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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.

Properads and Homotopy Algebras Related to Surfaces
Lada Peksova
Charles University
Wednesday, 12. June 2019 - 11:30 to 12:30
in IM building, ground floor
Barannikov showed how an algebra over the Cobar construction over a
modular operad can equivalently be described as a solution of a master
 equation for certain generalized BV algebra. We give an analogous
description for properads which were first introduced by B. Vallette
 as connected parts of PROPs.

 In short, we introduce properads along with our main examples, the
closed (commutative) Frobenius properad, and the open (associative)
 Frobenius properad. We consider the construction of the Cobar complex
 of a properad as a free properad over its suspended linear dual
equipped with the differential induced by the duals of the structure
 operations. We describe algebras over the Cobar construction in terms
 of solutions of generalized master equations and recover the
 well-known result that the corresponding algebras for closed Frobenius
properad are $IBL_\... more

Almost formality of manifolds of low dimension
Lorenz Schwachhoefer
TU Dortmund
Wednesday, 5. June 2019 - 11:30 to 12:30
in IM building, ground floor
This is based on work with Domenico Fiorenza, Kotaro Kawai and Hong Van Le.

A classical result by Miller states that any $(k-1)$-connected closed oriented manifold of dimension $\leq 4k-2$ is formal. More recently, Crowley and Nordstr\”om defined the  Bianchi-Massey tensor and showed that its vanishing is the only obstruction to the formality of a $(k-1)$-connected closed oriented manifold of dimension $\leq 5k-3$. Moreover, recently Chan-Karigiannis-Tsang showed that closed $G_2$-manifolds are almost formal, meaning that their deRham algebra is equivalent to a DGA whose differential vanishes in all but one dimension.

... more

Structure functions and Spencer cohomology in zero and positive characteristics
Pasha Zusmanovich
University of Ostrava
Wednesday, 29. May 2019 - 11:30 to 12:30
in IM building, ground floor
Structure functions are obstructions to integrability of G-structures, where G is a Lie group,
on a real or complex manifold. These well known invariants are expressed in terms of Spencer
cohomology of the corresponding graded Lie algebra. Many important particular cases of structure
functions are known under the names of Riemann tensor, Nijenhuis tensor, etc.
... more

First lecture: Local entropy for finite time dynamics, Second lecture: On the behavior of the Diaconis-Freedman chain in a multi-dimensional simplex
Hoang Duc Luu and Tat Dat Tran
MPI MIS Leipzig
Wednesday, 22. May 2019 - 11:15 to 12:45
in IM building, ground floor
Local entropy for finite time dynamics (by  Hoang Duc Luu)

Abstract: This talk aims to introduce a concept of entropy for difference and differential equations which is a local-in-space and transient-in-time version of the classical concept of metric entropy. Based on that, a finite-time version of Pesin’s entropy formula arederived. I would also discuss about how to apply the finite-time entropy field to detect special dynamical behavior such as Lagrangian coherent structures, and how to relate the topic to information geometry.

On the behavior of the Diaconis-Freedman chain in a multi-dimensional simplex (by Tat Dat... more

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