# 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.

### Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2

##### Andrey Krutov
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 30. March 2022 - 13:30 to 14:30
in IM building, front building, ground floor, konirna seminar room +ZOOM meeting
As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2.
We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0|1, or 1|1; it is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic distinct from 2, see arXiv:1806.05505).

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We shall meet   this  time  at    the   seminar   room KONIRNA,  ground floor, front building  of IM.     We... more

### Four-manifolds and knots

##### Andras Stipsicz
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Wednesday, 23. March 2022 - 13:30 to 14:30
ZOOM meeting
Slice knots (which bound a disk in the four-space) play important role both
in knot theory and in smooth four-dimensional topology. I will explain some of these
common points, recall a simple way to construct slice knots and focus on obstructions
of sliceness provided by knot Floer homology.
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We open ZOOM at 13.15 for virtual coffee  and close  ZOOM  at 15.00
Join Zoom Meeting
https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09
Meeting ID: 995 9841 3922
Passcode: Galois

### CR-twistor spaces over manifolds with $G_2$ -and $Spin(7)$-structures

##### Hong Van Le
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 16. March 2022 - 13:30 to 14:30
in IM building, the blue lecture room, ground floor +ZOOM meeting
In 1984  LeBrun constructed  a  CR-twistor  space  over  an arbitrary conformal Riemannian  3-manifold and proved that  the  CR-structure  is formally integrable.   This   twistor  construction  has been    generalized by Rossi in 1985  for  $m$-dimensional Riemannian  manifolds endowed with a $(m-1)$-fold  vector cross product (VCP).  In 2011 Verbitsky   generalized    LeBrun's construction   of   twistor-spaces     to   $7$-manifolds  endowed  with    a $G_2$-structure.  In my talk I shall explain how to unify    and generalize     LeBrun's, Rossi's  and  Verbitsky's   construction of a CR-twistor  space to the case    where   a   Riemannian  manifold  $(M, g)$ ... more

### Lie algebra of operators on moduli space of Riemann surfaces

##### Alexander Zuevsky
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 9. March 2022 - 13:30 to 14:30
in IM building, the blue lecture room, ground floor +ZOOM meeting
We recall variational formulas for holomorphic elements on Riemann surfaces
with respect to arbitrary local coordinates on the moduli space of complex structures.
These formulas are written in terms of a canonical element on the  moduli space
which corresponds to the pairing between the space of quadratic differentials and
the tangent space to the  moduli space. Next, we recall the notion of continual
Lie algebras introduced by Saveliev and Vershik and provide several classical examples.
We show that canonical differential operators on moduli space $\mathcal M_{n, 3g-3}$
of Riemann surfaces form a continual Lie algebra with the base field given by domains
of points on $\mathcal M_{n, 3g-3}$, where $n$ is the number of punctured points.
General formulation of exactly solvable models associated to continual Lie algebras
will be given. As an application, we provide explicit formulas for solutions to solvable equations... more