This proposal consists of two main sections. The first section deals with coarse geometry and geometric group theory. The PI (together with D. Fisher and K. Whyte) has recently developed a new technique, "coarse differentiation", which can be viewed as a sort of differentiability substitute for quasi-isometries. Of course, conventional derivatives do not make sense for such maps, since they are not even defined on small scales; instead we must go to larger and larger scales. Using this technique, we were able to resolve three longstanding open problems in the field, namely proving the quasi-isometric rigidity of the three-dimensional solvable group Sol, exhibiting a transitive graph which is not quasi-isometric to any Cayley graph, and showing that the two lamplighter groups Z wreath product Z mod 2 and Z wreath product Z mod 3 are not quasi-isometric. We list some recent progress and other potential applications of the method, many of which are to problems which seemed completely out of reach before. The second section concerns the interrelated analytic study of billiards in rational polygons, moduli spaces of abelian and quadratic differentials, and the dynamics of the SL(2,R) action and the geodesic flow on these moduli spaces. In recent work with M. Mirzakhani, the PI was able to resolve a twenty year old conjecture by V. Veech in this area. Some of the techniques are based on a loose analogy with flows on locally symmetric spaces. Even though the moduli spaces of differentials are substantially different, the PI was and is involved in transferring some of the symmetric space techniques to this setting. We propose additional research in this direction.
Some of the coarse geometry in the the first part of the proposal has unexpected connections to computer science, in particular the existence of efficient algorithms for finding ways to disconnect a graph by cutting as few edges as possible. In fact, some of our ideas were already used to solve problems in this field. Some natural phenomena are "chaotic" (i.e. unpredictable). These are often studied by statistical methods. Others are "integrable" (i.e. predictable and regular). Other phenomena fit somewhere in between. The polygonal billiard system, which is one of our main subjects of study in the second section of the proposal, is a good model of intermediate behavior. As such it has been studied extensively in physics as well, in particular in connection to "quantum chaos".
This project deals with a long-standing problem related to ergodicity or randomness. Consider a billiard ball on a frictionless polygonal table. If the ball is set in motion, it will travel forever, making perfectly elastic collisions with the walls. If the table is a square or an equilateral triangle, one can easily show that there are only two possible behaviors: either the ball repeats the same path forever,or it travels completely randomly in the entire polygon, eventually visiting the neighborhood of any point. This project is directed toward the basic mathematical problem of understanding the behavior of a billiard ball in more general polygons. This becomes much more difficult because most polygons do not tile the plane. Such problems are motivated by problems in statistical mechanics, such as the hard ball gas. Given any physical system, one may ask whether the motion is ``ergodic'' i.e. provably random in a certain sense.This concept, which goes back to Boltzmann, has been highly fruitful for mathematics. In the case where the angles of the polygon are rational, there are connections between this problem and other fields of mathematics, namely Lie groups and Teichmuller theory. A Lie group is a smooth manifold which admits a group operation, e.g.multiplication. The canonical example is the group SL(n,R}) of $n-by-n real matrices of determinant one. Lie groups arise in many different contexts, e.g. the study of symmetries, gauge theory etc. These groups also have interesting discrete subgroups. For example the group SL(n,Z) of n-by-n integer matrices is a discrete subgroup of SL(n,R). There are deep connections between the properties of these discrete groups and classical questions in number theory. Lie groups and their discrete subgroups give rise to dynamical systems which one may study from an ergodic point of view. This connection has been used in the past to prove deep results about the discrete groups. In recent years there has been a lot of progress in this direction, notably in connection with unipotent flows. This is a special kind of dynamical system, which is ergodic, yet possesses striking regularity properties. For example, Raghunathan's conjectures, proved by Ratner ca. 1990, state that the orbit closures and the ergodic invariant measures are `` algebraic''. Even before Ratner's theorem was proved, Margulis used this circle of ideas to prove a long standing open problem in number theory: the Oppenheim conjecture on values of quadratic polynomials. There techniques are very powerful, and when they can be applied they yield a remarkable degree of control, which can often be used to solve problems outside the field. However, before our work, their scope is limited to the ``algebraic case'', i.e. actions of a Lie group G on spaces of the form G/Γ, where Γ is a discrete subgroup. (If G/Γ has finite volume, Γ is called a lattice). One may attempt to consider other actions of Lie groups, but for various reasons one would not expect the these rigidity results to extend to the case of arbitrary actions. However, there is a promising example from geometry: the moduli space of Riemann surfaces. A Riemann surface is a one-dimensional complex manifold. If one forgets the complex structure, one gets a real two-dimensional manifold, i.e a doughnut with holes. The number of holes is called the genus of the surface. If one fixes the genus, the surface is determined by a finite number of parameters, and the parameter space is called moduli space. Moduli space, which also arises in string theory, is the quotient of a simply connected space,called Teichmuller space, under the action of a certain finitely generated group called the mapping class group. The paradigm where one thinks of the unit tangent bundle of Teichmuller space as a Lie group and of the mapping class group as a lattice has been useful in the past. The Lie group SL(2,R) acts on the tangent bundle of Teichmuller space and its orbits, the Teichmuller disks, have been used by many authors in the study of Teichmuller geometry. The main result of our project is a proof of the analogue of the Raghunathan conjectures for the Teichmuller disks. For example, we proved that the closures of the Teichmuller disks, viewed as subsets of moduli space, always submanifolds invariant under the SL(2,,R) action. This gives a powerful new tool for the study of moduli space and the mapping class group. There is a remarkable connection, exploited by a number of authors, between the billiards in rational polygons and the flow on Teichmuller space described above. One expects that our theorem will yield many new results in this direction, in particular for the study of periodic trajectories.
