Email: | saketmshah (at) gmail.com |
---|---|
Home Institution: | Princeton University |
Project: | Hyperbolic Geometry and Number Theory: Pants decompositions of cusped surfaces |
Advisor: | Alexander Kontorovich |
I, along with Colin Fan and Kai Shaikh, will be working on a project related to the Ehrenpreis conjecture, which asserts that for any two closed Riemann surfaces, there exist finite covers for each surface which are close in geometry
to each other, in the sense that there is a diffeomorphism between the covers which has only small distortion in the angles. This conjecture has been proven by Kahn and Markovic, in 2015^{1}.
The ultimate goal of the project would be to strengthen the result of the Ehrenpreis conjecture to surfaces which may possibly have cusps. To accomplish this, the approach we will be attempting to use is a modification of the method of constructing covers via immersions of pants into the surface, from the paper of Kahn and Markovic^{1}. However, this is a difficult, long-term goal, so it will be useful for us to be able to compute examples of such decompositions, which will be our initial focus.
This week, we settled on the project we would be focusing on for the rest of the program. I skimmed a few papers related to my topic, including some material on geodesics of the complex upper half-plane. We were rather busy with making the decision on which topic to focus on, as well as orientation, so unfortunately we did not get much research done.
This week, we began our work on pants decompositions by focusing on the fundamental domain of the modular group SL_2(Z). To begin, we must work on finding closed geodesics on the modular domain, which will correspond to the cuffs of immersed pants on the modular domain. By the end of the week, we were able to compute and draw the closed geodesic associated to a hyperbolic conjugacy class of matrices in SL_2(Z). Our computations were made easier by the use of the python libraries sympy and numpy, as well as Github as a collaborative tool.
This week, we were able to understand how to reduce the problem of putting together geodesics on the fundamental domain of SL_2(Z) into the cuffs for immersed pairs of pants on the fundamental domain, to a relatively simple combinatorial problem. Moreover, by the end of the week, we were able to code up a program in Python, which given a set of geodesics as input yields a collection of pairs of pants which can be made from those geodesics. With our work from the previous week, we should be able to find many immersed pairs of pants on the domain of SL_2(Z) with specified cuff-lengths.
This week, we worked on figuring out how to reduce the problem of finding a set of pants and eyes to glue together and cover the modular domain to finding an element in the kernel of a large matrix, and we managed to use mixed integer linear programming to find minimal integer solutions to this matrix.
Building off of our work from the previous week, we tried to use linear programming and quadratic programming to find minimal (in the L^{1} and L^{2} norms) real vectors in the kernel of the matrix we found in the previous week, in an attempt to find a relatively "balanced" solution to the equation.
This week, we began learning about the shear. This is a numerical invariant between two pants which we join togethrer along a shared geodesic cuff, and due to certain technical conditions regarding the Teichmuller metric and gluing together pants to form a cover, we would like this invariant to be close to 1. We learned this week how to compute the shear using methods from inversive geometry, but are still working out the kinks in our code.
We have essentially finished up coding how to compute the shear, using inversive coordinates, modulo possible small edge cases. However, the computation of the shear using inversive coordinates only gave us the magnitude, and we figured out an algorithm for quickly computationally determining the correct sign for this numerical invariant.
This week, we started writing up a method to construct covers of the modular surface with the condition that the shears are close to 1, once again using linear programming. Apparently, one can use Hall's marriage theorem to reduce this condition to a set of linear inequalities in the number of pants in a cover. By the end of the week, we in principle finished coding this method up, but the solutions it outputted were not valid, and it is a possibility that we need the cuffs of all the pants involved to be of a longer length to construct the desired covers.
For our last week at DIMACS, we focused on presenting our work and writing up a paper, focusing on the algorithms we used. The experience of doing mathematics research was not exactly what I had imagined, and for us was quite computational, but I enjoyed it nonetheless.