Type
Text
Type
Dissertation
Advisor
Milnor, John | Chas, Moira | Rocek, Martin. | Sullivan, Dennis
Date
2013-12-01
Keywords
Dynamics, Gaussian, Intersection, Statistics, Surface, Topology | Mathematics
Department
Department of Mathematics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76413
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
ABSTRACT. Oriented loops on an orientable surface are, up to equivalence by free homotopy, in one-to-one correspondence with the conjugacy classes of the surface's fundamental group. These conjugacy classes can be expressed (not uniquely in the case of closed surfaces) as a cyclic word of minimal length in terms of the fundamental group's generators. The self-intersection number of a conjugacy class is the minimal number of transverse self-intersections of representatives of the class. By using Markov chains to encapsulate the exponential mixing of the geodesic flow and achieve sufficient independence, we can use a form of the central limit theorem to describe the statistical nature of the self-intersection number. For a class chosen at random among all classes of length n, the distribution of the self intersection number approaches a Gaussian when n is large. This theorem generalizes the result of Steven Lalley and Moira Chas to include the case of closed surfaces. | 41 pages
Recommended Citation
Wroten, Matthew Murray, "The Eventual Gaussian Distribution of Self-Intersection Numbers on Closed Surfaces" (2013). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2336.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2336