL 60608 Office: S.E.O. Room 616 Email: mobrie06@math.uic.edu

My name is Mark O'Brien, and I am currently a first year postdoc at the University of IL at Chicago (UIC). I received my PhD in August of 2008. I started this webpage as a junior at Lycoming College in Williamsport, PA. Since then I have basically just thrown on a lot of random stuff that I've managed to write over the past 10 years.
My advisor's name was Kim Ruane, and we worked in a subject called geometric group theory. Here is a page on geometric group theory which keeps one up-to-date with geometric group theory.

I am mostly interested in some of the more classical aspects of it (geometric group theory), namely, CAT(0) spaces. These spaces are spaces where the notions of angles and triangles make sense (versus on the set of blades of grass in your backyard or the set of kids under the age of 5) and these triangles have angles that are smaller than they would be if the triangle lived on the plane (the space you studied in high school geometry). These spaces rock because their set of symmetries have many nice properties.

Some Pictures of our baby (kitty that is)
Cat Pic 1
Cat Pic 2
Cat Pic 3
2 1 1 1 on the Torus (1 iteration, 2 iterations, original).

Some pics I've made with maple over the years.
Original Torus (2/19/06)	Torus after one application of cat map. (2/19/06)	Torus after two applications of cat map.(2/19/06)
This picture consists of nothing but color and circles	A hyperbolic tessellation	Another hyperbolic tessellation
Riem Zeta of x + it plotted as a curve from t=0 to t=34 and animated in x from 0 to 3/4.