DIMACS REU 2024

A picture of me with Grant
Sanderson (3blue1brown).

Student:	Edwin Lu
Office:	CoRE 450
School:	Department of Mathematics, Brown University
Contact:	edwin_lu@brown.edu
Project:	Curves on Topological Surfaces
Working with:	Shiv Yajnik
Mentors:	Feng Luo, Hongbin Sun

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Project Description

We are interested in counting isotopy classes of curves on compact, topological, orientable surfaces of the form Σ_g,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(Σ_g,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.

Research Log

Week 1 (5/27-5/31)

This week was mostly orientation and moving in. We met with one of our mentors Professor Hongbin Sun and discussed two potential projects. We ended up deciding on the project about counting essential loops on a surface with combinatorial length less than or equal to L. We prepared a presentation for next week, and did some reading on hyperbolic geometry using the first three chapters of Athanase Papadopoulos' notes "Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichmüller space".

Week 2 (6/3-6/7)

We gave our initial presentations at the beginning of the week. We mostly discussed informal definitions necessary for the statement of our problem, such as 'surface', 'triangulation', 'combinatorial length'. We had several meetings with both Professor Sun and Professor Luo, and learned a bit more about the problem we chose to work on. They suggested we start by working with the 4-holed sphere case, due to its similarity with the punctured torus. We were able to get some preliminary results on some of the simpler triangulations of the 4-holed sphere, but nothing has been completely proven yet.

Week 3 (6/10-6/14)

With an insight from Professor Sun, we were able to work out results for all the triangulations of the 4-holed sphere. We also did some work on the twice punctured torus, and made some conjectures about generalizing our results about simple curves to arbitrary surfaces.

Week 4 (6/17-6/21)

Unfortunately, our conjectures turned out to be false, though we were able to show an upper bound for our results. We spent quite some time trying to lower bound, but without success, thanks to Professor Sun's keen eye for counterexamples.

Week 5 (6/24-6/28)

We worked a bit more on lower bounding, then pivoted to the curve system case. Professor Luo showed us a paper with similar work done in the three-dimensional case, and we worked on trying to produce a similar result in the surface case.

Week 6 (7/1-7/5)

We were able to conclude the case about surfaces with punctures, and spent some time (briefly) writing up our results. We then began working on the closed surface case.

Week 7 (7/8-7/12)

We began our week by visiting Nokia Bell Labs. We attended several talks and a panel about research in industry, and also got to visit the 'anechoic chamber', a room which blocks out nearly all sound. The rest of the week was spent working on the closed surface case, and we were able to translate some results done in past work to our specific case. During the last couple hours of the week, we attended a graduate school panel consisting of several graduate studies directors and some grad students, and got to ask questions about graduate school and applying to graduate school.

Week 8 (7/15-7/19)

We first worked on our final presentation, then focused on completing the closed surface case. On Wednesday and Thursday we had final presentations for the program. A particular presentation I found really interesting is one about 'Triple product L-functions', given by Dmitriy Shvydkoy.
We drafted an argument for finishing the closed surface case, then focused our attention on finishing the final report.

Week 9 (7/22-7/26)

We completed our final report, then tried to finish writing up the closed surface case. We unfortunately found another issue with our argument, so we will need to work a bit more to wrap everything. We decided to continue working on the project after the REU, hopefully finishing things up before the end of the summer. Any further work will be linked below.

Presentations and Papers

• First Presentation
• Final Presentation
• Final Report

References

• [1]  (insert citation)

Acknowledgements

This research was done at the 2024 DIMACS REU program, supported by NSF grant no. DMS 2220271. Special thanks to our mentors Professor Feng Luo and Professor Hongbin Sun!