Universität zu Köln

Leipold, Karla Halla Marlene (2026). Combinatorial Approaches to the EHZ Capacity of Simplices. PhD thesis, Universität zu Köln.

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Abstract

This thesis investigates how methods from discrete optimization can be used to study problems in symplectic geometry. Its central object is the Ekeland--Hofer--Zehnder capacity \(\cEHZ\), a fundamental symplectic invariant that is monotone under symplectic embeddings and plays an important role in Hamiltonian dynamics and convex symplectic geometry. Although \(\cEHZ\) is conceptually well understood, explicit computations are notoriously difficult. A central theme of this thesis is that new connections to combinatorial optimization not only clarify the computational complexity of these problems, but also open the door to new and more effective algorithmic approaches. By specializing a combinatorial formula for the Ekeland--Hofer--Zehnder capacity of polytopes to the case of simplices, we obtain a finite combinatorial formula for \(\cEHZ\) in this setting. This reveals a discrete structure underlying the symplectic problem and creates a bridge to methods from combinatorial optimization, integer programming, and computational complexity. One of our main results is that computing the Ekeland--Hofer--Zehnder capacity of convex polytopes is \(\NP\)-hard. This is proved via a reduction from the feedback arc set problem on directed bipartite tournaments. In addition, we relate the problem of deciding affine symplectomorphism of full-dimensional simplices, via polynomial-time reductions, to weighted directed graph isomorphism. On the algorithmic side, these connections allow us to reformulate the simplex formula as an exact integer linear program for evaluating \(\cEHZ\) of simplices, providing an effective alternative to brute-force enumeration. They also motivate the algorithmic study of optimization problems that would otherwise be computationally out of reach, such as the search for simplices with large systolic ratio. To this end, the exact evaluation procedure is combined with optimization on the special linear group \(\SL(2n,\mathbb R)\). Since the objective is nonsmooth, we develop a retraction-based manifold optimization approach using Clarke subdifferentials, active branch gradients, and Armijo-type line search. Together, these ideas yield a computational framework that combines exact evaluation, nonsmooth optimization on manifolds, and symmetry detection via weighted graph isomorphism. Computational experiments produce many simplices with systolic ratio exceeding that of the standard simplex in several dimensions. Overall, this thesis shows that discrete optimization provides an effective framework for studying symplectic capacities.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Leipold, Karla Halla Marlene karla.leipold@gmail.com UNSPECIFIED UNSPECIFIED
Contributors:	Contribution Name Email Censor Vallentin, Frank frank.vallentin@uni-koeln.de
URN:	urn:nbn:de:hbz:38-805663
Date:	2026
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language Symplectic geometry UNSPECIFIED EHZ- capacity UNSPECIFIED Complexity UNSPECIFIED Maximizer of systolic ratio of simplices UNSPECIFIED
Date of oral exam:	8 June 2026
Referee:	Name Academic Title Vallentin, Frank Professor
Funders:	DFG, German Research Foundation) through the Collaborative Research Centre / Transregio 191, GreenEPS project, funded by the Federal Ministry for Research, Technology and Space (BMFTR)
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/80566