Nonlinear PDEs, Their Geometry, and Applications: Proceedings of the Wisła 18 Summer School (Tutorials, Schools, and Workshops in the Mathematical Sciences) PDF
Preview Nonlinear PDEs, Their Geometry, and Applications: Proceedings of the Wisła 18 Summer School (Tutorials, Schools, and Workshops in the Mathematical Sciences)
Tutorials, Schools, and Workshops in the Mathematical Sciences Radosław A. Kycia Maria Ułan Eivind Schneider Editors Nonlinear PDEs, Their Geometry, and Applications Proceedings of the Wisła 18 Summer School Tutorials, Schools, and Workshops in the Mathematical Sciences Thisserieswillserve asa resource for thepublicationofresultsanddevelopments presentedatsummerorwinterschools,workshops,tutorials,andseminars.Written in an informal and accessible style, they present important and emerging topics in scientific research for PhD students and researchers. Filling a gap between traditionallecturenotes,proceedings,andstandardtextbooks,thetitlesincludedin TSWMS present material from the forefront of research. More information about this series at http://www.springer.com/series/15641 ł ł Rados aw A. Kycia Maria U an (cid:129) (cid:129) Eivind Schneider Editors Nonlinear PDEs, Their Geometry, and Applications ł Proceedings of the Wis a 18 Summer School Editors Radosław A.Kycia Maria Ułan TheFaculty ofScience Baltic Institute of Mathematics MasarykUniversity Warszawa, Poland Brno, Czech Republic Faculty of Physics Mathematics and Computer Science CracowUniversity of Technology Kraków,Poland EivindSchneider Department ofMathematics andStatistics UiT TheArctic University of Norway Tromsø,Norway ISSN 2522-0969 ISSN 2522-0977 (electronic) Tutorials, Schools,andWorkshopsin the Mathematical Sciences ISBN978-3-030-17030-1 ISBN978-3-030-17031-8 (eBook) https://doi.org/10.1007/978-3-030-17031-8 LibraryofCongressControlNumber:2019935998 MathematicsSubjectClassification(2010): 35Q30,45Gxx,35Q79,35J96,35K96 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland We would like to dedicate this book to the speakers and scientific committee: A. Kushner, V.V. Lychagin, I. Roulstone, V. Rubtsov, J. Slovák, S.N. Tychkov, M. Wolf, and organizing committee R.A. Kycia, J. Szmit, M. Ułan. Extra acknowledgements to O.A. Laudal, U. Persson. Foreword TheSummerSchoolWisla18:NonlinearPDEs,TheirGeometry,andApplications was organized by the Baltic Institute of Mathematics and took place in a beautiful mountainregionofWisłacityinthesouthofPolandbetween20and30ofAugust 2018.ThecityislocatedintheBeskidMountainsandissurroundedbymountains thatarecoveredbywildforest.Thenameofthecitycomesfromthebiggestriverin Polandwhichhasitssourcenearthetown.InPoland,itisawell-knowngoodplace for hiking, biking, and skiing and even ski-jumping in winter. Wisła is also a spa. The city is located near the border with the Czech Republic and a few kilometers from the border with Slovakia, with the unique triple point where the borders of Poland, The Czech Republic, and Slovakia meet. The school was devoted to geometric theory of differential equations and applications to physics. A special place is occupied by the theory of the Monge– Ampère equations and their applications in meteorology. There was also a session for learning how to use the Computer Algebra System Maple® in practical com- putations.Thesecondpartoftheschoolwasdevotedtoparticipant’scontributions. Thissplittingintolecturesandcontributionsofparticipantsisreflectedinthisbook. vii viii Foreword Fig.1SunsetoverthemountainssurroundingWisła(photographybyEivindSchneider) Brno, Kraków, Oslo, Tromsø, Warszawa Maria Ułan 2019 Eivind Schneider Radosław A. Kycia Preface This book is a summary of The Summer School Wisla 18: Nonlinear PDEs, Their Geometry, and Applications that took place on 20–30 August 2018 in Wisła, Poland.Itisdividedintotwocomplementaryparts:Thefirstpartthebookpresents geometric methods in nonlinear differential equations and their application to physics(includingthermodynamicsandmeteorology)ispresented.Theyaresorted in the ascending order from introductory to more advanced. It is required that the reader has basic knowledge of differential geometry (at the level of standard uni- versity course of Global Analysis and/or Differential Geometry) and some knowledgeinphysicsforlecturestouchinguponthissubject.Thesecondpartofthe book contains participant’s contributions. They are original research articles that show how to apply the methods present in the first part. Some of the papers are complementarytothelecturesandpresentdifferentviewpointsonthesamesubject. This part is more advanced and concise, however a motivated reader should not havebigproblemswithfollowingthemafterreadingthelecturenotesfromthefirst part of the book. The first part of the book contains lectures given at the School and focuses on various subjects from the geometric theory of differential equations. The first lecture set in the book by Valentin V. Lychagin explains interesting connections between the theory of measurements, contact geometry, and thermo- dynamics. Its novel approach should be interesting to anyone who wants to investigatetheintriguingconnectionbetweenprobabilityanddifferentialgeometry. The second set of lectures by Alexei Kushner, Valentin V. Lychagin, and Jan Slovák contains an introduction to geometric theory of differential equations, reformulation of the Monge–Ampère equations in this language, and examples of use of Maple CAS to computations. It should be interesting for anyone new to the subject. The following lectures by Volodya Rubtsov take the reader deeper into the Monge–Ampère equations and related structures including complex, Kähler and hyperkähler. The last chapter in this series can be treated as an introduction to the equations used in meteorology. ix x Preface The final set of lectures by Sergey N. Tychkov introduces, in the form of a hands-on approach, symbolic computations indifferential geometry using Maple®. The second part of the book contains research papers by participants of the school. They are more concise and the reader can treat them as an exercise in understanding the material from the first part. They describe various subjects and therefore they give reader a glimpse of current research in these areas. Wehopethatthemixoflecturesandoriginalresearcharticleswillgiveyou,dear reader, a good starting point in your journey to the world of geometric theory of differential equations and their applications, and that they will help you improve your skills in this field. Brno, Kraków, Oslo, Tromsø, Warszawa Maria Ułan January 2019 Eivind Schneider Radosław A. Kycia
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