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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.

One parameter Morse theory and Morse theory for manifolds with boundaries
Petr Pushkar
Higher School of Economics, Moscow
Wednesday, 15. March 2023 - 13:30 to 14:30
ZOOM meeting
Let M be a compact manifold with boundary N and g be a generic germ of a function along N.  I will explain how one can estimate from below a number of critical points of Morse extension of g to a function on M.  Construction lies in one parameter Morse theory.
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We shall open the ZOOM meeting at 13.15 for virtual coffee. Join Zoom Meeting
https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09

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 Passcode:Galois

Complex reflection groups and classification of $3$-forms in $R^9$ and of $4$-forms in $R^8$
Hông Vân Lê
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 1. March 2023 - 13:30 to 14:30
in IM rear building, ground floor, the blue lecture room +ZOOM meeting
In my talk   I shall    discuss  the role  of complex reflection  groups in  Vinberg-Elashvili's  classification of 3-forms  in $C^9$, Antonyan's classification  of  4-forms  in $C^8$ and Borovoi-De  Graaf-Lê's  classification of 3-forms  in  $R^9$. I shall  also  explain   the role  of complex reflection  groups  in   classification  of   4-forms  in $R^8$.  
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We shall open the  seminar room and  ZOOM at 13.15 for coffee and close at 15:00
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Perturbations of Fefferman’s conformal structures
Arman Taghavi-Chabert
Warsaw University
Wednesday, 22. February 2023 - 13:30 to 14:30
ZOOM meeting
In 1976, Charles Fefferman constructed, in a canonical way, a conformal structure of Lorentzian
signature on a circle bundle over any given strictly pseudo-convex Cauchy-Riemann (CR) manifolds
of hypersurface type.

It is also known, notably through the work of Roger Penrose and his associates, and of the Warsaw
group led by Andrzej Trautman, that CR three-folds underlie four-dimensional Einstein Lorentzian
metrics whose Weyl tensors are said to be algebraically special.

In this talk, I will show how these algebraically special Einstein metrics find a natural formulation
as exact perturbations of Fefferman’s original construction. The additional CR data required turns
out to be constrained by a non-linear, or gauged, analogue of a second-order (BGG) differential
operator, and is related to the existence of CR functions.... more

On numbers associated with a strong Morse function
Misha Temkin
Dartmouth College
Wednesday, 11. January 2023 - 13:30 to 14:30
ZOOM meeting

Morse function on a manifold M is called strong if all its critical points have different critical values. Given a strong Morse function f and a field F we construct a bunch of elements of F, which we call Bruhat numbers (they're defined up to sign). More concretely, Bruhat number is written on each bar in the barcode of f (a.k.a. Barannikov decomposition). It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f. We then construct the barcode and Bruhat numbers with twisted (a.k.a. local) coefficients and prove that the mentioned product equals the Reidemeister torsion of M. In particular, it's again independent of f. This way we link Morse theory to the Reidemeister torsion via barcodes. Time permitting, we will also discuss how parametric Morse theory comes into play. Based on a joint work with Petya... more

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