TRR 191:
Symplectic Structures in Geometry, Algebra and Dynamics
Subject Area
Mathematics
Computer Science, Systems and Electrical Engineering
Term
since 2017
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 281071066
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, Hansjörg
;
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, Hansjörg
;
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, 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, Alberto
;
Albers, Peter
;
Thäle, 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, Hansjörg
;
Sadlo, Filip
)
-
C07 - Associative Algebras from Symplectic Geometry
(Project Heads
Littelmann, Peter
;
Reineke, Markus
;
Schroll, Sibylle
)
-
Z - Central tasks
(Project Heads
Geiges, Hansjörg
;
Zehmisch, Kai
)
Completed projects