# Statistical Regularities of Self-Intersection Counts for Geodesics on Negatively Curved Surfaces

Wednesday, October 30, 2013 - 10:15am - 11:05am

Keller 3-180

Steven Lalley (University of Chicago)

Let U be a compact, negatively curved surface. From the

(finite) set of all closed geodesics on U of length less than L

choose one, and let N (L) be the number of its self-intersections.

There is a positive constant $K$ such that with overwhelming

probability as L grows,

N (L)/L^{2} approaches K.

This talk will concern itself with fluctuations. The main theorem

states that if U has variable negative curvature then

(N (L)-KL^{2})/L^{3/2}

converges in distribution to a Gaussian law,

but if U has constant negative curvature then

(N (L)-KL^{2})/L

converges in distribution to a (probably) non-Gaussian law.

(finite) set of all closed geodesics on U of length less than L

choose one, and let N (L) be the number of its self-intersections.

There is a positive constant $K$ such that with overwhelming

probability as L grows,

N (L)/L^{2} approaches K.

This talk will concern itself with fluctuations. The main theorem

states that if U has variable negative curvature then

(N (L)-KL^{2})/L^{3/2}

converges in distribution to a Gaussian law,

but if U has constant negative curvature then

(N (L)-KL^{2})/L

converges in distribution to a (probably) non-Gaussian law.

MSC Code:

53C22

Keywords: