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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.

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.

On embedded contact homology
Roman Golovko
Charles University, Praha
Wednesday, 5. December 2018 - 11:30 to 12:30
in IM building, ground floor
The embedded contact homology (ECH) of a closed, oriented 3-manifold with a contact form is a variant of the symplectic field theory of Eliashberg, Givental and Hofer. It is defined in terms of a contact form but is an invariant of the underlying 3-manifold. This invariance has been established by Taubes via the identification with Seiberg-Witten Floer (co-)homology as defined by Kronheimer and Mrowka and in particular implies the Weinstein conjecture in dimension three. There is also an isomorphism from embedded contact homology to Ozsvath-Szabo Heegaard Floer homology. We will discuss the construction of ECH and its applications.

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