Project 1:: Billiards and linear manifolds in moduli space
While the game of billiards is usually played on a rectangular table, the game becomes much more interesting if you allow tables shaped like other polygons, such as a triangle or regular pentagon. In recent years, incredible advances have been made in the study of billiards in polygons by studying dynamics on the space of all possible billiard tables (called a moduli space). There is an action of the group SL(2,R) on this space, and understanding the dynamics of this action can be used to understand billiards in many polygons.
Beautiful recent work of Eskin-Mirzakhani has shown that the closure of every SL(2,R) orbit in moduli space has a very simple structure called a linear manifold. It remains an important open problem to understand what types of linear manifolds can arise as closures of SL(2,R) orbits. In this project we will learn a bit about moduli spaces and understand the statement of Eskin-Mirzakhani's theorem as well as concrete examples of linear manifolds. We will then learn of some recent tools introduced to study linear manifolds, for example Alex Wright's cylinder deformation theorem which gives a very concrete method for exploring linear manifolds. We will learn of some recent applications of the cylinder deformation theorem to the problem of linear manifolds and try to use these methods to develop new results.
For students with an interest in programming and experimental math, we may instead attempt to discover new linear manifolds by a rigorous computer search.
Students interested in this project should have some knowledge of topology and abstract algebra. Alternatively, students with knowledge of abstract algebra and an interest in computer programming would be well suited for a more experimental project involving a computer search for new linear manifolds.
Project 2:: Searching for a Tumtum Tree?
It is quite easy to show that a closed negatively curved (i.e. everywhere saddle-shaped) surface cannot be isometrically embedded into Euclidean 3-space. On the other hand, there are many examples of (complete) noncompact negatively curves surfaces living in 3-space, even within a specified topological type. These embedded surfaces extend off to infinity in different directions. (Think of the hyperboloid as a prototypical example.)
So could it be possible to isometrically embed a complete noncompact smooth negatively curved surface inside a compact ball? At first blush, such an embedding may seem unlikely since it could never be complete in the ambient Euclidean metric. Nevertheless, in 1966 Èmil Rozendorn of Lomonosov Moscow State University did construct a smooth, intrinsically complete, nonpositively curved surface contained in a ball! This surface has a tree-like structure and completeness implies that there cannot be any finite-length curves on the surface which reach its set-theoretic boundary. All we need to do is make this negatively curved! This turns out to be more difficult than it first appears since the zero curvature of Rozendorn's example cannot be eliminated locally for topological (index) reasons. While the search for this branching negatively curved surface may turn out to be as ephemeral as the Tumtum tree, I suspect it is really out there...somewhere.
Enthusiasm to go surface hunting is a must. An introduction to differential geometry would be a very valuable asset, but not strictly necessary.
Project 3:: Knotted Spheres in Four-Dimensional SpaceFaculty Mentor:
Most people know that a string can be tied in a circular knot in many different ways. In fact, this idea been rigorously studied mathematically for over 100 years. Many beautiful theorems have been proven in the area of knot theory, yet, perhaps surprisingly, there remain many more interesting and elusive questions to be answered. On the other hand, very few people have even considered the possibility of taking, instead of a string, a 2-dimensional sphere, and trying to tie it in a knot! Of course, this is very hard to visualize, because it is only possible to do this if one puts the sphere in 4-dimensional space.
It may seem very difficult to study such knottings of sphere, and it is, in fact! However, we can make it easier on ourselves by looking for connections between normal 1-dimensional knots and these complicated 2-dimensional sphere knots. There are many known connections between these types of objects that have been discovered and studied over the years. In this project, we'll explore some of these connections. In the process, we will discover how knots, surfaces, and manifolds can be built using simple pieces, and we will see how studying simple objects like tangles and disks can give us deep insight into the mysterious world of knots and manifolds.
Some familiarity with topology at the intuitive level would be helpful, but not required. An interest in drawing pictures of knots, disks, and tori while using lots of colored chalk, is a must!
Project 4:: Classifying pillowcase covers as dynamical systems
Take a piece of graph paper, and draw a closed polygon, P. Choose four "vertices" of the graph paper that lie on the boundary of P, and label them a, b, c, d in the counterclockwise ordering on the boundary of P. Finally, draw the letter "Q" on one of the squares in P. The double DQ is a sphere with a length metric: this is the "square pillowcase". The double DP is also a pillowcase, though it is a bit lumpier than DQ. The data above describes two maps: an "origami" DP --> DQ, and an identification DQ --> DP. The composition DP --> DQ --> DP gives a map from a space to itself, and therefore a dynamical system on the sphere. This project will concern the classification of such systems from various points of view: combinatorial-topological, group-theoretic, and complex-analytic.
Enthusiasm is the only criterion.
Project 5:: Rubber Bands and Extremal LengthFaculty Mentor:
Given two rubber band networks, when is one "looser" than the other? Intuitively, this means that however the two networks are stretched, the energy of the first is less than the energy of the second. In some cases, good criteria are known, but not in general. The question also turns out to be related to when one surface embeds inside another conformally (i.e., without distorting angles), via the notion of extremal length.
Prerequisites: In this project, we will investigate this problem experimentally and theoretically. From the experimental point of view, good programming experience is helpful, preferably in Haskell. From the theoretical point of view, good backgrounds in analysis, topology, and/or complex anlysis are very helpful.