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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.

Deformed Donaldson-Thomas connections

Kotaro Kawai
Gakushuin University, Tokyo
Wednesday, 19. May 2021 - 11:30 to 12:30
ZOOM meeting
The deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a $G_2$-manifold satisfying certain nonlinear PDEs. This is considered to be the mirror of a calibrated (associative) submanifold via mirror symmetry. As the name indicates, the dDT connection can also be considered as an analogue of the Donaldson-Thomas connection ($G_2$-instanton). 
In this talk, after reviewing these backgrounds, I will show that dDT connections indeed have properties similar to associative submanifolds and $G_2$-instantons. I would also like to present some related problems. This is joint work with Hikaru Yamamoto. 

We shall  open ZOOM  at 11.15 and close at 13.00

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G(3) supergeometry and a supersymmetric extension of the Hilbert-Cartan equation

Andrea Santi
UiT The Artic University of Norway
Wednesday, 12. May 2021 - 11:30 to 12:30
ZOOM meeting
I will report on the realization of the simple Lie superalgebra G(3) as supersymmetry of various geometric
structures – most importantly super-versions of the Hilbert–Cartan equation and Cartan’s involutive PDE system
that exhibit G(2) symmetry – and compute, via Spencer cohomology groups, the Tanaka-Weisfeiler prolongation
of the negatively graded Lie superalgebras associated with two particular choices of parabolics. I will then discuss
non-holonomic superdistributions with growth vector (2|4, 1|2, 2|0) obtained as super-deformations of rank 2 distributions
in a 5-dimensional space, and show that the second Spencer cohomology group gives a binary quadric,
thereby providing a “square-root” of Cartan’s classical binary quartic invariant for (2, 3, 5)-distributions.
If time allows, I will outline an extension of Tanaka’s geometric prolongation scheme to the case of supermanifolds.
This is a joint work with B. Kruglikov and D. The.
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Information Geometry and Hamiltonian Systems on Lie Groups

Jean-Pierre Francoise
Sorbonne Université, Paris
Wednesday, 5. May 2021 - 11:30 to 12:30
ZOOM meeting
The link between Hamiltonian Integrable Systems and Information Geometry was discovered by Amari, Fujiwara and Nakamura (90s). In particular, Nakamura succeeded to define the tau-function for the open Toda Lattice by using Information Geometry .

We propose a more general study of Hamiltonian Systems related with the Information Geometry on Lie groups.

Fisher-Rao semi-definite metric is naturally induced as a left-invariant semi-definite metric on the Lie group, which is regarded as the parameter space of the family of probability density functions. For a specific choice of family of probability density functions on compact semi-simple Lie group, the equation for the geodesic flow is derived through the Euler-Poincaré reduction. Certain perspectives are mentioned about the geodesics equation on the basis of its similarity with the Bloch-Brockett –Ratiu double bracket equation and with the Euler-Arnol'd equation for a generalized free rigid body dynamics.... more

Hertz potentials and the decay of higher-spin fields

Jérémie Joudioux
Albert Einstein Institute, Golm
Wednesday, 28. April 2021 - 11:30 to 12:30
ZOOM meeting
The purpose of the talk is to illustrate how differential complexes can be used in relativity. Electromagnetism and linearized gravity (more generally higher-spin fields) are governed by hyperbolic systems of partial differential equations. Solutions to these systems can be generated by the mean of potentials (here, Hertz potentials) satisfying a wave equation. It is possible to recast the problem of representing a solution to these higher-spin fields by Hertz potentials in the context of the initial value problem. Initial data for higher-spin fields satisfy constraint equations, and cannot be chosen freely. The integrability conditions for these constraints are described by elliptic complexes. These elliptic complexes also happen to be those describing the relation between initial data for higher-spin fields and those for their Hertz potentials. The problem of describing the asymptotic behavior of generic solutions to higher-spin fields can then be completely deduced from the... more