- Enumerating curves on surfaces
- Project Year:
2024
- REU Student (s):
Edwin Lu | Brown University RI
| Shiv Yajnik | Columbia University in the City of New York NY
- Student 1 Institution:
Brown University
- Student 2 Institution:
Columbia University in the City of New York
- Project Mentor:
Feng Luo
- Project Mentor Area:
Mathematics
- Project Abstract:
We are interested in counting isotopy classes of curves on compact, topological, orientable surfaces of the form Σg,n, where g is the genus of the surface and n is the number of punctures. We are interested in surfaces which can be assigned a hyperbolic structure; namely, this requires the Euler characteristic χ(Σg,n) = 2-2g-n to be negative. In particular, we are not interested in spheres with at most 2 punctures or the genus 1 torus, as (1) those are not hyperbolic surfaces, and (2) those only contain boundary-parallel or nullhomotopic curves, which are not very interesting.
Starting in the late 2000s, M. Mirzakhani produced some major results concerning geodesic counting on hyperbolic surfaces. More recently, methods have been developed so that computations may not rely on hyperbolic geometry but rather a strictly topological and combinatorial point of view, even though the underlying structures may be hyperbolic. The notion of length that we use--called combinatorial length--is based on the number of intersections with special kinds of triangulations called ideal triangulations. Our goal is to find the number of curves on a given surface under a certain combinatorial length. A known and important fact is that isotopy classes of simple closed curves have a bijective correspondence with closed hyperbolic geodesics. Furthermore, while our definition of ideal triangulations is strictly topological, there is a construction of ideal triangulations in the language of hyperbolic geometry. One result of this is that combinatorial length is proportional to the geodesic length with respect to a hyperbolic metric. Therefore, our results will serve as estimations for geodesic counting problems.
