MAT 487 T01 - Independent study: Geometry and dynamics of surfaces
This is an "independent" study offered in preparation for Stony Brook's Enhanced Research Experience for Undergraduates during Summer 2022. In addition to the independent study portion of the course, there will be a weekly meeting held Thursday at 6:00 on Zoom (link).
Course materials can be found in the Google drive folder here.
Our main text for the independent study is:
R. Schwartz, Mostly surfaces. Student Mathematical Library, 60. American Mathematical Society, Providence, RI, 2011.
We will cover one chapter per week according to the schedule below. Each week in our Thursday meeting, one student will have the assignment to give a 30-40 minute presentation covering the highlights of that chapter (for example, a favorite theorem and an interesting exercise). Naturally, there will likely not be enough time to cover everything from the textbook, so the presenter will have to be selective about what to include.
In addition, you will be divided into two groups of four students each to discuss exercises from the assigned chapter. You should find a time to meet with your group during the week. You should talk about as many of the exercises as possible within the constraints of time. As a group, you should choose four of the exercises to write solutions for, each group member writing out one of the solutions. Have these ready in preparation for the Thursday meeting.
|Jan. 27||1 Overview|
|Feb. 3||2 Definition of a Surface|
|Feb. 10||3 The Gluing Construction|
|Feb. 17||4 The Fundamental Group|
|Feb. 24||5 Examples of Fundamental Groups|
|Mar. 3||6 Covering Spaces and the Deck Group|
|Mar. 10||7 Existence of Universal Covers|
|Mar. 24||8 Euclidean Geometry|
|Mar. 31||9 Spherical Geometry|
|Apr. 7||10 Hyperbolic Geometry|
|Apr. 14||11 Riemannian Metrics on Surfaces|
|Apr. 21||12 Hyperbolic Surfaces|
|Apr. 28||17 Flat Cone Surfaces|
|May 5||18 Translation Surfaces and the Veech Group|
Two additional references on general metric geometry:
- D. Burago, Y. Burago and S. Ivanov. A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001.
- M. Bridson, A. Haefliger. Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999.
Mathematical resources by topic
As we get further into the course, you will have the chance to specialize further in a direction of interest. We have come up with five potential project topics. For each topic, we have some specific research problems in mind. Once we get to the summer, the REU participants will will be split into a couple groups each focusing on a topic.
1. The illumination and finite blocking problems (with Yusheng)
- A. Wolecki. Illumination in Rational Billiards. arXiv:1905.09358
- S. Lelievre, T. Monteil and B. Weiss. Everything is illuminated. Geometry and Topology 20:1737–1762, 2016. arXiv:1407.2975
- P. Apisa and A. Wright. Marked points on translation surfaces. arXiv:1708.03411
- A video by Howard Masur on Numberphile: https://www.youtube.com/watch?v=xhj5er1k6GQ
The illumination problem is a classical problem first formulated in 1950s. It asks if a room with mirrored walls can always be illuminated by a single point light source. The original problem was first solved by Penrose using ellipses. The problem was also solved for polygonal rooms by Tokarsky in 1995. A related problem is called finite blocking problem. A pair of points are said to be finitely blocked if there is a finite collection of points B so that any light from one point to the other must pass through B. More recent work of Apisa and Wright gives geometric reasons why such phenomena can happen. These problems are closely related to the study of the translation surfaces and Teichmuller dynamics. It is an interesting project to understand these problems for some special classes of surfaces, such as square-tiled surfaces.
2. Alexandrov surfaces (with Matthew)
- S. Alexander, V. Kapovitch, and A. Petrunin. Alexandrov geometry: preliminary version no. 1, 2019.
- A. D. Aleksandrov and V. A. Zalgaller. Intrinsic geometry of surfaces. Translated from the Russian by J. M. Danskin Translations of Mathematical Monographs, Vol. 15 American Mathematical Society, Providence, R.I. 1967.
- Y. Machigashira. The Gaussian curvature of Alexandrov surfaces. J. Math. Soc. Japan 50 (1998), no. 4, 859–878.
- Y. Machigashira and F. Ohtsuka. Total excess on length surfaces. Math. Ann. 319 (2001), no. 4, 675–706.
- A. Lytchak and K. Nagano. Geodesically complete spaces with an upper curvature bound. Geom. Funct. Anal. 29 (2019), no. 1, 295–342.
Alexandrov geometry is an influential branch of geometry that considers spaces with "bounded curvature" in one sense or another. There is a very general and beautiful theory of these spaces. However, there are some aspects at the foundation of this theory not completely addressed in the current literature, and the main goal of this project is to fill these in.
3. Dilation surfaces and affine interval exchange maps (with Yusheng)
- E. Duryev, C. Fougeron and S. Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics 14: 121–151, 2019. arXiv:1609.02130
- S. Marmi, P. Moussa and J.-C. Yoccoz. Affine interval exchange maps with a wandering interval. Proc. Lond. Math. Soc. (3) 100: 639669, 2010.arXiv:0805.4737
- J. Wang. The realization problem for dilation surfaces. arXiv:2103.14720
- A Full Study of the Dynamics on One-Holed Dilation Tori by M. Haberle and J. Wang. arXiv:2012.04159
Dilation surfaces are a generalization of translation surfaces. They have only been studied systematically in the recent years. Many questions that are known to be true for translation surfaces have been proven false for dilation surfaces, and many problems are still wide open. Dynamics on dilation surfaces are related to affine interval exchange maps, which have been studied extensively in the literature. It is interesting to explore the analogous questions in the translation surface setting, such as illumination problem, closed geodesics and etc, for dilation surfaces.
4. Polyhedral approximation of surfaces (with Matthew)
- P. Creutz and M. Romney. Triangulating metric surfaces. preprint.
- D. Ntalampekos and M. Romney. Polyhedral approximation of metric surfaces and applications to uniformization. preprint.
A basic question is how to represent an arbitrary surface as a limit of simpler surfaces, such as polyhedral surfaces. This topic, along with applications, has been recently investigated in the papers listed. The objective of this paper is to study this approximation scheme in more depth. For example, if the original surface has a certain geometric property, then this should also be satisfied by the approximating space in a controlled way.
5. Geodesics in Grushin spaces (with Matthew)
- A. Bellaïche. The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 1–78.
- R. Monti and Morbidelli. Isoperimetric inequality in the Grushin plane, J. Geom. Anal. 14 (2004), no. 2, 355–368.
- M. Romney. Conformal Grushin spaces. Conform. Geom. Dyn. 20 (2016), 97–115.
A rather curious, though simple, geometric object is the Grushin plane. It is a standard example of a sub-Riemannian manifold and is a sort of interpolation between the Euclidean plane and hyperbolic plane. As part of my doctoral thesis, I studied a generalized class of "Grushin spaces". The purpose of this project is to investigate the geometry of these spaces in detail. It would be interesting to see which for which spaces one can compute geodesics exactly.
Doing undergraduate research