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Cohomology in algebra, geometry, physics and statistics

usually takes place every Wednesday at 11:30 AM Institute of Mathematics of ASCR, Žitná 25, Praha 1, konírna
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.

Symmetry, holonomy and special geometries
Petr Zima
Charles University
Wednesday, 26. May 2021 - 11:30 to 12:30
ZOOM meeting
Various types of geometrical structures can be described via their so
called structure group. This becomes especially apparent when studying
homogeneous spaces. Those spaces are of the form G/H where G is a
transitive symmetry group and H is the isotropy subgroup which plays the
role of structure group. A natural question is to ask whether we can
enlarge or reduce the structure group while preserving the geometrical
structure. Particular answer is given by the notion of holonomy that
provides the smallest possible structure group H. We will review these
notions and demonstrate them by examples of special Riemannian
geometries.

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We open ZOOM at 11.15 and close  at 13.00
Join Zoom Meeting

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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. 
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We shall  open ZOOM  at 11.15 and close at 13.00

Join Zoom Meeting

https://cesnet.zoom.us/j/99598413922?pwd=... more

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.
... more

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

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