General Information
| 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 |
Project Description
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Research Log
Week 1 (5/27-5/31)
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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)
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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)
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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)
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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)
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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)
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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)
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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.
Project Summary
References
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Acknowledgements
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This research was done at the 2024 DIMACS REU program, supported by NSF grant no. DMS 2220271.