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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.

Segal-Shale-Weil representation and connected global analysis
Svatopluk Krysl
MÚ UK
Wednesday, 20. March 2019 - 11:30 to 12:30
in IM building, ground floor

The Segal-Shale-Weil representation, which is a unitary representation of the double cover of a finite dimensional symplectic Lie group, was introduced by D. Shale a A. Weil in the beginning of sixties. This representation is unitary and splits into two irreducible (i.e., simple closed) modules. Its complex version can be induced to the principle bundle of the double cover of (complexified) symplectic frames defined over a symplectic manifold, similarly as spinor bundles are associated to spin structures on Riemannian manifolds. Resulting bundles are called symplectic spinor bundles.

For the induced bundle structures, one can define several operators of Dirac type (K. Habermann around 1995). However since the symplectic spin bundles are of infinite rank, analysis for these operators is rather difficult. Especially till now, just spectral problems over homogeneous symplectic manifolds can be treated and no qualitative results for more general cases are... more

Homogeneous 8-manifolds admitting invariant Spin(7)-structures
Ioannis Chrysikos
Hradec Kralove University
Wednesday, 13. March 2019 - 11:30 to 12:30
in IM building, ground floor

We study invariant Spin7-structures on 8-dimensional compact homogeneous spaces. We classify all simply-connected non-symmetric compact homogeneous spaces M = G/K of a compact almost effective Lie group G, which admit a G-invariant Spin... more

Information cohomology and Topological Information Data Analysis
Pierre Baudot
Median Technologies
Wednesday, 6. March 2019 - 11:30 to 12:30
in IM building, ground floor
We establish methods that quantify the statistical interactions structure within a given data set using the characterization of information theory in cohomology by finite methods, and provide their expression in term of statistical physic and machine learning.

In a first part, we will have a look at the formalism of Information Cohomology obtained with Daniel Bennequin and refined by Juan Pablo Vigneaux with extension to Tsallis entropies [1,2]. It considers random variables as partitions of atomic probabilities and the associated poset given by their lattice. The basic cohomology is settled by the Hochschild coboundary, with a left action corresponding to information conditioning. The first degree cocycle is the entropy  chain rule, allowing to derive the functional equation of information and hence to characterize entropy uniquely as the first group of the  cohomology. (minus) Odd multivariate mutual informations (MI, I2k+1) appears as even degrees... more

The rational homotopy type of k-connected manifolds of dimension < 5k+3
Johannes Nordstrom
University of Bath, UK
Wednesday, 12. December 2018 - 11:30 to 12:30
in IM building, ground floor
The main aim of the talk is to introduce the "Bianchi-Massey tensor" of a topological space: a certain linear map on a subspace of the fourth tensor power of the rational cohomology, with symmetries analogous to the Riemann curvature tensor. In the case of closed k-connected manifolds of dimension at most 5k+2 (with k > 0), the cohomology algebra and Bianchi-Massey tensor turn out to be enough to completely determine the rational homotopy type, i.e., the equivalence class up to continuous maps that induce isomorphisms on the rationalised homotopy groups. This is joint work with Diarmuid Crowley.

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