Abstract
dc:descriptionOne major question in the study of symplectic capacities is the strong Viterbo conjecture: all normalized symplectic capacities agree on convex domains. This conjecture has spurred extensive research and numerous interesting results about this conjecture have been established. In this thesis, we contribute to this area by focusing on the comparison of symplectic capacities. We explore the interplay between symplectic and convex geometry. More specifically, we study convex bodies in \RR2n with the property that their mean width, one of the most fundamental measurements in convex geometry, cannot be decreased by the action of natural classes of symplectomorphisms. Our main finding is that toric symmetry is a preferred feature of convex bodies that are in optimal symplectic position with respect to the mean width. We then study an S1-equivariant version of Varolgunes' relative symplectic cohomology. As an application, we construct a relative version of the Gutt-Hutchings capacities and a relative version of the symplectic (co)homology capacity. We show that these relative symplectic capacities can detect the diplaceability and the heaviness of compact subsets of a symplectic manifold. We also compare the first relative Gutt-Hutchings capacity and the relative symplectic (co)homology capacity and prove that they are equal to each other under a natural convexity assumption.
Degree
thesis:*- Name thesis:degree_name
- Ph.D.
- Level thesis:degree_level
- Dissertation
- Discipline thesis:degree_discipline
- Mathematics
- Grantor
- University of Illinois Urbana-Champaign
- Year dc:date
- 2025
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Ahn, Jonghyeon
- Contributors dc:contributor
-
- Kerman, Ely
- Pascaleff, James
- Fernandes, Rui
- Hind, Richard
Subjects
dc:subject × 1Rights
dc:rights- Statement dc:rights
-
- Copyright 2025 Jonghyeon Ahn
- Language dc:language
- en, eng
Identifiers
dc:identifier.*- Handle dc:identifier
- https://hdl.handle.net/2142/129426