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

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% 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}{978-3-99078-070-1} % 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.

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% 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.
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% Required data.
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% 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.
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% Define convenience function for an ISTA affiliation.
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% 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{}{Albert Chern}{}{UCSD, California, USA} % Optional for Master's and PhD theses.
% \setcommitteememberI{}{Albert Chern}{}{UCSD, California, USA}
\setcommitteememberI{}{Jan Maas}{}{\istaustriaaffiliation} % Optional for Master's thesis.
\setcommitteememberII{}{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}


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%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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
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\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 simulation method for the dynamics of space curves. To handle chaotic systems such as vortex filaments in hydrodynamics, we exploit the infinite degrees of freedom, hidden in both the configuration and dynamics of implicit representations.

In the third part, we introduce new symplectic structures on the space of space curves, which generalize the only previously known symplectic structure on this 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}
% In the past couple of years, I had a great (not only PhD) life. Here I want to acknowledge those who made it possible. 

My PhD journey started (actually didn't start) when I finished my master’s study in mathematics with poor grades and no chance of doing a PhD. But after some time, I'm now completing a PhD with a thesis that lies somewhat in the union of math, physics, and computer graphics.\footnote{On the way, I got some thoughts about math, computer graphics, and science in general. If you are interested, see \autoref{ch:postface}.} 
 % 
 During these times, I had a great (not only PhD) life. 
  % I had a great (not only PhD) life In the past couple of years. 

These are all due to the people who made it possible. Here I want to acknowledge them.

My first thanks go to those who maintain the great working environment and the beautiful campus of ISTA. Specifically, I thank the Cleaning staff, Construction, IT, Admin, Campus services, Gradschool office, ISTA Family, and everyone who contributes to keeping ISTA a great place to work. I love chatting over free coffee in a clean common space. But moments like this are not automatic. Their hard work and dedication let it happen. I am truly grateful for their commitment, which has let me focus on research and have fun in such a pleasant environment.

Next I thank my advisors and collaborators. I am doing research across multiple fields. This was made possible thanks to the huge support I received from my advisors and collaborators. They opened the doors of research for me, who was (and still is) a novice in these fields, welcomed me as a colleague, guided me through, and shared their expertise. They spent an immense amount of time for me. Thanks to their support, I am now completing my PhD.

I thank Albert Chern. His influence on me is huge and will remain throughout my scientific life. He does science seamlessly across multiple fields without being trapped by labels like mathematics, graphics, physics, and engineering. I’m always impressed by his command of mathematics, which is not just fancy mathematics for itself, but is strongly driven by his deep physical and geometric intuition, and the way he can talk about it using the language of different fields. I want to do science this way.

I thank Hugo Lavenant, who guided me through my first research experience in the field of optimal transport. His responses to my questions and ideas were always quick and spot-on. I also learned from him how to write a mathematical paper in a reader-friendly way.

I thank Martin Bauer. He welcomed me into the field of infinite-dimensional geometry, showing that one can naturally enter research in this seemingly background-heavy and intimidating field. Among the many mathematicians I have met, he stood out for his low barrier to working with. I learned a lot from his exceptional ability to find interesting research questions and make reasonable decisions.

I thank Peter W. Michor, a math giant and a founder of the modern theory of infinite-dimensional geometry. I have always been amazed by the way he can turn my vague thoughts into theorems, and never hesitates to get his hands dirty with manual computations. I also like coffee you make for us after lunch.

I thank Fumika Suzuki. At a time when I had a difficult time keeping motivation for my PhD and didn’t know what to do, she invited me to a project on quantum mechanics, a field I was always interested in but didn't know what to start from. Her open-mindedness and hands-on support saved me during that period. I look forward to completing our suspended project with you.

I thank Toshiya Hachisuka. My very first research started with him. He showed me how to perform scientific research while respecting my original ideas. Toshiya is known to be a photo-realistic rendering guy, but to me he is like a scientist, who can think outside the Cornell box. This inspires me to become a scientist rather than either of a mathematician, physicist, or computer scientist. 

I thank Ryoichi Ando. He was my first teacher in fluid dynamics and physics simulation, from whom I learned how fluids behave intuitively and visually. I hope to become an advisor like him who is always accessible and offers help in an hands-on manner.

I thank my official advisor, Chris Wojtan. People in the field know Chris Wojtan is a great scientist. But I was very lucky to be in a position where I could learn about him as a person too. In my opinion, he is a rare figure in academia who maintains a great sense of humanity and fairness. 

I learned from him many things like how to turn ideas into tractable problems, how to present them appropriately depending on the audience—whether sharing with potential collaborators, explaining to experts, or entertaining a broader audience—and how to set reasonable goals in research. But that's not all.
I was not a typical graphics student. In particular, I was trying to research many things even beyond graphics, which I think made it hard to advise me. Yet he celebrated my ambition and supported me in trying to make it come true, both scientifically and non-scientifically. This made me want to become an advisor like Chris who can commit to realizing someone’s dreams come true. 

I thank my former supervisors, Antti Herva at Remedy Entertainment and Toru Fujii at Nikon Corporation. They always tried to find a way for me to engage in work that I could be passionate about and enjoy, and backed me in doing a PhD.

I thank my coauthors, Peter Heiss-Synak, Fumiya Narita, Masafumi Yamamoto. I learned a lot through collaborating with them, especially the joy of sharing excitement, challenges, fun, and sometimes even pain.

I owe a lot to the members of the visual computing group at ISTA during my PhD—Peter, Manas, Alexei, Christian, Micka\"el, Evgeny, Georg, Carmen, Cristina, Dian, Michaela, Mathilde, Samara, Mau, Camille, Arian, Ruslan, Malina, Stefan, Tomáš, Thomas, Martin, Eder, Ihor, Desmond, Benjamin, Dafne, and Christine. My kind of research was quite different from the group's main direction and I didn’t actively join group activities like daily lunch. But I never felt alone or isolated. I always felt cared, liked, supported, respected, and accepted.
I loved those spontaneous moments of just dropping by someone’s office and procrastinating over chats. With you guys, I could stay honest and be myself, and had a wonderful PhD life.

I thank the students in my cohort. I especially enjoyed the first PhD year with Alice and Anton, who shared the same office. I remember doing the analysis course homework together—not only did it help me learn analysis, but I also had so much fun with you.

I thank my friends at and outside ISTA, in Klosterneuburg, and across many countries. However, due to space limitations, it is unfortunately impossible to mention everyone by name. Thanks to them, my life has been so much richer. 
% \TODO{I'm surprised that I now feel like Austria, in particular, Maria-Gugging and Kierling in Klosterneuburg, is my home, which I didn't expect when I came to here to start my PhD. But it happed thanks to friends (and friends of my family) here.}

I thank my (extended) family—Sadao, Hiroko, Akira, Dominique, Tomoaki, Miku, Mari, Naoki, Satomi, Mio, Yuri, Ichizo, Teiko---for their constant support. This includes their financial help. This made it possible for me to receive a university education and study overseas, which eventually enabled me to get a job abroad. And whenever we go back to Tokyo, they always welcome us and let us stay. Having a place I can always return to has given me a great feeling of comfort. 
This extends to Linda, Emil, and Luna, friends of both my family and me, 
% (though I cannot name everyone due to the space limitation),
who made us feel that Klosterneuburg, Austria is also our home.

I would also like to acknowledge my grandfather, Tsunehiro, who had a huge influence on the way I am now. He was almost the only person in my family involved in science or engineering and traveling around the world. His sense of art and science, as well as his honesty and love for everyone, shaped me.

I think my PhD was quite different from a typical one for several reasons. A major factor was my children, Ai Amelia and Yuu Noa, who made my PhD actually more challenging and full of hassle, while making my life happier, adding new colors and adventures.
Science was just one of the things I learned during my PhD. I thank Ai Amelia and Yuu Noa, from whom I am learning a lot—about being a parent, being a human, a form of happiness, and about life.

And Shizuka Claire. For me, she is like a wizard. From her, I learned that magic is not a supernatural power. Magic is love for everyone and everything. I admire how your love always makes people happy and smile. 

And thank you for always trusting me. I tend to be unplanned and spontaneously change the direction of life. You let me take on the adventure of leaving a stable job and moved to Austria with me while having a nine month old baby. Thanks to that, we are now in Klosterneuburg, Austria, having our time with people here and sharing moments as our children grow.

\newpage
\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 .
今この瞬間、今の日々をあなたと共有できていることが嬉しいです。いつも一緒に冒険してくれてありがとう。これからもあなたと楽しい時も苦しい時も人生の冒険を続けてゆきたいです。\hfill 2025年10月　 定繁
\end{CJK}

\newpage
\paragraph{Grant support}
Projects contained in this thesis were financially supported in part by the European Research Council with grants 1. ERC Consolidator Grant 101045083 CoDiNA, and 2. the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 638176.
\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 at 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}


\cleardoublepage
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% List of figures
% \listoffigures


% Create a group and deactivate all page breaks and blank pages.
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\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}
% \todo{delete here}

\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}

\input{istaustriathesis-master/ch_postface}

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


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%%% BACK MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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}