We will start with a brief introduction to a relative version of symplectic field theory (SFT) for Legendrian submanifolds in a particular class of contact manifolds. This will provide us with Chekanov-Eliashberg differential graded algebra (DGA), whose homology is an invariant of Legendrian isotopy. This algebraic structure is difficult to compute and so we will simplify the differential using (bi)linearization. Then we will sketch the proof of DGA-homotopy criterion for augmentations using a duality long exact sequence, which may be seen as a version of Poincare duality for this relative SFT. We will mention the geometric origin of those augmentations coming from exact Lagrangian fillings of our Legendrian submanifold. If time permits, we will show the application of the mentioned results to the question of geography for bilinearized Legendrian contact homology of disconnected Legendrian submanifolds in a one jet space, which is a generalization of work of Bourgeois and... more
Cohomology in algebra, geometry, physics and statistics
usually takes place every Wednesday Institute of Mathematics of ASCR, Žitná 25, Praha 1Chair: Anton Galaev, Roman Golovko, Igor Khavkine, Alexei Kotov, Hong Van Le and Petr Somberg
In this seminar we shall discuss topics concerning constructions and applications of cohomology theory in algebra, geometry, physics and statistics. In particular we shall discuss in first four seminars the relations between vertex algebras and foliations on manifolds, Gelfand-Fuks cohomology on singular spaces, cohomology of homotopy Lie algebras. The expositions should be accessible for all participants.
Augmentations of Chekanov-Eliashberg Algebra and Geography of Bilinearized Legendrian Contact Homology
Filip Strakoš
Uppsala University
Wednesday, 29. March 2023 - 13:30 to 14:30
ZOOM meeting
We will start with a brief introduction to a relative version of symplectic field theory (SFT) for Legendrian submanifolds in a particular class of contact manifolds. This will provide us with Chekanov-Eliashberg differential graded algebra (DGA), whose homology is an invariant of Legendrian isotopy. This algebraic structure is difficult to compute and so we will simplify the differential using (bi)linearization. Then we will sketch the proof of DGA-homotopy criterion for augmentations using a duality long exact sequence, which may be seen as a version of Poincare duality for this relative SFT. We will mention the geometric origin of those augmentations coming from exact Lagrangian fillings of our Legendrian submanifold. If time permits, we will show the application of the mentioned results to the question of geography for bilinearized Legendrian contact homology of disconnected Legendrian submanifolds in a one jet space, which is a generalization of work of Bourgeois and... more
Twistor complexes in symplectic geometry
Svatopluk Krysl
Charles University
Wednesday, 5. April 2023 - 13:30 to 14:30
in IM building, ground floor +ZOOM meeting
For a manifold with a vanishing second Stiefel--Whitney class and equipped with a symplectic form, it is possible to define the so-called symplectic spinor bundle (B. Kostant) that is a parallel notion to the spinor bundle on a Riemannian manifold. The fibre of the bundle is an infinite dimensional complex vector space which is called the space of symplectic spinors. It is a direct sum of two irreducible representations of the connected double cover of the symplectic group.
The tensor product of exterior forms on the manifold with the symplectic spinor bundle ("twisted" deRham complex) splits into subbundles and the symplectic twistor operators are defined with help of them. We describe their construction. Computing a representation-theoretic characteristic (Schur--Weyl--Howe type duality), we can determine the symbols of the symplectic twistor operators and prove the ellipticity of the cohomological complexes formed by these operators.... more
The tensor product of exterior forms on the manifold with the symplectic spinor bundle ("twisted" deRham complex) splits into subbundles and the symplectic twistor operators are defined with help of them. We describe their construction. Computing a representation-theoretic characteristic (Schur--Weyl--Howe type duality), we can determine the symbols of the symplectic twistor operators and prove the ellipticity of the cohomological complexes formed by these operators.... more
TBA
Zoran Skoda
University of Zadar and University of Hradec Kralove
Wednesday, 19. April 2023 - 13:30 to 14:30
in IM building, ground floor
TBA