Geometric Methods in Physics XXXVII: Workshop and Summer School, Białowieża, Poland, 2018

271 Pages·2019·4.961 MB·English

by  Piotr Kielanowski, Anatol Odzijewicz, Emma Previato

Trends in Mathematics
Piotr Kielanowski
Anatol Odzijewicz
Emma Previato
Editors
Geometric
Methods in
Physics XXXVII
Workshop and Summer School,
Białowiez˙a, Poland, 2018

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Piotr Kielanowski • Anatol Odzijewicz
Emma Previato
Editors
Geometric Methods in Physics
XXXVII
Workshop and Summer School, Białowieża,
Poland, 2018
Editors
Piotr Kielanowski Anatol Odzijewicz
Departamento de Física Institute of Mathematics
CINVESTAV University of Białystok
Ciudad de México, Mexico Białystok, Poland

Emma Previato
Department of Mathematics and Statistics
Boston University
Boston, MA, USA
ISSN 2297-0215 ISSN 2297-024X (electronic )
Trends in Mathematics
ISBN 978-3-030-34071-1 ISBN 978-3-030-34072-8 (eBook )
https://doi.org/10.1007/978-3-030-34072-8

Mathematics Subject Classification (2010): 01-06, 20N99, 58A50, 58Z05, 81P16, 33D80, 51P05

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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
In Memoriam Bogdan Mielnik. . . . . . . . . . . . . . . . . . . . xi
G. Dito
Some aspects of the work of Daniel Sternheimer . . . . . . . . . . xiii
I Differential equations and integrable systems 1
M.V. Babich
On canonical parametrization of phase spaces of
Isomonodromic Deformation Equations . . . . . . . . . . . . . . . 3
A. Dobrogowska, G. Jakimowicz, and K. Wojciechowicz
On some deformations of the Poisson structure associated
with the algebroid bracket of differential forms . . . . . . . . . . . 13
D. Bermudez, D.J. Fernández, and J. Negro
Generation of Painlevé V transcendents . . . . . . . . . . . . . . 24
M.N. Hounkonnou and M. J. Landalidji
Hamiltonian Dynamics for the Kepler Problem
in a Deformed Phase Space . . . . . . . . . . . . . . . . . . . . . 34
G. Nugmanova, Zh. Myrzakulova, and R. Myrzakulov
Notes on integrable motion of two interacting curves
and two-layer generalized Heisenberg ferromagnet equations . . . . 49
A.K. Prykarpatski
About the solutions to the Witten–Dijkgraaf–Verlinde–Verlinde
associativity equations and their Lie-algebraic
and geometric properties . . . . . . . . . . . . . . . . . . . . . . 57
vi Contents
N. Yoshimi
2+2-Moulton Configuration – rigid and flexible . . . . . . . . . . 68
P. Lubowiecki and H. Żołądek
Melnikov functions in the rigid body dynamics . . . . . . . . . . . 75
II Quantization 83
R. Fresneda, J.P. Gazeau, and D. Noguera
E(2)-covariant integral quantization
of the motion on the circle and its classical limit . . . . . . . . . . 85
Y. Hirota, N. Miyazaki, and T. Taniguchi
On Deformation Quantization using
Super Twistorial Double Fibration . . . . . . . . . . . . . . . . . 92
G. Sharygin
Deformation Quantization of
Commutative Families and Vector Fields . . . . . . . . . . . . . . 100
S.B. Sontz
Co-Toeplitz Quantization: A Simple Case . . . . . . . . . . . . . 121
III Quantum groups and non-commutative geometry 127
T. Brzeziński and W. Szymański
On the quantum flag manifold SU (3)/T2 . . . . . . . . . . . . . 129
q
S. Halbig and U. Krähmer
A Hopf algebra without a modular pair in involution . . . . . . . 140
IV Infinite-dimensional geometry 143
E. Chiumiento
Hopf–Rinow theorem in Grassmann manifolds of C∗-algebras . . . 145
G. Larotonda
Short geodesics for Ad invariant metrics
in locally exponential Lie groups . . . . . . . . . . . . . . . . . . 153
Contents vii
V Miscellaneous 161
T. Hayashi, J.H. Hong, S.E. Mikkelsen, and W. Szymański
On Conjugacy of Subalgebras of Graph C∗-Algebras . . . . . . . . 163
T. Kanazawa
A Direct Proof for an Eigenvalue Problem
by Counting Lagrangian Submanifolds . . . . . . . . . . . . . . . 174
P. Moylan
Applications of the Fundamental Theorems
of Projective and Affine Geometry in Physics. . . . . . . . . . . . 181
B.S. Shalabayeva, N.Zh. Jaichibekov, B.N. Kireev, and Z.A. Kutpanova
Modeling the dynamics of a charged drop of a viscous liquid . . . 188
M. Myronova and M. Szajewska
The orthogonal systems of functions
on lattices of SU(n+1), n<∞ . . . . . . . . . . . . . . . . . . . 195
G.M. Tuynman
The Super Orbit Challenge . . . . . . . . . . . . . . . . . . . . . 204
T.Ł. Żynda
Weighted generalization of the Szegö kernel
and how it can be used
to prove general theorems of complex analysis . . . . . . . . . . . 212
VI Abstracts of the Lectures
at “School on Geometry and Physics” 219
A.Ya. Helemskii
Amenability, flatness and measure algebras . . . . . . . . . . . . . 221
G. Larotonda
Functional Analysis techniques
in Optimization and Metrization problems . . . . . . . . . . . . . 234
A. Sergeev
Twistor Geometry and Gauge Fields . . . . . . . . . . . . . . . . 240
A. Skalski
Quantum Dirichlet forms and their recent applications. . . . . . . 246
viii Contents
A. Tyurin
Lagrangian approach to Geometric Quantization . . . . . . . . . . 255
GeometricMethodsinPhysics.XXXVIIWorkshop2018
TrendsinMathematics,ix–x
(cid:2)c SpringerNatureSwitzerlandAG2019
Preface
ThisvolumecontainsaselectionofpaperspresentedduringtheThirty-Seventh
WorkshoponGeometricMethodsinPhysics in2018,organizedbytheInstitute
of Mathematics of the University of Białystok. About 70 physicists and math-
ematicians from important scientific centers from all over the world attended
the workshop. The Workshop was accompanied by the School of Geometry
and Physics, where several cycles of didactic lectures on important and new
subjects for advanced students and young scientists were presented. Abstracts
oftheselecturesarealsoincludedinthisvolume. Informationonpreviousand
upcoming schools and workshops, and related materials, can be found at the
URL: http://wgmp.uwb.edu.pl.
ThegeometricalmethodsinPhysicsconstituteaverywidebranchofmath-
ematical physics. The main topics that have been discussed this year are:
quantum groups, non-commutative geometry, integrable systems, differential
equations, operator algebras, quantization and infinitely dimensional geome-
try.
An important event during the workshop was a session dedicated to the
scientific activity of professor Daniel Sternheimer on the occasion of his 80th
birthday.
Białowieża,whichisatraditionalplace,wheretheWorkshopstakeplacede-
servesspecialmention. ItislocatedontheborderbetweenPolandandBelarus
andistheonlyplaceinEuropewheretherearetheremainsofprimevalforests
and was designated a UNESCO World Heritage Place. Such close contact
with nature creates a special atmosphere during the workshop and scientific
discussions take less formal character.
The Organizing Committee of the 2018 Workshop on Geometric Method
in Physics gratefully acknowledges the financial support of the University of
Białystok.
The Editors