CRC TRR 191
Symplectic Structures in
Geometry, Algebra and Dynamics
Second Funding Period
2021 – 2024
University of Cologne
Ruhr-University Bochum
Heidelberg University
University of Cologne
Ruhr-University Bochum
Heidelberg University
The German Research Foundation has granted the funding of a Collaborative Research Centre / Transregio (CRC/TRR 191) on Symplectic Structures in Geometry, Algebra and Dynamics. This CRC is based at the universities of Bochum, Cologne and Heidelberg, and it includes mathematicians from Aachen University and Karlsruhe Institute of Technology.
Within this CRC, 20 individual projects will be funded. The funding comprises 15 full-time postdoctoral positions (pd) and 27 positions for doctoral students (ds).
Currently all positions are filled.
The CRC provides an excellent infrastructure for training and research in an internationally visible network, and we offer intensive support for international students and postdocs. There are specially dedicated funds to support researchers with children.
The positions are based at the universities of the project leaders of the relevant project. Many of the projects are based at more than one university.
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.
Research Area A comprises those projects where topological aspects are in the foreground. A central role is played by symmetries, that is, group actions of various kinds.
Reineke, Sabatini
no vacancies at present
In the proposed project we continue and expand our study of topological properties of symplectic and Kähler manifolds endowed with a compact Lie group action, aiming at information on their equivariant topological invariants. Our key objects of study are compact symplectic manifolds with a torus action, and manifolds associated with representations of quivers.
More specifically, we will study the topological invariants and diffeomorphism type of compact symplectic manifolds of dimension 6 with a Hamiltonian action of a torus of dimension 2, which goes in the direction of proving a conjecture posed by Fine and Panov, and we will study the Mukai inequality for certain categories of manifolds, using new techniques developed during the first funding period. Moreover, we will continue the development of a GKM type theory for moduli spaces of quiver representations, with applications to Donaldson–Thomas and Gromov–Witten invariants.
Lytchak, Marinescu
no vacancies at present
The aim of the project is to study geodesics and geodesic flows in singular metric spaces, in particular, in spaces with one-sided curvature bounds, in singular Kähler spaces and in some infinite-dimensional spaces, especially in completions of the space of Kähler potentials.
Heinzner, Marinescu
no vacancies at present
The main idea in the theory of geometric quantization (introduced by Kostant and Souriau) of a manifold X is to associate with X ‘quantum’ Hilbert spaces defined by means of some canonical geometric constructions. In the presence of a group action, an important meta-principle is ‘reduction commutes with quantization’; this principle describes how the quantization of the quotient relates to the quantization of the original manifold.
Given a symplectic manifold (X, ω) with a Kähler polarization such that ω has integer cohomology class, it is possible to fix a holomorphic Hermitian line bundle L whose Chern curvature is ω. The sequence of line bundles Lp provides a sequence of quantum Hilbert spaces H0(X,Lp), consisting of holomorphic sections of the tensor power Lp.
We will consider the following aspects of quantization: (1) Berezin–Toeplitz quantization; (2) Holomorphic Morse inequalities; (3) Ergodic complex geometry; (4) Quantization and reduction. One of the tools employed will be Szegõ and Bergman kernel expansions.
Albers, Geiges, Zehmisch
no vacancies at present
The main focus of this project lies on topological constructions in Reeb dynamics, such as surgery, contact cuts, plugs, fillings, or cobordisms. Filling and surgery questions are also studied in the context of the classification of Legendrian knots and links.
Albers, Pozzetti, Reineke, Wienhard
no vacancies at present
Representation varieties and (Nakajima) quiver varieties provide families of higher-dimensional symplectic varieties which allow the interaction of representation-theoretic techniques with the study of their (symplectic) geometry. The main focus of this project lies on studying these classes of symplectic varieties with modern tools of symplectic geometry, in particular the study of Lagrangian submanifolds, the investigation of rigidity questions, and the study of magnetic deformations of their geometric structures.
In Research Area B we assemble those projects where dynamical questions are in the focus, and where variational methods feature prominently.
Bramham, Hryniewicz, Knieper
no vacancies at present
The main focus of this project is to investigate which entropy related dynamical phenomena of geodesic flows on surfaces are geometric in nature and which are symplectic. For example, which phenomena generalize to three-dimensional Reeb flows? In three subprojects we will further develop the connection between finite energy foliations and symbolic dynamics to understand orbit travel, also in the context of the restricted three-body problem, and use chord contact homology to study measures of maximal entropy. Questions about non-finite energy holomorphic curves, so-called feral curves, will also be studied.
Knieper, Kunze
no vacancies at present
We plan to continue our study of low-dimensional Hamiltonian systems. More specifically, we want to extend certain results, among others by Mather and Neishtadt, for periodic twist maps to the case of non-periodic twist maps as well as to geodesic and Finsler flows on the 2-torus T2. This also motivates the slight change of the title of our project from ‘Minimal geodesics’ to ‘Twist maps and minimal geodesics’.
Twist maps are area-preserving maps φ: S1×R → S1×R (in the periodic case) or φ: R×R → R2 (in the non-periodic case) such that the image of the fibres are graphs over the real line. Such maps frequently occur as section maps of Hamiltonian systems with two degrees of freedom. They are called integrable if their domain is foliated by φ-invariant graphs on which φ acts by rotations.
Abbondandolo, Benedetti, Bramham, Hryniewicz
no vacancies at present
Let (M, ξ) be a closed (2n − 1)-dimensional contact manifold. The systolic ratio ρ(α) of a contact form α supporting ξ is the ratio between the nth power of the minimal period of closed orbits of the Reeb flow of α and the volume of M calculated with respect to α ∧ (dα)n−1. The goal of this project is to study global and local properties of the function α → ρ(α).
This construction recovers two classical settings: the systolic ratio of a Riemannian metric on S2 (taking M = T^1S^2), and the systolic ratio of convex bodies C in R2n (by taking M = ∂C). In both cases it is known that the restriction of ρ is bounded. However, determining what the optimal upper bound is and what objects achieve it, is a deep open problem. For example, in the convex case, a famous conjecture by Viterbo asserts that the optimal upper bound is 1 and the optimal objects are symplectomorphic to a ball. What makes the ball special is that the Reeb flow on its boundary is Zoll, that is, all orbits are periodic and with the same minimal period.
Knieper, Pozzetti, Wienhard
no vacancies at present
In this project we continue to study interactions of hyperbolic dynamics and hyperbolic geometry. In the last period the focus was on non-compact harmonic manifolds and questions about the growth rate and distribution of closed geodesics on Riemannian manifolds with some hyperbolicity.
Furthermore we plan to study the following subprojects:
Pressure metrics and distance functions on the space of Riemannian metrics of negative curvature.
The pressure metric and the space of Anosov representations.
Visualization and Hitchin components.
Kunze, Suhr
no vacancies at present
The area of infinite-dimensional dynamical systems (mostly governed by PDEs) knows many instances where symplectic and advanced Hamiltonian system methods are applicable, be it rigorous or formal. Prominent examples include KAM theory in infinite dimensions, integrable systems, growth of Sobolev norms/non-squeezing for equations such as KdV, Marsden-Weinstein reduction with respect to symmetry groups, etc. It is the purpose of this project to develop two of those promising topics further.
Nemirovski, Suhr
no vacancies at present
The project will continue to investigate the relations between contact and Lorentzian geometry using ideas originating in the other field. The meeting point of both fields is the natural contact structure existing on the space of null geodesics of a spacetime whenever that space happens to be a smooth Hausdorff manifold. The geometry of the space of null geodesics reflects important properties of the spacetime. The space of null geodesics is smooth and Hausdorff for causally simple subsets of globally hyperbolic spacetimes. This may yield new types of contact manifolds arising as spaces of null geodesics in addition to the classical cases of globally hyperbolic and ‘Zollfrei’ spacetimes.
An important notion in contact topology is that of a loose Legendrian introduced by Murphy. The Legendrians arising from Lorentz geometry seem to be always non-loose. This implies a variety of geometric restrictions on the structure of wave-fronts in general relativity that should be properly investigated.
Albers, Bramham, Hryniewicz
no vacancies at present
The aim of this project is to combine the theory of finite energy foliations for 3-dimensional Reeb flows and 2-dimensional billiard dynamics. One goal is to achieve a deeper understanding specifically of non-convex billiard tables and of generalized billiard systems. The use of holomorphic curve methods for 3-dimensional Reeb flows, in particular on S3, was pioneered by Hofer–Wysocki–Zehnder.
The link between the projects in Research Area C is provided by their reliance on algebraic and combinatorial methods, or by the relevance of the symplectic approach to addressing questions in these areas. Various aspects of polytopes come into play, including optimization-theoretic ones. Several of the projects have a strong algorithmic component.
Abbondandolo, Thäle, Vallentin
no vacancies at present
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:
Higher-order symplectic capacities
Symplectic capacities of random polytopes
Algorithms for computing symplectic capacities
Cupit-Foutou, Heinzner, Littelmann, Reineke
no vacancies at present
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.
Bringmann, Sabatini
no vacancies at present
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.
Bringmann, Suhr, Zehmisch
no vacancies at present
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.
Abbondandolo, Albers, Geiges, Sadlo
no vacancies at present
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.
Littelmann, Reineke, Schroll
no vacancies at present
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.