{"id":{"repo_id":"mississippi","oai_identifier":"oai:egrove.olemiss.edu:etd-1671"},"canonical_url":"https://search.dev.ndltd.org/etd/mississippi/oai:egrove.olemiss.edu:etd-1671","repository":{"repo_id":"mississippi","name":"University of Mississippi","base_url":"https://egrove.olemiss.edu/do/oai/"},"display":{"title":"Generalized Characteristics Of A Generic Polytope","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Naugle, Tommy"],"institution":null,"degree_name":"Ph.D. in Mathematics","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Samuel Lisi","Farhad Farzbod","Micah B. Milinovich"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-01-01T08:00:00Z","date_published":"2018-01-01T08:00:00Z","updated_at":"2026-07-24T03:05:43Z","subjects":["Capacity","Geometry","Hamiltonian","Hofer-Zehnder","Polytope","Symplectic","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://egrove.olemiss.edu/etd/672","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Samuel Lisi","Farhad Farzbod","Micah B. 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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."]},{"key":"dc:title","label":"Title","values":["Generalized Characteristics Of A Generic Polytope"]}]}],"canonical_facts":{"dc:contributor":["Samuel Lisi","Farhad Farzbod","Micah B. Milinovich"],"dc:creator":["Naugle, Tommy"],"dc:date.available":["2019-06-28T07:00:00Z"],"dc:description.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."],"dc:identifier":["https://egrove.olemiss.edu/etd/672"],"dc:subject":["Capacity","Geometry","Hamiltonian","Hofer-Zehnder","Polytope","Symplectic","Mathematics"],"dc:title":["Generalized Characteristics Of A Generic Polytope"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D. in Mathematics"]},"updated_at":"2026-07-24T03:05:43Z"}