Questions around symplectic capacities

Ahn, Jonghyeon

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https://hdl.handle.net/2142/129426 Copy

Description

Title

Questions around symplectic capacities

Author(s)

Ahn, Jonghyeon

Issue Date

2025-04-25

Director of Research (if dissertation) or Advisor (if thesis)

Kerman, Ely

Doctoral Committee Chair(s)

Pascaleff, James

Committee Member(s)

• Fernandes, Rui
• Hind, Richard

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Symplectic geometry

Abstract

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.

Graduation Semester

2025-05

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/129426

Copyright and License Information

Copyright 2025 Jonghyeon Ahn

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