Summary of the research programme

Since their inception, the study of symplectic structures and the applications of symplectic techniques (as well as their odd-dimensional contact geometric counterparts) have benefited from a strong extraneous motivation. Symplectic concepts have been developed to solve problems in other fields that have resisted more traditional approaches, or they have been used to provide alternative and often conceptionally simpler or unifying arguments for known results. Outstanding examples are property P for knots, Cerf's theorem on diffeomorphisms of the 3-sphere, and the theorem of Lyusternik-Fet on periodic geodesics.

The aim of the CRC is to bring together, on the one hand, mathematicians who have been socialized in symplectic geometry and, on the other, scientists working in areas that have proved important for the cross-fertilization of ideas with symplectic geometry, notably dynamics and algebra. In addition, the CRC intends to explore connections with fields where, so far, the potential of the symplectic viewpoint has not been fully realized or, conversely, which can contribute new methodology to the study of symplectic questions (e.g. optimization, computer science).

The CRC bundles symplectic expertise that will allow us to make substantive progress on some of the driving conjectures in the field, such as the Weinstein conjecture on the existence of periodic Reeb orbits, or the Viterbo conjecture on a volume bound for the symplectic capacity of compact convex domains in R2n. The latter can be formulated as a problem in systolic geometry and is related to the Mahler conjecture in convex geometry.

The focus on symplectic structures and techniques will provide coherence to what is in effect a group of mathematicians with a wide spectrum of interests.

In this project we wish to study symplectic properties of convex polytopes from a quantitative point of view. The project consists of three strongly interrelated subprojects:

1. Higher-order symplectic capacities 

2. Symplectic capacities of random polytopes 

3. Algorithms for computing symplectic capacities 

Actions of Lie groups by symplectic transformations are quite often Hamiltonian. By definition, this means that there exists an equivariant momentum map on the manifold with values in the dual Lie algebra of the acting group. In this project, we mainly focus on multiplicity-free manifolds (e.g. toric varieties, flag varieties and more generally spherical varieties), representation spaces of quivers and Nakajima quiver varieties. In the cases under study, the momentum map has remarkable convexity properties. One of our goals is to understand the images of these momentum maps in terms of the geometry of the varieties and vice versa.

Elliptic genera of a certain level are modular forms associated with almost complex manifolds that encode much information about certain invariants of the manifold. If the latter admits a circle action preserving the almost complex structure, one can define its equivariant elliptic genus, which additionally depends on a parameter belonging to the unit circle. One of the fundamental results in the theory of elliptic genera is the Rigidity Theorem due to Witten, Bott and Taubes, and Hirzebruch, which asserts that — under suitable conditions — the equivariant elliptic genus agrees with the non-equivariant one.

The goal of this project is, on the one side, to study the implications — at a geometric and number theoretical level — of the rigidity and vanishing of certain elliptic genera associated with an almost complex manifold acted on by a circle, which will give results in the direction of the Mukai conjecture. On the other side we will define elliptic genera for combinatorial objects, such as cones and abstract GKM graphs, and generalize related results by Borisov and Gunnels for the so-called toric modular forms.

Gromov–Witten invariants count isolated stable holomorphic maps from a Riemann surface into a symplectic manifold subject to point-wise constraints. This count leads to a sequence of numbers labelled by the genus, the homology class, and a finite collection of cohomology classes. This sequence can be organized in a formal power series in several variables — the so-called Gromov–Witten potential. In order to compute the Gromov–Witten invariants, one is looking for algebraic relations between the coefficients of the Gromov–Witten potential. In some cases this can be done by invoking modularity properties of the Gromov–Witten potentials.

The aim of this project is to develop novel concepts and techniques for the visual analysis of billiard dynamics and contact geometry. These shall help reveal and explore new phenomena, raise new questions and guide theoretical considerations.

The central topic of this project is on the interplay of the algebraic and geometric properties of the of associative algebras arising in the context of different settings of symplectic geometry. A particular focus is on the associative algebras related to ((partially) wrapped) Fukaya categories and certain quantum field theories, in particular N=2 4d gauge theories. The main goal is to study the insight the geometric origin of these algebras give on their representation theory and in turn to feed this back into our understanding of the underlying geometry.