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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.

Losik classes for codimension one foliations

Yaroslav Bazaikin and Anton Galaev
University Hradec Kralove
Wednesday, 28. November 2018 - 11:00 to 12:30
UHK, Building S, Room S48, Faculty of Science, Hradecká 1285, 500 03 Hradec Králové

Following Losik's approach to Gelfand formal geometry, certain
characteristic classes for codimension one foliations coming from
Gelfand-Fuchs cohomology are considered. Sufficient conditions for
non-triviality in terms of the dynamical properties of generators
of the holonomy groups are found. The non-triviality for the Reeb
foliation is shown; this is in contrast with some classical theorems
on the Godbillon-Vey class, e.g, the Mizutani-Morita-Tsuboi Theorem
about triviality of the Godbillon-Vey class of foliations almost
without holonomy is not true for the classes under consideration. It
is shown that the considered classes are trivial for a large class
of foliations without holonomy. The question of triviality is
related to ergodic theory of dynamical systems on the circle and to
the problem of smooth conjugacy of local diffeomorphisms. Certain
classes are obstructions for the existence of... more

Lyapunov cohomology in random dynamical systems: theory and applications

Luu Hoang Duc
MIS Leipzig, Germany \& VAST Hanoi, Vietnam
Wednesday, 21. November 2018 - 12:00 to 12:45
in IM building, ground floor

In this talk I will give a survey on the notion of cohomology in the theory of random dynamical systems, which have its origin in homological algebra. A classical result is that the Lyapunov spectrum and random attractors are invariant under a Lyapunov cohomology. Cohomologies therefore help to define the structural stability of stochastic systems and to study the bifurcation phenomena. For applications, I will present our new results on random attractors for rough differential equations using the cohomology method.

(Co)homology vs. Information Geometry in Mathematical Population Genetics

Tran Tat Dat
MIS Leipzig, Germany
Wednesday, 21. November 2018 - 11:15 to 12:00
in IM building, ground floor

In this talk, I will first recall some constructions of topological invariants on networks based on (co)homology theory. I will then give a brief introduction to geometric invariants on networks based on information geometry. I will discuss about a connection between these invariants and apply it in the context of population genetics.

Exponential maps, ODE-s and a Hopf algebroid

Zoran Škoda
University of Zadar, Croatia and University of Hradec Kralove
Wednesday, 14. November 2018 - 11:30 to 12:30
in IM building, ground floor

Hopf algebroids over noncommutative base are generalizations of convolution algebras of groupoids. In a series of works with collaborators we have constructed a completed Hopf algebroid of differential operators on a Lie group and related examples of Drinfeld-Xu twists. Comparing the twist formulas in an arbitrary coordinate chart with the twist formulas in the normal coordinates (i.e. given by the exponential map) I have observed a new formulas for passage between the coordinates, inverse to the passage given by solving flow ODEs. It is interesting to compare these formulas with other series for ODEs including the ones coming from Lie integrators, Runge-Kutta integrators and the related considerations around Butcher group. As the latter combinatorics is parallel to the combinatorics of renormalization of QFTs, I expect that our approach could be eventually useful there as well.