Date of Award
Ph.D. in Mathematics
Micah B. Milinovich
For a smooth hypersurface S ⊂ R 2n given by the level set of a Hamiltonian function H, a symplectic form ω on R2n induces a vector field XH which flows tangent to S. By the nondegeneracy of ω, there exists a distinguished line bundle LS whose characteristics are the integral curves of XH. When S is the boundary of a smooth convex domain K˜ ⊂ R 2n, then the least action among closed characteristics of LS is equal to the Ekeland-Hofer-Zehnder capacity, a symplectic invariant. From a result due to Artstein-Avidan and Ostrover, there exists a continuous extension of this capacity to nonsmooth convex domains K˜ ⊂ R2n, and from the work of Künzle, there is a generalization of the notion of characteristics of K˜. The existence of corners in @K˜ , however, prevents the analogous uniqueness/existence result found in the smooth case, coming from the characteristic initial value problem. First, we will define a generic class of polyhedral sets, called “symplectic-faced”, which avoid certain obstructions to uniqueness. We will show that, for symplectic-faced 4-polytopes ∑, we have the existence and local uniqueness of generalized characteristics of ∑. Then, we will show that symplectic-faced polytopes ∑ ⊂ R2n admit only characteristics with piecewise-linear trajectories. Finally, we will extend our existence/uniqueness result from 4-polytopes to the relative interior of low-codimension faces of symplectic-faced 2n-polytopes.
Naugle, Tommy, "Generalized Characteristics Of A Generic Polytope" (2018). Electronic Theses and Dissertations. 672.