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.
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, konírnaChair: 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.
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
(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.
Contact Geometry of Hydrodynamic Integrability (will be cancelled because of open door days)
Artur Sergyeyev
University of Opava
Wednesday, 7. November 2018 - 11:30 to 12:30
in IM building, ground floor
The search for integrable partial differential systems in four independent variables (4D) is a longstanding problem in mathematical physics. In the present talk we address this problem by introducing a new construction for integrable 4D systems which are dispersionless (a.k.a. hydrodynamic-type) using nonisospectral Lax pairs that involve contact vector fields. In particular, we show that there is significantly more integrable 4D systems than it appeared before, as the construction in question produces new large classes of integrable 4D systems with Lax pairs which are polynomial and rational in the spectral parameter. For further details please see A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, https://arxiv.org/abs/1401.2122