Periodic orbits in the Kepler-Heisenberg problem
Corey Shanbrom
Journal of Geometric Mechanics 2014, 6(2): 261-278 doi: 10.3934/jgm.2014.6.261
One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.
keywords: Heisenberg group Periodic orbits fundamental solution to Laplacian. integrable system Kepler problem Carnot group
Complete spelling rules for the Monster tower over three-space
Alex Castro Wyatt Howard Corey Shanbrom
Journal of Geometric Mechanics 2017, 9(3): 317-333 doi: 10.3934/jgm.2017013

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.

keywords: Goursat multi-flags prolongation Semple tower subtowers Monster tower
Bridges between subriemannian geometry and algebraic geometry: Now and then
Alex L Castro Wyatt Howard Corey Shanbrom
Conference Publications 2015, 2015(special): 239-247 doi: 10.3934/proc.2015.0239
We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.
keywords: topological invariants. Subriemannian geometry normal form theory local singularities enumerative geometry

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