Nonlinear PDEs, Their Geometry, and Applications: Proceedings of the Wisła 18 Summer School (Tutorials, Schools, and Workshops in the Mathematical Sciences)

289 Pages·2019·1.52 MB·English

by  Radosław A. Kycia (editor), Maria Ułan (editor), Eivind Schneider (editor)

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
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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