Electronic Theses and Dissertations

Date of Award

2018

Document Type

Dissertation

Degree Name

Ph.D. in Mathematics

Department

Mathematics

First Advisor

Samuel Lisi

Second Advisor

Farhad Farzbod

Third Advisor

Micah Milinovich

Abstract

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.

Included in

Mathematics Commons

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