Shiv Yajnik (Columbia University, Mathematics)
Email: sry2111@columbia.edu
Office: CoRE 450
Coresearcher: Edwin Lu (Brown University, Mathematics)
Mentors: Hongbin Sun and Feng Luo (Rutgers University, Mathematics Department)
This week was focused on getting to know what the project was about (plus trying to figure out how to upload a file to the DIMACS server). Edwin and I met with Professor Sun, who told us about two possible projects we could take on. We have decided that we are going to explore the number of curves on topological surfaces with a fixed upper bound on their "combinatorial length." Hiding behind this question is some nontrivial geometric structure. The kinds of curves we are looking for are (simple) essential and closed curves, and each curve of this type is isotopic to a unique geodesic under a hyperbolic metric. The kinds of surfaces that we are interested in are g-holed tori with n missing points; for n > 0, these surfaces can be endowed with a hyperbolic metric. Hence it becomes convenient to understand some aspects of hyperbolic geometry; as well as reading about our project, we are also reading some material on mapping class groups and hyperbolic surfaces.
We have two different notions of length: one is given directly by the hyperbolic metric (which for our purposes is assumed to be geodesically complete); the other is given by what is called an "ideal triangulation"--a triangulation formed by triangles whose sides are determined by the hyperbolic geodesics, and another notion of length, called combinatorial length, is given by how many edges the curve intersects.
Finding the number of curves corresponding to a given combinatorial length involves trying to find an associated generating function--that is, we find a power series where coefficients corresponding to t^L are the number of curves with combinatorial length L.
On Tuesday, Edwin and I gave a short presentation whose purpose was to briefly introduce the problem to an audience that included Professor Lazaros, our fellow student researchers in the REU, and other professors who attended the event. Since Wednesday, we have begun to attempt our curve-counting problem for specific examples. First, we, together with Professor Luo and Professor Sun, went through through the computation for a punctured torus (which has only one possible ideal triangulation), then Edwin and I have attempted to work through the computations for different ideal triangulations of the four-holed sphere. We have some evidence (based on our more arithmetic devices) of what the generating functions and the associated sequences are, but we may need to more rigorously justify our results. I have also slightly improved the formatting on my webpage by adding headers, links, and other basic things...
This week, the DIMACS REU hosted a series of talks called "Beyond the Freshmen Horizons," where speakers from Rutgers University, Princeton University, the University of Bonn, and the University of Michigan gave presentations on a wide variety of topics in mathematics. A few moments stood out to me especially: (1) In the presentation about enumerative geometry, I really enjoyed learning about the rich structures that are involved in the Schubert calculus of Grassmanians, and the question about trying to derive the Littlewood-Richardson coefficients in a combinatorial way seems very interesting. It reminds me of how the set up of our project involves some discussion hyperbolic geometry and Teichmüller spaces (c.f. the moduli spaces from Schubert calculus) but the actual research ends up being much more combinatorial. (2) I was surprised to learn about how versatile a tool random matrices are to study basically... anything! Their presence in functional analysis, group theory, and hyperbolic geometry was something that I would have never expected to hear about.
This week, Edwin and I finished computing the generating functions which correspond with each triangulation of the four-holed sphere as well as the corresponding asymptotic growths of the sequence c_L, consisting of the number of curves less of less than or equal to combinatorial length L. To get these results for the four-holed sphere, we used the Dehn-Thurston coordinates and corresponding train tracks (see the paper by F. Luo and R. Stong for more details). Then we moved onto the case of the two-holed torus. We may have stumbled onto a promising result about the asymptotic growth of the length in terms of the Dehn-Thurston coordinates that, if correct, will be very useful.
The result that we found last week did not work the way we expected it to. It required one to prove that a curve is simple if the set of all coordinates given by the train tracks given by pants decomposition / Dehn Thurston (these are not the same in general, but for the case of the two-holed torus, they are similar, if not the same) have greatest common divisor 1, but it turns out that this is not true; there are multicurves that have this property as well. So we thought that perhaps what we did find was an upper bound, and we should start looking for a lower bound. We spent a lot of this week fiddling around with pants decomposition, the idea of piecing surfaces together using n-punctured tori and spheres and more gcd conditions. So far, nothing has seemed to work.
This week, there was a nice presentation about generative artificial intelligence; I thought the speaker was very engaging and brought out the nuances of the subject quite well. Culture day also happened this week; it was really fun to see what the students had put together.
This week, we continued to attempt to find a lower bound, though without much success. Later in the week, Professor Sun wrote an argument that addressed whether the generating function that we are looking for has to be rational. Professor Luo introduced us to a paper which discusses an analagous surface counting problem on 3-manifolds. The goal for next week be to try to use their methods for our curve counting problem on 2-manifolds; now, we will be counting curve systems, which allow for multicurves and boundary parallel curves.
This week, there was a presentation, "Fair Claims Resolution"--this subject seems to have very interesting problems.
This week, we went through the first four sections of the paper from last week in greater detail and are trying to adapt their methods. We used this to get a result for the case with at least one puncture. The closed case is proving to be much more difficult; we spent Thursday trying to read through parts of another source in an effort to prove some analogues to the paper from last week. We have also attempted to prove a lemma that rules out the nullhomotopic curve class on closed surfaces.
On Tuesday, we saw a very engaging presentation about complexity theory, and the REU from New York came to visit us; I had a chance to meet a few of the students.
This week, in continuation of our efforts regarding the closed surface case, we have attempted to compile and prove a series of lemmas which would serve as a classification of isotopy classes of curves, which would ultimately lead quite directly (i.e. using much of the same reasoning as in parts the paper we discovered last week) to the result we are looking for.
On Monday, we visited Nokia Bell Labs, a fascinating place with a myriad of different research projects in technology. We visited their anechoic chamber--which was the most silent room on Earth for a long time--and an exhibition regarding the history of Bell Labs. On Friday, there was a panel of graduate students and graduate department directors that discussed topics related to graduate school, including the structure of the program and the application requirements.
Farb, B., & Margalit, D. (2011). A primer on mapping class groups. In Princeton University Press eBooks. https://doi.org/10.1515/9781400839049
Luo, F., & Stong, R. (2004). Dehn-Thurston coordinates for curves on surfaces. In Communications in Analysis and Geometry (pp. 1–41). https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2004/0012/0001/CAG-2004-0012-0001-a003.pdf
Mirzakhani, M. (2008). Growth of the number of simple closed geodesics on hyperbolic surfaces. Annals of Mathematics, 168(1), 97-125. https://doi.org/10.4007/annals.2008.168.97
Papadopoulos, A. (2021). Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichmüller space. arXiv.org. https://arxiv.org/abs/2103.10066
Wilf, H. S. (2005). generatingfunctionology. In A K Peters/CRC Press eBooks. https://doi.org/10.1201/b10576