TRR 191 - Symplectic structures in geometry, algebra and dynamics

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 CRC/TRR 191 fosters the cooperation of, 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 explores 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, stochastics, visualization). 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.

DFG Programme CRC/Transregios

International Connection Netherlands

Current projects

A01 - Topological aspects of symplectic manifolds with symmetries (Project Heads Heinzner, Peter; Reineke, Markus; Sabatini, Silvia)

A02 - Geometry of singular spaces (Project Heads Geiges, Hansj?rg; Lytchak, Alexander; Marinescu, George Teodor; Zehmisch, Kai)

A03 - Geometric quantization (Project Heads Alldridge, Alexander; Heinzner, Peter; Marinescu, George Teodor)

A05 - Reeb dynamics and topology (Project Heads Albers, Peter; Geiges, Hansj?rg; Zehmisch, Kai)

A08 - Symplectic geometry of representation and quiver varieties (Project Heads Albers, Peter; Pozzetti, Maria Beatrice; Reineke, Markus; Wienhard, Anna)

A09 - Symplectic dynamics - celestial mechanics & billiards (Project Heads Albers, Peter; Hryniewicz, Umberto; Moreno, Agustin)

B01 - Topological entropy and geodesic flows on surfaces (Project Heads Bramham, Barney; Hryniewicz, Umberto; Knieper, Gerhard)

B02 - Twist maps and minimal geodesics (Project Heads Knieper, Gerhard; Kunze, Markus)

B03 - Systolic inequalities in Reeb dynamics (Project Heads Abbondandolo, Ph.D., Alberto; Benedetti, Gabriele; Bramham, Barney; Hryniewicz, Umberto)

B05 - Hyperbolicity in dynamics and geometry (Project Heads Knieper, Gerhard; Kunze, Markus; Pozzetti, Maria Beatrice; Wienhard, Anna)

B06 - Symplectic methods in infinite-dimensional systems (Project Heads Burban, Igor; Kunze, Markus; Suhr, Stefan)

B07 - Lorentz and contact geometry (Project Heads Nemirovski, Stefan; Suhr, Stefan)

B08 - Symplectic methods for generalized billiards (Project Heads Albers, Peter; Bramham, Barney; Hryniewicz, Umberto)

B09 - Hamiltonian dynamics of surface deformations (Project Heads Farre, James; Knieper, Gerhard; Wienhard, Anna)

C01 - Symplectic capacities of polytopes (Project Heads Abbondandolo, Ph.D., Alberto; Albers, Peter; Th?le, Christoph; Vallentin, Frank)

C03 - Momentum polytopes, string polytopes and generalizations (Project Heads Cupit-Foutou, Stéphanie; Heinzner, Peter; Littelmann, Peter; Reineke, Markus)

C04 - Combinatorics of manifolds with symmetries and modularity properties (Project Heads Bringmann, Kathrin; Sabatini, Silvia)

C05 - Modular forms and Gromov-Witten theory (Project Heads Bringmann, Kathrin; Suhr, Stefan; Zehmisch, Kai)

C06 - Visualization in billiards and geometry (Project Heads Albers, Peter; Geiges, Hansj?rg; Sadlo, Filip)

C07 - Associative Algebras from Symplectic Geometry (Project Heads Littelmann, Peter; Reineke, Markus; Schroll, Sibylle)

Z - Central tasks (Project Heads Geiges, Hansj?rg; Zehmisch, Kai)

Completed projects

A06 - Rabinowitz Floer homology (Project Heads Abbondandolo, Ph.D., Alberto; Albers, Peter)

A07 - Derived categories of singular curves (Project Heads Burban, Igor; Marinescu, George Teodor)

B04 - Loop groups and the path model (Project Heads Littelmann, Peter; Lytchak, Alexander)

C02 - Algorithmic symplectic packing (Project Heads Geiges, Hansj?rg; Jünger, Michael; Vallentin, Frank)

Applicant Institution Universit?t zu K?ln

Co-Applicant Institution Ruhr-Universit?t Bochum; Ruprecht-Karls-Universit?t Heidelberg

Participating University Rheinisch-Westf?lische Technische Hochschule Aachen

Spokespersons Professor Hansj?rg Geiges, until 7/2021; Professor Dr. Kai Zehmisch, since 7/2021