Abstract. The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.The monster tower, also known in the algebro-geometric literature as the Semple tower, is a tower of smooth spaces (varieties) over a specified smooth base M. Each space M(k) in the tower, called a monster space, is a parameter space for curvilinear data up to order k on M. We will describe a coarse stratification of each monster space, with each stratum corresponding to a code word created out of a certain alphabet according to rules that we will specify. These strata parametrize curvilinear data "of the same type." The monster space can be regarded as an especially nice compactification of the parameter space for curvilinear data of nonsingular curves on M (as explained in Section 2), with the added points representing the data of singular curves; in our stratification the nonsingular data points will form a single open dense stratum.Versions of this coarse stratification have been observed by virtually everyone who has studied the monster construction. Here we develop the theory in full generality, beginning with a base space of arbitrary dimension and at all levels. A finer stratification would result from a thorough analysis of the orbits under the action of a suitable group acting on the base (whose action can be lifted to the tower) or, working locally, of the pseudogroup of local diffeomorphisms at a selected point. Results such as those in Section 5.7 of [12] show that one can expect there to be infinitely many strata, i.e., that there are moduli. We seem to be very far, however, from a full understanding of where and why moduli occur.

Posing Kepler's problem of motion around a fixed "sun" requires the geometric mechanician to choose a metric and a Laplacian. The metric provides the kinetic energy. The fundamental solution to the Laplacian (with delta source at the "sun") provides the potential energy. Posing Kepler's three laws (with input from Galileo) requires symmetry conditions. The metric space must be homogeneous, isotropic, and admit dilations. Any Riemannian manifold enjoying these three symmetry properties is Euclidean. So if we want a semblance of Kepler's three laws to hold but also want to leave the Euclidean realm, we are forced out of the realm of Riemannian geometries. The Heisenberg group (a subRiemannian geometry) and lattices provide the simplest examples of metric spaces enjoying a semblance of all three of the Keplerian symmetries. We report success in posing, and solving, the Kepler problem on the Heisenberg group. We report failures in posing the Kepler problem on the rank two lattice and partial success in solving the problem on the integers. We pose a number of questions.

Abstract. Germs of Goursat distributions can be classified according to a geometric coding called an RVT code. Jean ([3]) and Mormul ([7]) have shown that this coding carries precisely the same data as the small growth vector. Montgomery and Zhitomirskii ([5]) have shown that such germs correspond to finite jets of Legendrian curve germs, and that the RVT coding corresponds to the classical invariant in the singularity theory of planar curves: the Puiseux characteristic. Here we derive a simple formula, Theorem 3.1, for the Puiseux characteristic of the curve corresponding to a Goursat germ with given small growth vector. The simplicity of our theorem (compared with the more complex algorithms previously known) suggests a deeper connection between singularity theory and the theory of nonholonomic distributions.

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity, with all collisions occurring in finite time.

Abstract. The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the sub-Riemannian Heisenberg group. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. This system is known to admit closed orbits, which all lie within a fundamental integrable subsystem. Here, we develop a computer program which finds these closed orbits using Monte Carlo optimization with a shooting method, and applying a recently developed symplectic integrator for nonseparable Hamiltonians. Our main result is the discovery of a family of flower-like periodic orbits with previously unknown symmetry types. We encode these symmetry types as rational numbers and provide evidence that these periodic orbits densely populate a one-dimensional set of initial conditions parametrized by the orbit's angular momentum. We provide links to all code developed.