2026-09-22T09:29:54Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/129426 2025-10-20 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms The student, Jonghyeon Ahn, accepted the attached license on 2025-04-22 at 14:38. The student, Jonghyeon Ahn, submitted this Dissertation for approval on 2025-04-22 at 14:42. This Dissertation was approved for publication on 2025-04-25 at 10:27. DSpace SAF Submission Ingestion Package generated from Vireo submission #21895 on 2025-10-19 at 18:18:39 Questions around symplectic capacities Ahn, Jonghyeon 2025-04-25 Kerman, Ely Pascaleff, James Fernandes, Rui Hind, Richard Symplectic geometry eng One 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 $\RR^{2n}$ 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 $S^1$-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. 2025-05 Thesis https://hdl.handle.net/2142/129426 Copyright 2025 Jonghyeon Ahn Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation