# 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.

### Smooth vector spaces

##### Enxin Wu
Shantou University, China
Wednesday, 27. April 2022 - 13:30 to 14:30
ZOOM meeting
Vector spaces are fundamental objects in mathematics. In practice,
vector spaces from functional analysis and vector bundle theory carry smooth
information. In this talk, I will present a general homology theory of such vector
spaces in the setting of diffeology. The connection to topological vector spaces
and vector bundle theory will be discussed.

---------------------------------------------------------------
We  open ZOOM at 13.15 for virtual cafe and close at  15.00.
Join Zoom Meeting

https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09

Meeting ID: 995 9841 3922

Passcode: Galois

### Structures of G(2) type in super setting and in positive characteristic, and related curvature tensors

##### Dimitri Leites
Wednesday, 20. April 2022 - 13:30 to 14:30
ZOOM meeting
Cartan and Killing described finite-dimensional simple Lie algebras (over fields of real or complex numbers) in terms of the distributions they preserve. The technique of root system and Dynkin (Coxeter) graphs was discovered several decades later. Two o the four series of simple infinite-dimensional Lie algebras of vector fields are Cartan prolongations of non-positive parts of simple finite-dimensional Lie algebras. For any $\mathbb{Z}$-grading of any simple finite-dimensional Lie algebra $\mathfrak{g}$ (bar the two series of examples), the Cartan prolongation of the non-positive part of $\mathfrak{g}$ returns $\mathfrak{g}$. This is not so for the exceptional Lie algebra $\mathfrak{g}_2$ in characteristic 5, whose Cartan prolongation is called Melikyan algebra. Recall that the Lie superalgebras appeared not in high energy physics in 1970s, but in topology, and either over $\mathbb{Z}$ as super Lie rings, or over finite fields. Lately, modular Lie (super)algebras became of interest... more

### Combinatorics of multilinear differential operators, or still another explanation of the ubiquity of Lie and strongly homotopy Lie algebras

##### Martin Markl
Institute of Mathematics of the Czech Academy of Sciences
Wednesday, 13. April 2022 - 13:30 to 14:30
in IM building, ground floor +ZOOM meeting
As a motivation, we start with an analysis of the interplay between the classical Jacobi identity and differential operators, and
compare it with the effect of the associator.  Moving to the `quantized' level, we compare the nature of the big bracket and
IBL(=infinitesimal Lie bialgebras)-infinity algebras with Terilla's quantization of associative algebras.
In the second part, we introduce a filtration mimicking combinatorial properties of multidifferential operators, and
the associated notion  of tight operads. We then come back to Lie algebras and give another reason why they deserve
to be, along with commutative and associative algebras, recognized as one of the Three Graces.

The talk will be based on the paper   "Calculus of multilinear differential operators, operator $L_\infty$-algebras and $IBL_\infty$-algebras"
of Denis Bashkirov and mine. Its preprint is available at
https://arxiv.org/abs/2108.12158... more

### Symmetry and Separation of variables

##### Stepan Hudecek
Charles University
Wednesday, 6. April 2022 - 13:30 to 14:30
in IM rear building, ground floor, blue lecture room +ZOOM meeting

We present a condition under which a differential operator on a two dimensional manifold admits a so-called separated solution and the separation is non-trivial in a sense, that we explain. Along the way we "develop" definitions in order to make these propositions precise, such as of a symmetry generating an operator and of a function that does not depend on a set of variables with respect to a coordinate chart.

We are motivated by problems in Physics, where the separation of variables is often used, e.g., in specific problems of electromagnetic waves, quantum mechanics (hydrogen atom), or in general relativity.  In mathematical Physics the notion of separation was studied in many works, including the works of Kalnins, Winternitz, Miller and Koornwinder. In a part of the Physics literature, the notion of the separation is studied without giving a definition of a separated solution.

In... more