% Copyright (C) 2020-2024 by Thomas Auzinger <thomas@auzinger.name>

\PassOptionsToPackage{usenames,dvipsnames,table}{xcolor} % Ensure that the document loads packages with all required options.
\documentclass[draft,final]{istaustriathesis}


% Set the copyright license. The possible choices are 'restricted' to reserve all rights or a Creative Commons license.
% See https://creativecommons.org/choose/ for information on choosing a license.
% The supported Creative Commons licenses are 'cc-by', 'cc-by-sa', 'cc-by-nd', 'cc-by-nc', 'cc-by-nc-sa', and 'cc-by-nc-nd'.
\setlicense{cc-by} % Choose 'restricted' or any of the 'cc-by...'.
% If a Creative Commons license is chosen, the corresponding license information can be added to the PDF metadata.
% For this, the package 'doclicense' needs to be uncommented below.
% Furthermore, the 'modifier' option of 'doclicense' needs to mirror the chosen Creative Commons license.
\usepackage[type={CC},modifier={by-nc-sa},version={4.0},hyperxmp=false]{doclicense} % Sets the license metadata. See copyright licensing below.
% If no Creative Commons license is chosen, then the appropriate copyright text and (potentially empty) URL must be entered manually.
% For this, edit the command \Copyright{...} and \CopyrightURL{...} in the filecontents* environment of \jobname.xmpdata below.

% The pdfx package throws an error when utilizing the \xspace command inside the XMP metadata generation.
% Since \xspace is utilized by the \doclicenseLongTextForHyperref command of the doclicense package, a workaround is implemented below.
% It defines a \doclicenseLongTextForHyperrefFixed command that can be used instead of \doclicenseLongTextForHyperref.
\begingroup
\renewcommand{\xspace}{\space} % Temporarily replace \xspace with \space.
\xdef\doclicenseLongTextForHyperrefFixed{\doclicenseLongTextForHyperref} % Expand the \doclicenseLongTextForHyperref and define the fixed version globally.
\endgroup

% Define convenience functions to use the author name and the thesis title in the PDF document properties.
\newcommand{\authorname}{Sadashige Ishida} % The author name without titles.
% \newcommand{\thesistitle}{Symplectic  structures and dynamics on the codimension-2 shape space } % The title of the thesis.

% \newcommand{\thesistitle}{Symplectic, prequantum, and dynamic aspects of the codimension-2 shape space}

% \newcommand{\thesistitle}{Symplectic and prequantum structures and dynamics on the codimension-2 shape space}

\newcommand{\thesistitle}{Symplectic--prequantum structures and dynamics on the codimension-2 shape space}



% \newcommand{\thesistitle}{Symplectic and prequantum structures on the codimension-2 shape space (tentative)} % The title of the thesis.


\newcommand{\issn}{2663-337X} % Standard ISSN of all ISTA PhD theses.
\newcommand{\isbn}{978-0-00000-000-0} % Leave empty if no ISBN is used.
% Please contact IST Library library@ist.ac.at if you want to register an ISBN for your thesis.

% Create the XMP metadata file for the creation of PDF/A compatible documents.
\begin{filecontents*}[overwrite]{\jobname.xmpdata}
\Author{\authorname}                                    % The author's name in the document properties.
\Title{\thesistitle}                                    % The document's title in the document properties.
\Language{en-US}                                        % The document's language in the document properties. Select 'en-US' or 'en-GB'.
\Keywords{a\sep list\sep of\sep keywords}               % The document's keywords in the document properties (separated by '\sep ').
\Publisher{Institute of Science and Technology Austria} % The document's publisher in the document properties.
\Subject{PhD Thesis}                                    % The document's subject in the document properties.
\Copyright{\doclicenseLongTextForHyperrefFixed}         % The document's copyright message in the document properties.
\CopyrightURL{\doclicenseURL}                           % The document's copyright URL in the document properties.
\Copyrighted{True}
\ISBN{\isbn}                                            % The document's ISBN in the document properties.
\Journalnumber{\issn}                                   % The document's ISSN in the document properties.
\end{filecontents*}

% Extended LaTeX functionality is enabled by including packages with \usepackage{...}.
\usepackage[english]{babel} % The language of the thesis. Use 'british' for British English.
\usepackage{natbib}     % Extend bibliography management.
\usepackage{bibentry}   % Provide full citation support for single references.
% \usepackage{amsmath}    % Extended typesetting of mathematical expression.
% \usepackage{amssymb}    % Provides a multitude of mathematical symbols.
% \usepackage{mathtools}  % Further extensions of mathematical typesetting.
\usepackage{microtype}  % Small-scale typographic enhancements.
\usepackage[inline]{enumitem} % User control over the layout of lists (itemize, enumerate, description).
\usepackage{multirow}   % Allows table elements to span several rows.
\usepackage{booktabs}   % Improves the typesetting of tables.
\usepackage{subcaption} % Allows the use of subfigures and enables their referencing.
\usepackage[ruled,linesnumbered,algochapter]{algorithm2e} % Enables the writing of pseudo code.
\usepackage[usenames,dvipsnames,table]{xcolor} % Allows the definition and use of colors. This package has to be included before TikZ.
\usepackage{nag}        % Issues warnings when best practices in writing LaTeX documents are violated.
\usepackage{todonotes}  % Provides tooltip-like todo notes.
\usepackage{morewrites} % Increases the number of external files that can be used.
\usepackage[a-2b,mathxmp]{pdfx}  % Enables PDF/A compliance. Loads the package hyperref and has to be included second to last.
\usepackage[acronym,toc]{glossaries} % Enables the generation of glossaries and lists of acronyms. This package has to be included last.

% Set PDF document properties
\hypersetup{
	pdfpagelayout   = TwoPageRight,            % How the document is shown in PDF viewers (optional).
	linkbordercolor = {Melon},                 % The color of the borders of boxes around hyperlinks (optional).
}

\setpnumwidth{2.5em}        % Avoid overfull hboxes in the table of contents (see memoir manual).
\setsecnumdepth{subsection} % Enumerate subsections.

\nonzeroparskip             % Create space between paragraphs (optional).
\setlength{\parindent}{0pt} % Remove paragraph indentation (optional).

\nobibliography* % Allow the use of full citation in the list of publications.

\makeindex      % Use an optional index.
\makeglossaries % Use an optional glossary.
%\glstocfalse   % Remove the glossaries from the table of contents.

% Comment the following line to switch the document to a serif font, i.e., Latin Modern Roman
% (see https://tug.org/FontCatalogue/latinmodernroman/)
\renewcommand{\familydefault}{\sfdefault} % Switch document to sans serif font, i.e., Latin Modern Sans.
% (see https://tug.org/FontCatalogue/latinmodernsans/)

% Set the page numbers to bottom centered.
\makeevenfoot{plain}{}{\thepage}{} % For front and back matter.
\makeoddfoot{plain}{}{\thepage}{} % For front and back matter.
\makeevenfoot{Ruled}{}{\thepage}{} % For main matter.
\makeoddfoot{Ruled}{}{\thepage}{} % For main matter.

% Required data.
\setthesis{phd} % Select 'master' or 'phd'.
\settitle{\thesistitle}
\setsubtitle{} % Comment this line if no subtitle is used.
% If your subtitle contains commas, enclose it with additional curvy brackets (i.e., \setsubtitle{{subtitle text}}) or define it as a macro as done with \thesistitle.
\setdate{2025}{07}{10} % Set signing date with 3 arguments: {day}{month}{year}.
\setissn{\issn}
\setisbn{\isbn} % Comment this line if no ISBN is used.

% Define convenience function for an ISTA affiliation.
\newcommand{\istaustriaaffiliation}{ISTA, Klosterneuburg, Austria}

% Set persons with 4 arguments:
%  {title(s) before name}{name}{title(s) after name}{affiliation}
%  where both titles are optional (i.e. can be given as empty brackets {}).
% Some of the following persons are optional for Master's or PhD theses and can be commented if they do not apply.
\setauthor{}{\authorname}{}{\istaustriaaffiliation}
\setsupervisor{}{Chris Wojtan}{}{\istaustriaaffiliation}
% \setcosupervisor{}{Name Surname}{}{Institution, Place, Country} % Optional for Master's and PhD theses.
\setcommitteememberI{}{Albert Chern}{}{UCSD, California, USA}
\setcommitteememberII{}{Jan Maas}{}{\istaustriaaffiliation} % Optional for Master's thesis.
\setcommitteememberIII{}{Cornelia Vizman}{}{West University of Timișoara, Romania} % Optional for Master's and PhD theses.
% \setcommitteememberIIII{}{Name Surname}{}{Institution, Place, Country} % Optional for Master's and PhD theses.
\setdefensechair{}{Julian Fischer}{}{\istaustriaaffiliation} % Optional for Master's thesis.

\input{istaustriathesis-master/macro}

\begin{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frontmatter % Switch to roman page numbering and prevent chapter numbering.
\pagestyle{plain} % Set simple page numbering.
% The structure of the thesis has to conform to the guidelines at
%  https://phd.pages.ist.ac.at/student-resources/.

\addtitlepage
\addcommitteepage
\addstatementpage

\begin{abstract}
The space of codimension-2 shapes, such as curves in 3D and surfaces in 4D, is an infinite-dimensional manifold. This thesis explores geometric structures and dynamics on this space, with emphasis on their implications for physics, particularly hydrodynamics.

Our investigation ranges from theoretical studies of infinite-dimensional symplectic and prequantum geometry to numerical computation of the time evolution of shapes. The thesis presents four main contributions.

In the first part, we introduce implicit representations of codimension-2 shapes using a class of complex-valued functions, and prove that the space of these implicit representations forms a prequantum bundle over the codimension-2 shape space. This reveals a new geometric interpretation of the canonical symplectic structure on the codimension-2 shape space.

In the second part, we use implicit representations to develop a computational method for simulating the dynamics of space curves that can handle chaotic systems such as vortex filaments in hydrodynamics. For this purpose, we exploit the infinite degrees of freedom, hidden in both the configuration and dynamics of implicit representations.

% Our method automatically takes care of topological changes of curves like reconnection events.

In the third part, we introduce symplectic structures on the space of space curves, which generalize the only previously known symplectic structure on the space, allowing for new Hamiltonian dynamics of space curves.

In the fourth part, we apply a symplectic viewpoint to a differential geometric problem with practical applications. We derive a new area formula for spherical polygons via prequantization. 
\end{abstract}

% The Acknowledgement chapter is optional and can be removed.
\begin{acknowledgements}
\todo[inline]{ In the final version, I will write acknowledgements here. It is going to be a few pages of somewhat long acknowledgements. }
% \todo[inline]{
% 	Mention grants if you were funded.
% 	Consider Scientific Service Units (SSUs) at ISTA and other staff if you got their support.}

% \begin{CJK}{UTF8}{ipxm}
% Overleafで使う方法. 最も手軽なのは、Overleaf（無料プランあり）でLaTeXを動かす方法です。ブラウザ上ですぐに論文執筆を始められ、日本語対応も比較的 .
% \end{CJK}
\end{acknowledgements}

\begin{aboutauthor}
% \todo[inline]{
% 	Include paragraph of text summarizing your educational background, your academic profile (e.g. relevant research projects or work experience, list of publications), to give readers a snapshot idea of you as a person/academic.
% 	It is not meant to be a CV in the sense that it is not intended for job applications.
% 	This means things like your email address, your nationality, your current address, etc., should be left out.}
 Before starting the PhD program at ISTA, Sadashige Ishida studied mathematics in Japan and Switzerland, then worked in the optics and game industries. At ISTA, he researched the union of mathematics, physics, and computer graphics arising from natural phenomena. He especially likes to describe and study dynamics through the lens of infinite-dimensional geometry. This spans purely theoretical to applied aspects of dynamics like fluids, optimal transport, deformation of shapes, and geometric structures behind these dynamics. Sadashige is also passionate about scientific and non-scientific education. An example is the foundation of the course \emph{geometric mechanics} at ISTA. Aside from research, he enjoys interacting with nature, like plants and animals (including humans) around Kierling, Klosterneuburg where he lives with his family. 
%  he enjoys spending time with animals (including humans) and plants in Kierling, Klosterneuburg where he lives with his family. 
% 
% This ranges from 
\todo[inline]{At ISTA, the level of formality/informality in the biography is left up to the author’s preference.}
\end{aboutauthor}

\begin{publicationlist}  % Remove if not applicable for a Master's thesis.

This thesis comprises the following articles:
\begin{enumerate}
    \item \bibentry{chernIshida2024sphericalPolygons} \label{item:spherical}
    \item \bibentry{iwc2022implicit_filaments}\label{item:hidden_degree}
    \item \bibentry{chernIshida2025prequantum} \label{item:prequantum}
    \item \bibentry{bauerMichorIshida2024symplectic}\label{item:more_symplectic}
\end{enumerate}

The following works were also produced during the author's PhD studies:
\begin{enumerate}[start=5]
    \item  \bibentry{ishidaLavenant2024quantitative}
    \item \bibentry{isnhw2020soap}\label{item:soapfilm}
\end{enumerate}

I am the first author of the article \ref{item:hidden_degree} and a joint first author with Peter Synak (now Peter Heiss-Synak) of the article \ref{item:soapfilm}, both published in ACM Transactions on Graphics.
For the other articles, authors are listed alphabetically, following the convention of mathematics journals.

% \paragraph{Contributions of the authors}
% \todo[inline]{
% 	write heres.
% }
% \textit{Short version}

% In \ref{item:hidden_degree}, I contributed in part to the development of the theoretical settings and the implementation of the main algorithm. I performed all the numerical experiments in the article and contributed to writing.

% In [2], the original impetus comes from all the three authors. I generalized the initial result that Martin Bauer and Peter Michor obtained for a special case.

% In [3], I contributed to the development of . writing. I developed the theory using the language of currents. 

% \textit{Detailed version}



\end{publicationlist}

\input{istaustriathesis-master/ch_preface}

\cleardoublepage
\renewcommand{\contentsname}{Table of Contents} % Specify chapter heading for table of contents.
\tableofcontents

% List of figures
% \listoffigures


% Create a group and deactivate all page breaks and blank pages.
% This puts the lists of figures, tables, etc. on the same pages.
% To reintroduce page breaks or blank pages, take the lists out of the group.
\begingroup % Starting a group keeps the deactivations restricted to the group.
\let\newpage\relax
\let\clearpage\relax
\let\cleardoublepage\relax

% \listoftables
% \listofalgorithms\addcontentsline{toc}{chapter}{List of Algorithms}
\printglossary[title={List of Terms}]
\printglossary[type=\acronymtype,title={List of Abbreviations}]
\endgroup % End the deactivation group.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% MAIN MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mainmatter % Switch to Arabic page and chapter numbering.
\pagestyle{Ruled} % Show chapter and section name at the top of the page.
%\pagestyle{plain} % Alternative page style without the names at the top.

% \section*{Chapters for Latex edit}
\input{istaustriathesis-master/ch_introduction}
\input{istaustriathesis-master/ch_preliminary}
\input{istaustriathesis-master/ch_implicit_prequantum}
\input{istaustriathesis-master/ch_hidden_degrees}
\input{istaustriathesis-master/ch_more_symplectic}
\input{istaustriathesis-master/ch_spherical}

\input{istaustriathesis-master/ch_temp}


% REMOVE THE FOLLOWING LINE IN THE FINAL VERSION OF THE THESIS.
% \input{intro.tex} % A short introduction to LaTeX.
% \input{istaustriathesis-master/ch_appendix}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% BACK MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\backmatter
\pagestyle{plain}

\bibliographystyle{alpha} % Define the citation style.

\bibliography{cIntro} % Add the bibliography file 
% Sadashige: I changed it from intro.bib to cIntro.bib

\mainmatter* % Remove this line to prevent numbering of appendix chapters.
\appendix % Number appendix chapters with letters instead of numbers.

\backmatter
% REMOVE THE FOLLOWING THREE LINES IN THE FINAL VERSION OF THE THESIS.
% \makeatletter\@todonotes@SetTodoListName{List of Todos}\makeatother % Set name of list of todos.
% \todototoc % Add the list of todos to the table of contents.
% \listoftodos % Add a list of all todos.

\end{document}