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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.

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

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

G2 vs. Kahler

Tommaso Pacini
University of Torino, Italy
Wednesday, 30. November 2022 - 13:30 to 14:30
ZOOM meeting

We shall compare two categories of manifolds, known as G2 and Kahler, focusing on (i) their submanifold geometry, (ii) their function theory. We shall also discuss interactions between these topics. The talk is intended for a general audience.
In particular, we shall survey results in the preprints arXiv:2107.14117, arXiv:2207.13956,  arXiv:2208.125
We shall open ZOOM meeting at 13.15 for virtual coffee and close  ZOOM  at 15.00
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Condensed mathematics and set theory

Chris Lambie-Hanson
Institute of Mathematics, Czech Academy of Sciences
Wednesday, 16. November 2022 - 13:30 to 14:30
Institute of Mathematics, the blue lecture room in the rear building, + ZOOM meeting
Recently, Clausen and Scholze initiated the study of condensed mathematics, providing a framework in which to do algebra in situations in which the algebraic structures under consideration also carry topological information. The fundamental idea is to replace the categories of topological spaces or topological abelian groups, which are poorly behaved algebraically, with the more algebraically robust categories of condensed sets or condensed abelian groups. In this talk, we will give a very brief introduction to condensed mathematics and sketch a couple of very basic applications of set theoretic techniques to foundational questions therein. Time permitting, we will also briefly touch on related applications of these set theoretic ideas to other topics in homological algebra
We shall open the seminar room+ZOOM meeting at 13.15 for coffee  and close at 15.00
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