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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 invariant differential calculus

Jan Slovak
Masaryk University
Wednesday, 17. May 2023 - 13:30 to 14:30
in IM building, ground floor + ZOOM meeting
For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian and projective geometries. More recently, several general tools were presented for the entire class of parabolic geometries, i.e., the Cartan geometries modelled on homogeneous spaces G/P with P a parabolic subgroup in a semi-simple Lie group G. Similarly to conformal Riemannian and projective structures, all these geometries determine a class of distinguished affine connections, which carry an affine structure modelled on differential 1-forms . They correspond to reductions of P to its reductive Levi factor, and they are called the Weyl structures similarly to the conformal case. The standard definition of differential invariants in this setting is as affine invariants of these connections, which do not depend on the
choice within the class. In this talk, we describe a universal calculus which... more

On 3-nondegenerate CR manifolds in dimension 7

Andrea Santi
University of Rome "Tor Vergata"
Wednesday, 10. May 2023 - 13:30 to 14:30
ZOOM meeting
I will report on joint works with B. Kruglikov on CR hypersurfaces in C 4 with a degenerate Levi form. I will discuss the symmetry dimension bound 8 for all the 3-nondegenerate 7-dimensional CR real-analytic structures and present the classification of the locally homogeneous ones. The bound 8 is achieved on the homogeneous model, which is locally the only homogeneous 3-nondegenerate 7-dimensional CR manifold.
We shall open the seminar room+ZOOM meeting at 13.15 for virtual coffee
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Massey tensors and the rational homotopy type of 7- and 8-manifolds

Johannes Nordstroem
University of Bath
Wednesday, 3. May 2023 - 13:30 to 14:30
ZOOM meeting

Defining tensors on the cohomology of a differential graded algebra by multiplying Massey products by a further element gives an object that has less dependence on choices than the Massey products, making them easier to work with. On the other hand, all Massey products can be recovered from these tensors when the cohomology algebra satisfies Poincare duality, like for the de Rham complex of a closed manifold. Moreover, suitable interpretations of these tensors can capture information about the rational homotopy type even when the Massey products are undefined. For closed simply-connected 7-manifolds these tensors (along with the cohomology algebra itself) suffice to determine the rational homotopy type. Conjecturally the same is true for closed 8-manifolds, but at least the vanishing of the tensors is equivalent to formality in that case. This is based on joint work with Diarmuid Crowley and Csaba Nagy.
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On local and global invariants of flat bundles

Sam Nariman
Purdue University
Wednesday, 26. April 2023 - 15:30 to 16:30
ZOOM meeting

Characteristic classes of flat bundles are related to the group cohomology of diffeomorphism groups.
But computing the group cohomology of these large groups as discrete groups is too complicated.
Even calculating the second cohomology of these groups will resolve deep problems in foliation theory.
However, it is easier to define certain natural cohomology classes (characteristic classes) in group
cohomology of diffeomorphism groups. We will discuss that they are rarely bounded classes but some
of them (e.g. Godbillon-Vey classes) are continuous cocycles. We will discuss some ways to detect their
nontriviality inspired by the work of Thurston and Bott-Segal.
Please  notice the TIME  SHIFT   THIS  WEDNESDAY  ONLY.
We shall open the ZOOM meeting at 15.15 for virtual coffee
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