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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.

Regularized information geometric and optimal transport distances between covariance operators and Gaussian processes
Ha Quang Minh
RIKEN Institute, Tokyo
Wednesday, 24. November 2021 - 11:30 to 12:30
ZOOM meeting

Information geometry (IG) and Optimal transport (OT) have been attracting much research attention in various fields, in particular machine learning and statistics. In this talk, we present results on the generalization of IG and OT distances for finite-dimensional Gaussian measures to the setting of infinite-dimensional Gaussian measures and Gaussian processes. Our focus is on the Entropic Regularization of the 2-Wasserstein distance and the generalization of the Fisher-Rao distance and related quantities. In both settings, regularization leads to many desirable theoretical properties, including in particular dimension-independent convergence and sample complexity. All of the presented formulations admit closed form expressions that can be efficiently computed and applied practically.
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ZOOM meeting shall be... more

Shortest and straightest geodesics in sub-Riemannian geometry
Dmitri Alekseevsky
Institute for Information Transmission Problems, Moscow
Wednesday, 10. November 2021 - 11:30 to 12:30
ZOOM meeting
We present a short introduction to sub-Riemannian geometry, concentrating
on the various denitions of sub-Riemannian geodesics and their relationships.
E. Herz remark that there are two main characterisations of geodesics in
Riemannian geometry: geodesics as shortest curves, based on the Mopertrui's
principle of least action ( variational approach ) and
geodesics as straightest curves based on d'Alembert's principle of virtual
work.
These lead to dierent, but equivalent denitions of geodesics in Riemannian
geometry. These denitions can be generalized to sub-Riemannian geometry,
but they become non equivalent. We consider 3 denitions of sub-Riemannian
geodesics as shortest curves (Euler-Lagrange, Hamilton and Pontryagin),
which mostly used in control theory and 3 denitions of geodesics as straightest
curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka-Morimoto ),
used in nonholonomic mechanics.... more

A categorical approach to quantum probability
Arthur J. Parzygnat
IHES, Paris
Wednesday, 3. November 2021 - 11:30 to 12:30
ZOOM meeting
Recent advances in categorical probability theory suggest ideas on how to make inference in quantum mechanics. I will focus on two cases, which are Bayesian updating and disintegrations. Bayesian updating can be viewed as an algorithm for making decisions or guesses based on evidence. Disintegrations are special cases and are closely related to conditional expectations and error correction in classical and quantum computation. This will be an introduction to the subject, and I will give many examples.
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Our ZOOM meeting shall be opened at 11.15 and closed at 1 PM.
Join Zoom Meeting
https://cesnet.zoom.us/j/99598413922?pwd=YXNFbk50aVhleXhWSGtISFViLytRUT09
Meeting ID: 995 9841 3922
Passcode: Galois
 

On the differential geometry of groups of diffeomorphisms and of non-formal pseudo-differential operators
Jean-Pierre Magnot
University d'Angers, France
Wednesday, 27. October 2021 - 11:30 to 12:30
ZOOM meeting
Summary:
After reviewing a class of infinite dimensional groups based on the central extension of a group of diffeomorphisms by a group of pseudo-differential operators (PDOs), I will explain:
1) how the action of the group of diffeomorphisms generates the dressing operator of a KP hierarchy, which is shown to be well-posed in a class of NON FORMAL PDOs
2) How renormalized traces enables to define pseudo-Riemannian metrics on some of these groups of PDOs, different from the classical sobolev metrics present in the literature
3) how the geodesic equation of one of these metrics admit an infinite number of independent  integrals of the motion
Part of the results of this talk are obtained from works in collaboration with Enrique G. Reyes. Arxiv identifiers of related publications/preprints are: 2104.08159 ; 2007.00387 ; 1808.03791 and 1407.1427... more

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