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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.

Almost hypercomplex/quaternionic skew-Hermitian structures and their intrinsic torsion
Ioannis Chrysikos
University of Hradec Kralove
Wednesday, 23. February 2022 - 13:30 to 14:30
ZOOM meeting CANCELLED because of speaker's illness
We discuss the  differential geometry of  4n-dimensional manifolds admitting a SO*(2n)-structure, or a SO*(2n)Sp(1)-structure, where SO*(2n) denotes the quaternionic real form of SO(2n, C).  Such G-structures form the symplectic analog of the well-known hypercomplex/quaternionic Hermitian structures, and hence we cal them  hypercomplex / quaternionic skew-Hermitian structures, respectively.  We will describe the basic data encoding such geometric structures, their intrinsic torsion, related 1st-order integrability conditions and some classification examples, if time permitted.   This talk is based on  joint works with J. Gregorovič (UHK) and H. Winther (Masaryk)

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We open  ZOOM  at  13.15  for    coffee  and   close  at  15.... more

Cosmic Censorship and Smooth Structures, Legendrian linking and black holes
Vladimir Chernov
Dartmouth College
Wednesday, 15. December 2021 - 11:30 to 12:30
ZOOM meeting
Jointly with Nemirovski we showed that Strong Cosmic Censorship conjecture of Penrose prohibits all exotic smooth structures in the case where the spacetime is 4-dimensional and the Cauchy surface is either compact or contractible. The result in particular is based on the Thurston geometrization conjecture proved by Perelman.

We briefly recall the interpretation of Causality in globally hyperbolic spacetimes as Legendrian linking of the spheres of light rays through the two event points proved jointly with Nemirovski. Then we formulate similar work in progress results with Sadykov in the case where the spacetimes are not globally hyperbolic and the topology of the level set of a time like function changes with time. Here we use the virtual Legendrian isotopy results with Sadykov.

Finally we explain how black holes and their event horizon can be studied in terms of... more

Morita equivalence of singular Riemannian foliations and I-Poisson geometry

Thomas Strobl
University of Lyon
Wednesday, 8. December 2021 - 11:30 to 12:30
ZOOM meeting

We recall the notion of singular foliations and show how to extend it in a compatible way to the presence of a Riemannian metric. Morita equivalence of such structures provides an equivalence relation on the geometry transverse to the leaves. Finally we extend  ``coisotropic submanifolds of a Poisson manifold” to potentially singular subspaces, yielding what we call an I-Poisson structure, and use this notion to construct an invariant of singular Riemannian foliations under the above-mentioned Morita equivalence. 
This is joint work in progress with Hadi Nahari.
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Our ZOOM meeting shall be open at 11.15 and closed at 1 PM. Join Zoom Meeting
    https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09
    Meeting ID:... more

Relative Differential Invariants
Boris Kruglikov
UiT the Arctic University of Norway
Wednesday, 1. December 2021 - 11:30 to 12:30
ZOOM meeting
Relative invariants help to understand singularities of group actions on manifolds. Their weights are cocycles modulo coboundaries and thus correspond to the first Gelfand-Fuks cohomology. Relative differential invariants correspond to prolongation of the action to jet-spaces, and are important in the equivalence problem. I will discuss the weight lattice and the finiteness theorem for relative differential invariants.

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Our ZOOM meeting shall be open at 11.15 and closed at 1 PM. Join Zoom Meeting
  https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09
  Meeting ID: 995 9841 3922 Passcode: Galois

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