Geometric Methods in Physics XXXV: Workshop and Summer School, Białowieża, Poland, June 26 – July 2, 2016 PDF
Preview Geometric Methods in Physics XXXV: Workshop and Summer School, Białowieża, Poland, June 26 – July 2, 2016
Trends in Mathematics Piotr Kielanowski Anatol Odzijewicz Emma Previato Editors Geometric Methods in Physics XXXV Workshop and Summer School, Białowie˙za, Poland, June 26 – July 2, 2016 Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/4961 Piotr Kielanowski • Anatol Odzijewicz Emma Previato Editors Geometric Methods in Physics XXXV Workshop and Summer School, Białowieża, Poland, June 26 – July 2, 2016 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 978-3-319-63593-4 ISBN 978-3-319-63594-1 (eBook) https://doi.org/10.1007/978-3-319-63594-1 Library of Congress Control Number: 2017962618 Mathmatics Subject Classification (2010): 01-06, 01A70, 20N99, 58A50, 58Z05, 81P16, 33D80, 51P05 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents Preface .................................................................. ix G.A. Goldin In Memory of S. Twareque Ali ...................................... xi Part I: Geometry & Physics A. Antonevich Quasi-periodic Algebras and Their Physical Automorphisms ......... 3 I. Belti¸ta˘, D. Belti¸ta˘ and B. Cahen Berezin Symbols on Lie Groups ..................................... 11 T. Brzezin´ski and L. Da¸browski A Curious Differential Calculus on the Quantum Disc and Cones .... 19 M. Fecko Nambu Mechanics: Symmetries and Conserved Quantities ........... 27 L. Jeffrey, S. Rayan, G. Seal, P. Selick and J. Weitsman The Triple Reduced Product and Hamiltonian Flows ................ 35 B. Mielnik The Puzzle of Empty Bottle in Quantum Theories .................. 51 V.F. Molchanov Poisson Transforms for Tensor Products in Compact Picture ........ 61 T. Kobayashi, T. Kubo and M. Pevzner Conformal Symmetry Breaking Operators for Anti-de Sitter Spaces ............................................... 69 A.K. Prykarpatski Hamilton Operators and Related Integrable Differential Algebraic Novikov–Leibniz Type Structures ......................... 87 vi Contents C. Roger An Algebraic Backgroundfor Hierarchies of PDE in Dimension (2|1) .................................................. 95 A.I. Shafarevich LagrangianManifolds and Maslov Indices Corresponding to the Spectral Series of the Schr¨odinger Operators with Delta-potentials ............................................... 103 J. Smotlacha and R. Pincak Electronic Properties of Graphene Nanoribbons in a Uniform Magnetic Field ...................................................... 111 E. Stro´z˙yna Formal Normal Forms for Germs of Vector Fields with Quadratic Leading Part. The Rational First Integral Case ..................... 119 H. Z˙ol(cid:3)a¸dek The Poncelet Theorems in Interpretation of Rafal(cid:3) Ko(cid:3)lodziej ......... 129 Z. Pasternak-Winiarski and T.L. Z˙ynda Weighted Szego˝ Kernels ............................................. 145 Part II: Integrability & Geometry I. Cheltsov On a Conjecture of Hong and Won .................................. 155 C.D. Guevara and S.P. Shipman Short-time Behavior of the Exciton-polariton Equations ............. 175 I. Horozov Periods of Mixed Tate Motives over Real Quadratic Number Rings ...................................................... 181 W.X. Ma and X. Lu¨ Soliton Hierarchies from Matrix Loop Algebras ...................... 191 C.A. Tracy and H. Widom On the Ground State Energy of the Delta-function Fermi Gas II: Further Asymptotics ................................................ 201 D. Zakharov and V. Zakharov Non-periodic One-gap Potentials in Quantum Mechanics ............ 213 Contents vii Part III: Abstracts of the Lectures at “School on Geometry and Physics” A. Bolsinov Integrable Geodesic Flows .......................................... 229 M. Boz˙ejko Anyonic Fock Spaces, q-CCR Relations for |q|=1 and Relations with Yang–Baxter Operators ........................................ 237 T. Brzezin´ski Differential and Integral Forms on Non-commutative Algebras ....... 249 J. Kijowski General Relativity Theory and Its Canonical Structure .............. 255 B. Mielnik Exponential Formulae in Quantum Theories ......................... 261 E. Previato Complex Algebraic Geometry Applied to Integrable Dynamics: Concrete Examples and Open Problems ............................. 269 Th. Voronov Volumes of Classical Supermanifolds ................................ 281 TrendsinMathematics,ix–x (cid:2)c 2018SpringerInternational Publishing Preface ThisbookcontainsaselectionofpaperspresentedduringtheThirty-Fifth“Work- shoponGeometricMethodsinPhysics”(WGMPXXXV)andabstractsoflectures given during the Fifth “School on Geometry and Physics”,which both took place in Bia(cid:3)lowiez˙a, Poland, in the summer 2016. These two coordinated activities are an annual event. Information on the previous and the upcoming occurrences and related materials can be found at the URL: http://wgmp.uwb.edu.pl. Thevolumeisdividedintofourparts.Itopenswithapaperdedicatedtothe memory of S. TwarequeAli – for many yearsan active member of the Organizing Committee of ourworkshopwho diedsuddenly in2016.The secondpart,“Geom- etry and Physics”, includes papers based on talks delivered during the workshop. The third part, “Integrability and Geometry”, is based on the eponymous special session, organized by G.A. Goldin, A. Odesskii, E. Previato, E. Shemyakova and Th.Voronov.Thefinalpartcontainsextendedabstractsofthelecture-seriesgiven during the Fifth “School on Geometry and Physics”. The WGMP is an international conference organized each year by the De- partment of Mathematical Physics in the Faculty of Mathematics and Computer Science of the University ofBial(cid:3)ystok,Poland.The maintheme ofthe workshops, consistentwith the title, is the application of geometric methods in mathematical physics and it includes a study of non-commutative systems, Poisson geometry, completely integrable systems, quantization, infinite-dimensional groups, super- groupsandsupersymmetry,quantumgroups,Liegroupoidsandalgebroidsaswell asrelatedtopics.Participationintheworkshopsisopen;thetypicalaudiencecon- sists of physicists and mathematicians from many countries in several continents with a wide spectrum of interests. Workshop and School are held in Bia(cid:3)lowiez˙a, a village located in the east of Poland near the border with Belarus. Bia(cid:3)lowiez˙a is situated on the edge of the Bial(cid:3)owiez˙a Forest, shared between Poland and Belarus, which is one of the last remnants of the primeval forest that covered the European Plain before human settlement and was designated a UNESCO World Heritage Site. The peaceful at- mosphereofasmallvillage,combinedwithnaturalbeauty,yieldsauniqueenviron- mentforlearningandcooperating:asaresult,thecoreaudienceoftheWGMPshas become a strong scientific community, documented by this series of Proceedings. The Organizing Committee of the 2016 WGMP gratefully acknowledges the financialsupportoftheUniversityofBia(cid:3)lystokandtheBelgianSciencePolicyOf- x Preface fice (BELSPO),IAP GrantP7/18DYGEST. Thanks alsogoto the U.S. National ScienceFoundationforprovidingsupportto participantsinthe“Integrabilityand Geometry” session of the event, Grant DMS 1609812. Last but not least, credit is due to early-careerscholarsand students from the University of Bia(cid:3)lystok, who contributed limitless time and effort to setting up and hosting the event, aside from being active participants in the scientific activities. The Editors Participants of the XXXV WGMP (Photo byTomasz Golin´ski)
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