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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 and Hong Van Le

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.

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

TBA

Ioannis Chrysikos
Hradec Kralove University
Wednesday, 13. March 2019 - 11:30 to 12:30
in IM building, ground floor
TBA

TBA

Svatopluk Krysl
Charles University Prague
Wednesday, 20. March 2019 - 11:30 to 12:30
in IM building, ground floor
TBA

Combinatorial homotopy theory for operads

Jovana Obradovic
Institute of Mathematics, CAS
Wednesday, 27. March 2019 - 11:30 to 12:30
in IM building, ground floor
We introduce an explicit combinatorial characterization of the minimal model of the
coloured operad encoding non-symmetric operads, introduced in [3]. The polytopes
of our characterization are hypergraph polytopes [1, 2] whose hypergraphs arise in a
certain way from rooted trees – we refer to them as operadic polytopes. In particular,
each operadic polytope displays the homotopy relating different ways of composing
the nodes of the corresponding rooted tree. In this way, our operad structure gener-
alizes the structure of Stasheff’s topological A ∞ -operad: the family of associahedra
corresponds to the suboperad determined by linear rooted trees. We then further
generalize this construction into a combinatorial resolution of the coloured operad
encoding non-symmetric cyclic operads.
References
[1] K. Došen, Z. Petrić, Hypergraph polytopes, Topology and its Applications 158,1405–1444, 2011.
[2] P.-L. Curien, J.... more