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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, konírna
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.

Koszul algebras and one-dependent random 0-1 sequences
Leonid Positselski
IM, CAS
Wednesday, 1. April 2020 - 11:30 to 12:30
https://cesnet.zoom.us/j/239249755 Meeting ID 239-249-755
Koszul algebras are a natural class of graded algebras with
quadratic relations, defined by a series of homological conditions.
To a Koszul algebra over a field with finite-dimensional components,
one can assign a one-dependent stochastic 0-1 sequence, which carries
information about the dimensions of the algebra's grading components.
... more

Cyclic homology for bornological coarse spaces CANCELLED because of COVID-19
Luigi Caputi
Institute of Informatics of the Czech Academy of Sciences, Praha
Wednesday, 25. March 2020 - 11:30 to 12:30
in konirna seminar room of the front IM building, ground floor
Bornological coarse spaces are "large scale" generalizations of metric spaces (up to quasi-isometry). Homological invariants of such spaces are given by coarse homology theories, which are functors from the category of bornological coarse spaces to a stable cocomplete ∞-category, satisfying additional axioms. Among the main examples of coarse homology theories, there are coarse versions of ordinary homology, of topological
and algebraic K-theory. In the talk we define G-equivariant coarse versions of the classical Hochschild and cyclic homologies of algebras. If k is a field, the evaluation at the one point space induces equivalences with the classical Hochschild and cyclic... more

The cohomological characterization of information functions >

Juan Pablo Vigneaux
Max-Planck-Institute for Mathematics in Sciences
Wednesday, 11. December 2019 - 11:30 to 12:30
in konirna seminar room, IM front building, ground floor
Information structures are categorical objects that serve as models of systems of measurements in physics and
analog concepts in  computer science and logic. "Information cohomology" is a homological invariant naturally attached to the presheaves on them along the lines of SGA IV. Several functions appear as cocycles (for adapted modules of coefficients): Shannon entropy and some of its generalizations, the multinomial coefficients and their generalizations, the determinant of gaussian covariant matrices, the dimension of affine subspaces... The  cocycle conditions encode remarkable recurrence relations of these functions, and some of them can be put in correspondence (e.g. the multiplicative relations among multinomial coefficients imply the  chain rule for the corresponding entropy). These are the first steps  towards a general "topology of statistical systems" in the vein of topos theory.

Multiple sewn cohomology for vertex algebras
Sasha Zuevsky
IM, CAS, Praha
Wednesday, 4. December 2019 - 11:30 to 12:30
seminar room Konirna in IM front building, ground floor
In 2010 Y.Z. Huang has introduced the notion of a cohomology of grading-restricted vertex algebras. The main construction of coboundary operator was given using considerations of rational functions obtained as matrix elements for such vertex algebras.  As we know from the theory of correlation functions for vertex algebras, matrix elements correspond to the choice of formal parameters for vertex operators to be local coordinates on the complex sphere. One can consider more complicated situation when local coordinates for vertex operators are taken on a complex sphere sewn to itself. This procedure would change coboundary operators and enrich the cohomological structure of corresponding vertex algebras. In this lecture we introduce the cohomology... more

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