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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.

Some constructions from graded geometry

Vladimir Salnikov
La Rochelle University, France
Wednesday, 7. December 2022 - 13:30 to 14:30
in IM building, ground floor + ZOOM meeting
In this talk I introduce some natural constructions from the "graded world", paying particular attention to the differences between N- and Z- graded manifolds. I will start by the construction of the sheaf of functions on graded manifolds and describe its structure. The intrinsic properties of this functional space are conveniently given using the language of filtrations, allowing to formulate the analog of Batchelor’s theorem. Afterwards I will briefly introduce graded Hopf algebras and Harish-Chandra pairs, which in turn provide the result of equivalence of categories between graded Lie groups and algebras. These constructions are then used to solve the integration problem of differential graded Lie algebras to differential graded Lie groups. Time permitting, I will also say a few words on canonical forms of differential graded manifolds.

This talk is based on:
[1] B. Jubin, A. Kotov, N. Poncin, V. Salnikov, Differential graded Lie groups and... more

Jordan algebras, coadjoint orbits, and information geometry

Florio Ciaglia
Universidad Carlos III de Madrid,
Wednesday, 14. December 2022 - 13:30 to 14:30
ZOOM meeting
The purpose of this talk is to present a connection between the mathematical entities mentioned in the title. It will be argued that Jordan algebras provide a suitable playground in which parametric models of classical and quantum information geometry can joyfully play (and hopefully thrive). In order to recover the Riemannian geometry of parametric models extensively used in classical and quantum information geometry, the method of coadjoint orbits will be adapted to Jordan algebras. Indeed, given the symmetric nature of the Jordan product, the analogue of the Konstant-Kirillov-Souriau symplectic form becomes a symmetric covariant tensor field. When suitable choices of Jordan algebras are made, it is possible to recover the Fisher-Rao metric tensor characteristic of classical information geometry or the Bures-Helstrom metric tensor appearing in quantum information geometry. This instance tells us that geometrical structures in information geometry can be found looking at... more


Hoang Duc Luu
Max-Planck-Institute for Mathematics in Sciences, Leipzig
Wednesday, 4. January 2023 - 13:30 to 14:30
ZOOM seminar


Fedor Part
Institute of Mathematics, Czech Academy of Sciences
Wednesday, 1. March 2023 - 13:30 to 14:30
in IM building, ground floor +ZOOM meeting