# Cohomology in algebra, geometry, physics and statistics

usually takes place every Wednesday Institute of Mathematics of ASCR, Žitná 25, Praha 1
Chair: 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.

### Floer-Novikov (co)homology associated with non-abelian coverings and symplectic fixed points

##### Hong Van Le
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 19. October 2022 - 13:30 to 14:30
in IM rear building, blue lecture room, ground floor + ZOOM meeting
In my talk   I shall explain   our    with Kaoru Ono   construction   of  Floer-Novikov  cohomology  groups $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$ defined on a regular covering $M^{\Gamma_\xi \times H}$ of a  compact   symplectic  manifold   $(M, \omega)$ with  transformation group  $\Gamma_\xi \times H$  and associated  to  a    locally symplectic isotopy ${\{\varphi_t\}}$ of $(M, \omega)$ with  flux $\xi \in H ^1 (M, R)$. Then  $H$ acts naturally on $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$.  For a subgroup $G \subset H$  denote  by $(HFN^* (M^{\Gamma_\xi \times H},\xi, Q))^G$  the   subgroup of $HFN^* (M^{\Gamma_\xi \times H}, \xi, Q)$  consisting   of the fixed  points of the $G$-action.  We  prove that  the   rank   of  \$(HFN^* (M ^{\Gamma_\... more

### On non-geometric augmentations in high dimensions and torsion of Legendrian contact homology

##### Roman Golovko
Charles University
Wednesday, 12. October 2022 - 13:30 to 14:30
in IM building, ground floor of the rear building, blue lecture room + ZOOM meeting
We construct the augmentations of high dimensional Legendrian submanifolds of the contact Euclidean vector space which are not induced by exact Lagrangian fillings. Besides that, for an arbitrary finitely generated abelian group G, we construct the examples of Legendrian submanifolds whose integral linearized Legendrian  contact (co)homology realizes G.

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We  shall  open  the seminar  room  +  ZOOM  meeting  at   13.15    for  coffee

Join Zoom Meeting
https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09
Meeting ID: 995 9841 3922
Passcode: Galois

### Classification of four qubit and rebit states

##### Willem de Graaf
University of Trento
Wednesday, 11. May 2022 - 13:30 to 14:30
ZOOM meeting
We consider the problem of classifying the orbits of SL(2, C) ^4 on the space
C^2 ⊗ C^2 ⊗ C^2 ⊗ C^2. In quantum information theory this is known as the
classification of four qubit states under SLOCC operations. We approach
the problem by constructing the representation via a symmetric pair of max-
imal rank. This makes it possible to apply the theory of θ-representations
developed by Vinberg in the 70’s. The orbits are devided into three types:
nilpotent, semisimple and mixed. The orbits of each type are classified sep-
arately. We also obtain the stabilizers of representatives of the orbits. The
talk will end with some comments on the same problem over R, known as
the classification of four rebit states. This is joint work with Heiko Dietrich,
Alessio Marrani and Marcos Origlia.

... more

### Every symplectic manifold is a (linear) coadjoint orbit

##### Patrick Iglesias-Zemmour
The Hebrew University of Jerusalem, Israel
Wednesday, 4. May 2022 - 13:30 to 14:30
ZOOM meeting
I will show that every symplectic manifold is a (linear) coadjoint orbit of the group of automorphisms of the integration bundle, independently of the group of periods of the symplectic form. This result generalizes the Kirilov- Kostant-Souriau theorem when the symplectic manifold is homogeneous under the action of a Lie group and the symplectic form is integral.
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We open ZOOM at 13.15 for virtual cafe and close at 15.00.
Join Zoom Meeting
https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09
Meeting ID: 995 9841 3922
Passcode: Galois