\chapter{Introduction}
% --------------

% This thesis is concerned about infinite-dimensional symplectic geometry of codimension-2 shape spaces. 

% There are two fundamental concepts that revolve around this thesis.  Codimension-2 shape space and symplectic geometry. Let us break down one by one. 

% --------------

This thesis explores geometric structures and dynamics on the codimension-2 shape space, with emphasis on their implications for physics. 

Our investigation spans theoretical studies of infinite-dimensional symplectic and prequantum geometry to numerical computation of the time evolution of shapes.


\section{Background}
\paragraph{Codimensional shapes and physics}

The history of much of physical science is the history of the study of shapes.
Humans have long sought to describe physical phenomena in terms of the deformation of shapes. For example, some models—mathematical descriptions such as equations or rules that aim to replicate nature—express the motion of soap bubbles or biological cell membranes as the time evolution of surfaces in 3D. These surfaces are called codimension-1 because they are two-dimensional objects living in three-dimensional space; that is, the difference between the shape’s dimension and the ambient space’s dimension is one.

\begin{figure}[h]
\centering
        \centering
        \includegraphics[width=0.4\linewidth, trim={0cm 0cm 0cm 0cm},clip]{istaustriathesis-master/images_introduction/doublebubble_207.png}
    \caption{Evolution of soap bubbles modeled as deformation of surfaces.
    }\label{fig:double_bubble}
\end{figure}

Physical phenomena also involve shapes of other codimensions. Consider codimension-0, for instance: the motion of water droplets or elastic bodies that occupy a full 3D volume, so the codimension is zero. On the other hand, codimension-2 appears in situations such as electrically charged particles moving on a surface or tiny balls rolling on a billiard table—here the objects are zero-dimensional points moving within a 2D domain.



Hydrodynamics, the science of the motions of fluids, namely liquids and gases, provides particularly interesting examples involving codimension-1 and codimension-2 shapes. A fluid state is often described as a velocity field over a 3D region like a box. A quantity called \emph{vorticity} measures how much the fluid velocity is swirling locally. In common situations, \footnote{Technically, when the first homology group of the ambient space is trivial.} the velocity field can be recovered from the vorticity. Mathematically, this is expressed by the relation \( d\eta = \omega \), where \(\eta\) is the velocity field represented as a differential 1-form, and \(\omega\) is a differential 2-form representing vorticity. Hence, knowing how vorticity evolves over time allows us to track how the entire fluid state changes.

This draws our attention back to codimensional shapes. Vorticity often concentrates around ``thin'' codimension-1 or codimension-2 objects like surfaces or curves in 3D. This observation led to idealized models known as vortex sheets and vortex filaments—localizations of vorticity on surfaces and space curves, respectively. Since velocity can be recovered from vorticity, tracking the evolution of these shapes effectively tracks the evolution of the whole fluid state, illustrated as in \autoref{fig:jet_introduction}. This shows how understanding the time evolution of shapes, especially those of various codimensions, offers descriptions of physical phenomena.

\begin{figure}[h]
\centering
    \begin{minipage}[b]{.45\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 2cm 0cm 2cm},clip]{{images_hidden_degrees/jet/jet_curves.0189}.png}
    \end{minipage}
    \begin{minipage}[b]{.45\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 2cm 0cm 2cm},clip]{{images_hidden_degrees/jet/jet_smoke_intensity5.0189}.png}
    \end{minipage}
    \caption{Vorticity of a jet of fluid represented as a collection of space curves (left), inducing a velocity field which can transport smoke density (right). Simulation and visualaization were performed using a method presented in \autoref{ch:hidden_degrees}.
    }\label{fig:jet_introduction}
\end{figure}

Among these, codimension-2 shapes are significant in symplectic geometry, a framework for describing dynamics geometrically. I explain this now.


\paragraph{Symplectic geometry and physics}

% Symplectic geometry is a playground for Hamiltonian systems, which are descriptions of how physical objects evolve over time while conserving ``energy''. Here, energy means a quantity assigned to the system that we expect to stay constant in time, like kinetic energy, potential energy, or something more abstract depending on the type of phenomenon we want to describe.
% Symplectic geometry offers a geometric framework for describing how physical states evolve over time: instead of watching a system move, we may cast it into a path in an abstract space encoding possible states of an object. 

% A geometric structure called \emph{symplectic structure} plays an important role in symplectic geometry.
% When a space has a symplectic structure, it takes as input a quantity called the energy or Hamiltonian and outputs a dynamical system on that space.

% It is a sandbox for energy-preserving dynamics, called \emph{Hamiltonian systems}.

Symplectic geometry is a playground for Hamiltonian systems, which are descriptions of how physical objects evolve over time while conserving ``energy''.  Here, energy---also called the Hamiltonian---is a quantity assigned to the system that we expect to remain constant in time, like kinetic energy, potential energy, or something more abstract, depending on the type of phenomenon we want to describe. Rather than tracking the motion of objects directly, symplectic geometry views such evolution as a path in a static space of possible physical states.

A key ingredient in this framework is a geometric object called a \emph{symplectic structure}. It takes the Hamiltonian as input and outputs a dynamical system on the state space.

Mathematically, this idea is formulated as follows. A symplectic manifold $X$ is a manifold equipped with a closed and non-degenerate 2-form $\omega$ called a symplectic structure. Once we give the Hamiltonian $H$ as a function on $X$, the symplectic structure $\omega$ generates a vector field $V_H$ on $X$ through the relation \(dH= \iota_{V_H} \omega\). Then the Hamiltonian system is the flow along the vector field $V_H$, that is, solutions to the differential equation \(\partial_t x(t)=V_H(x(t))\) where $x$ represents a state of the object as a point of $X$. 
Then, the time evolution is a path on which the Hamiltonian is constant.
% This setup ensures the preservation of the Hamiltonian in the time evolution as a path in the state space.

The symplectic manifold $X$ here typically represents the collection of all the possible states of a physical phenomenon. For example, if we consider particles moving in a domain $M$, then the space $X$ is the cotangent bundle $T^*M^n$,  which contains all possible positions and momenta of $n$ particles. In quantum mechanics, it is often the space of wave functions, representing all possible quantum states. These spaces are known to be symplectic manifolds, on which phenomena like celestial mechanics and the Schrödinger equation are modeled as Hamiltonian systems. 

In this way, if the space of possible states is a symplectic manifold, we can model dynamics as Hamiltonian systems. This setup offers further advantages. For example, the Hamiltonian is often not the only constant in motion. If the system has symmetries, the symplectic structure reveals other conserved quantities. For instance, if the Hamiltonian is invariant under rotations, a quantity often called angular momentum remains constant in time. This symmetry-to-conservation link is known as Noether’s theorem.


% For example, it helps us study motions that repeat over time—geometrically periodic orbits. 

\paragraph{The codimension-2 shape space as an infinite-dimensional symplectic manifold}

Earlier, we saw that many physical phenomena can be phrased in terms of the deformation of codimensional shapes. Among these, the case of codimension-2  stands out, as the space of codimension-2 shapes is an infinite-dimensional symplectic manifold, equipped with the so-called Marsden–Weinstein (MW) symplectic structure. We postpone the precise definitions of these notions to the next chapter. 

Many phenomena in fields such as fluid dynamics and elasticity can be modeled as Hamiltonian systems with respect to the MW symplectic structure. 
% 
% 
% For example, the binormal equation, a localized approximation of vortex filaments, arises as a Hamiltonian system on the space of space curves where the Hamiltonian function is the length of the curve.
Important examples include vortex filaments and their localized approximation called the binormal equation. Using the framework of symplectic geometry, we can study properties of these dynamics, such as conserved quantities, integrability, periodic orbits, and connections between seemingly unrelated dynamics.

% integrability, conserved quantities, reduction. stability.

This rich interplay motivates the present thesis to explore the geometric structures and dynamics on the codimension-2 shape space. By doing so from several angles, the author hopes to offer a glimpse into the beautiful bridge 
 % that symplectic geometry builds 
between shapes and dynamics.  

%  Alternative: a look at fragments of the beautiful bridge

% This thesis explores the infinite-dimensional symplectic geometry of codimension-2 shape space from different angles.

\section{Research projects}

\subsection{Implicit representations}
The first major part of this thesis is concerned with \emph{implicit representations} of shapes.

A submanifold (mostly synonymous with shape in this thesis \footnote{Whenever unnecessary, we stay imprecise about whether a submanifold or shape may be an immersion or must be an embedding, and whether it is parametrized or unparametrized.}) in an ambient space $M$ is often described \emph{explicitly} as an embedding of a manifold $S$ into $M$, modulo reparametrizations.

An alternative is the \emph{implicit} representation. For example, a codimension-1 shape, like a surface in $\RR^3$, can be represented as a preimage of a function, often called the level set function. This representation naturally extends to higher codimensions using multiple functions. When the codimension is 2, submanifolds can be expressed as the zero sets of complex-valued functions (\autoref{fig:level_surface_introduction}). 

\begin{figure}[h]
\centering
    \begin{minipage}[b]{.45\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 2cm 0cm 2cm},clip]{istaustriathesis-master/images_hidden_degrees/illustrations/level_surfaces_and_ribbons/level_surface_only_rings.png}
    \end{minipage}
    \begin{minipage}[b]{.45\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 2cm 0cm 2cm},clip]{istaustriathesis-master/images_hidden_degrees/illustrations/level_surfaces_and_ribbons/level_surface_re_im.jpg}
    \end{minipage}
    \caption{Linked curves, known as the Hopf link (left) and the level sets of a complex function $\psi=a+ib$ consisting of real-valued functions $a$ and $b$ (right). The blue and red surfaces are zero level sets of $a$ and $b$ respectively.  Their intersection, the zero level set of $\psi$, agrees with the curves.
    }\label{fig:level_surface_introduction}
\end{figure}

The implicit representation for each codimension-2 submanifold is not unique, as multiple (in fact, infinitely many) complex-valued functions can share the same zero set.
% 
The following two projects investigate this redundancy of implicit representations. 

\subsection*{Project 1. Implicit representations of codimenion-2 shapes and their prequantum structures 
% \textnormal{\cite{chernIshida2025prequantum}}
}

This part is based on the article \cite{chernIshida2025prequantum}.

Implicit representations via complex functions have been widely applied to study the dynamics of submanifolds. 
 % For example, prior work has applied them to problems involving the motion of space curves, such as the curve-shortening flow \cite{MBO2001diffusion}, and vortex filament dynamics in fluid mechanics \cite{ogawa2002study,villois2016GrossPitaevskii,jerrard2018leapfrogging}.
However, the geometry of the space of these representations is largely unexplored. In this project, we investigate its geometric structures, focusing on symplectic geometric aspects.

The non-uniqueness of implicit representations for each shape makes the implicit shape space a fiber bundle over the space of codimension-2 submanifolds. Our main contribution is showing that a particular quotient space of this fiber bundle forms a prequantum bundle.

Roughly speaking, a prequantum bundle is a specific kind of fiber bundle over a base symplectic manifold. Each fiber over a point carries the additional information of quantum phase, in such a way that Hamiltonian flows on the base manifold are naturally and uniquely lifted onto this bundle. In this way, a prequantum bundle is a sandbox for quantum mechanics, aligned with the base symplectic manifold as a sandbox for classical mechanics. 
% Mathematically, it is a principal $G$-bundle $\pi\colon F \to B$ over the base symplectic manifold $(B,\omega)$, equipped with a connection form $\Theta$ such that $d\Theta=\pi^*\omega$.

% Prequantization is the process of constructing a prequantum bundle from a base symplectic manifold. It is part of geometric quantization, a framework for introducing quantum mechanical theory from classical mechanical theory. 


We do not, however, perform any quantum mechanical analysis using this framework. Rather, we focus on the geometric insights offered by our prequantum bundle. As illustrated in \autoref{fig:prequantum_schematic}, we present a new interpretation of the Marsden–Weinstein form:

\begin{quote}
% \fbox{%
  % \parbox{\textwidth}{%
    Each complex function \(\psi\) representing a codimension-2 shape \(\gamma\) carries phase information \(\phi = \psi / |\psi| \in \SS^1\). The level sets \(\{\phi^{-1}(s)\}_{s\in \SS^1}\) of the phase function \(\phi\) define a family of hypersurfaces in the ambient space, each bounded by \(\gamma\). Any motion of the base shape \(\gamma\) induces motions of these hypersurfaces, which sweep out volumetric domains. The MW form arises as the curvature of a connection of our prequantum bundle, which measures the average volume swept by these hypersurfaces.
  % }%
% }
\end{quote}

\begin{figure}[h]
\centering
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 0cm 0cm 0cm},clip]{istaustriathesis-master/images_introduction/prequantum_schematic_dark.png}
    \caption{\rev{A schematic image of our prequantum bundle. The fiber bundle is the space of implicit representations (top row) and the base space, referred to as the explicit shape space, is the space of codimension-2 shapes (bottom row). The tangent vector $\dot \psi$ at each implicit representation $\psi$ can be visualized as the  deformation of the phase hypersurface $\phi^{-1}(s)$, and the connection form $\Theta$ measures the infinitesimal swept volumes of these phase hypersurfaces, averaged over all the phases. The tangent vector $\dot \gamma$ at each codimension-2 shape $\gamma$ is a vector field on $\gamma$, representing a velocity of $\gamma$. The MW symplectic form $\omega$ measures the volume spanned by the velocity fields $\dot \gamma, \ring \gamma$ and the tangent of $\gamma$, integrated over $\gamma$. The 2-form $\omega$ is the curvature of the connection form $\Theta$, that is, $\pi^*\omega=d\Theta$.  }}\label{fig:prequantum_schematic}
\end{figure}


In the limiting case where the phase of the complex function is constant everywhere except for a $2\pi$ jump across a single hypersurface, this result reduces to the swept volume of that hypersurface, without explicit reference to the complex function.

This prequantum viewpoint thus unifies explicit and implicit descriptions of codimension-2 shapes.





\subsection*{Project 2. Hidden degrees of freedom in implicit vortex filaments}
This part is based on the article \cite{iwc2022implicit_filaments}.

% \TODO{Polish the following text}
% The dynamics of space curves have been both theoretically and numerically is an active 


% Rooted in several fields like physics, biology, and mathematics, the deformation of space curves has long been studied both theoretically and numerically.

The deformation of space curves is rooted in several fields like physics, biology, and mathematics, and has been studied both theoretically and numerically.

In this project, we develop a numerical method for space curve dynamics using implicit representations. Instead of simulating space curves explicitly, we compute the time evolution of implicit representations.

Our approach, in particular, exploits the non-uniqueness of these implicit representations in both their configurations and dynamics.
% 
As noted, multiple complex functions share the same zero level set. Such non-uniqueness also exists in the dynamics. The time evolution of curves represented by a complex function $\psi$ can be described by the transport of $\psi$ along a vector field, and the choice of the vector field yielding the same motion of the zero set is not unique. In fact, there are infinite degrees of freedom within the collection of both possible complex functions and vector fields.\footnote{Strictly speaking, the infinite degrees of freedom become finite when we perform a finite-dimensional approximation, a.k.a. discretization, for numerical purposes.}

Within these redundancies, we can make a specific choice for a particular purpose, such as improving the  robustness of numerical simulation. Specifically, to handle chaotic dynamics like vortex filaments in fluid dynamics, we introduce \emph{untwisted Clebsch variables} and \emph{non-swirling dynamics}. They successfully reduce the numerical instability, taming the twisting modes around the filaments.

Additionally, the level set description by implicit representations inherently supports topological changes of curves. Built on these features, our resulting  method stably simulates the dynamics of vortex filaments that split and merge (\autoref{fig:jet_introduction} and \autoref{fig:evolving_Hopf_link}).

\begin{figure}[h]
\centering
    \begin{minipage}[b]{.24\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={14cm 4cm 14cm 2cm},clip]{istaustriathesis-master/images_hidden_degrees/twoRings/2rings_0001.png}
    \end{minipage}
    \begin{minipage}[b]{.24\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={14cm 4cm 14cm 2cm},clip]{istaustriathesis-master/images_hidden_degrees/twoRings/2rings_ours.0045.png}
    \end{minipage}
        \begin{minipage}[b]{.24\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={14cm 4cm 14cm 2cm},clip]{istaustriathesis-master/images_hidden_degrees/twoRings/2rings_ours.0055.png}
    \end{minipage}
    \begin{minipage}[b]{.24\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={14cm 4cm 14cm 2cm},clip]{istaustriathesis-master/images_hidden_degrees/twoRings/2rings_ours.0070.png}
    \end{minipage}
    \caption{Time evolution of vortex filaments undergoing topological changes (left to right), simulated using our method.  }\label{fig:evolving_Hopf_link}
\end{figure}

In this project, we do not restrict ourselves to Hamiltonian systems. Our method can accommodate any first-order dynamics in time. As an example, we simulate the curve shortening flow, which is not a Hamiltonian flow (energy preserving flow) of the Marsden--Weinstein structure but rather a gradient flow (energy descent flow) with respect to the $L^2$-Riemannian metric.

% This project exploits the redundancy of implicit representations for developing a numerical method for space curve dynamics.


% We put emphasis on applications to vortex filaments from fluid dynamics, which is known to be highly turbulent and numerically instable.

% As we saw there are infinite degrees of freedom in the implicit representations for each codimension-2 shape. 

% In this project, we exploit the redundancy of implicit representations not only in the configuration of shapes, but also in their dynamics.

% That is, the time evolution of an implicit representation $\psi$ can be described by the advection of $\psi$ along a vector field, and there are infinite degrees of freedom in vector fields that give the same motion of the zero level set. 

% This can be tailored to specific purposes, such as taming the  . instability.

% e.g. untwisted Clebsch variables and non-swirling dynamics.


% A consequence is a simulation method, 

% -- Simulate input curve dynamics by simulating time evolution of complex functions. 


% Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we describe them using implicit representations. We leverage their redundant degrees of freedom in both the configuration and the dynamics, which can be tailored specifically to improve numerical robustness. This is in contrast to naive approaches for implicit curve dynamics  that suffer from overwhelming numerical stability problems.

% Furthermore, we note how these hidden degrees of freedom perfectly map to a {\em Clebsch variables} from symplectic geometry.
% Motivated by these observations, we  introduce {\em untwisted} Clebsch variables and {\em non-swirling} dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments.
% A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events. 


% As discussed earlier, complex functions have been used to study the dynamics of codimension-2 shapes, including simulation methods for them. Each of these studies is designed for a particular system, using specific kinds of complex functions and evolution laws. In contrast, we numerically simulate a given curve dynamics using implicit representations, leveraging their redundancy.

% A key feature of implicit representations is their hidden degrees of freedom, both in geometry and in dynamics. For any given codimension-2 shape, there are infinitely many complex functions that represent it. Similarly, there are infinitely many ambient vector fields whose advection of the complex function $\psi$ results in the same motion of its zero level set.

% This project leverages this observation. Among possible representations for configurations and dynamics, we can choose ones suited to our goals. When simulating systems exhibiting chaotic behaviors—like vortex filaments—we pick up untwisted Clebsch variables and non-swirling dynamics. 

% This enables numerically stable simulation of these turbulent systems and regularizes sources of instability, particularly in the twisting modes around curve filaments.

% Finally, because implicit representations by complex functions naturally support topological changes and reconnections, our method can handle these events seamlessly.


% We apply our method to curve dynamics such as shortening flow from differential geometry and the Rosenhead--Moore approximation of vortex filaments from fluid dynamics. In both cases, space curves undergo topological changes via reconnection events.


\subsection{No more symplectic structures?}

We now go back to explicit representations of shapes and focus on the space of space curves, which is the simplest infinite-dimensional instance of the codimenion-2 shape space.

The Marsden--Weinstein (MW) structure is known to be a canonical symplectic structure on this space. But are there no other symplectic structures? This following project answers this question. 

\subsection*{Project 3. Symplectic structures on the space of space curves}

This part is based on the article \cite{bauerMichorIshida2024symplectic}.

We derive new symplectic structures on the space of space curves. These new structures generalize the MW structure, which was the only previously studied symplectic structure on this space.

Our approach builds on two key ingredients. First, the MW form \(\omega\) admits a Liouville 1-form \(\eta\), \ie, \(d\eta = \omega\), and this \(\eta\) can be expressed in terms of the standard \(L^2\) Riemannian metric on the space of space curves. Second, in mathematical shape analysis, many alternative Riemannian metrics on this space have been developed by integrating a suitable operator on the tangent bundle of the space.

Motivated by this, we construct new 1-forms by modifying the standard Liouville form with such operators. Taking their exterior derivatives yields closed 2-forms, and we verify that these forms are non-degenerate, thus defining symplectic structures.

We then derive Hamiltonian systems induced by these new symplectic structures, as different symplectic structures $\omega'$ induce distinct Hamiltonian flows $V_H'$ from the same input Hamiltonian function $H$, according to the relation $dH=\iota_{V_H'}\omega'$ (\autoref{fig:different_Ham_flows}). 
We also numerically illustrate simple examples.

\begin{figure}[h]
\centering
    \begin{minipage}[b]{.32\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 2cm 0cm 2cm},clip]{istaustriathesis-master/images_more_symplectic/squaredScale_p0/Frame1201.png}
    \end{minipage}
    \begin{minipage}[b]{.32\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 2cm 0cm 2cm},clip]{istaustriathesis-master/images_more_symplectic/squaredScale_pm01/Frame1201.png}
    \end{minipage}
        \begin{minipage}[b]{.32\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 2cm 0cm 2cm},clip]{istaustriathesis-master/images_more_symplectic/squaredScale_p2/Frame1201.png}
    \end{minipage}
    \caption{Snapshots of a space curve evolving under Hamiltonian flows of the same Hamiltonian function, but induced by the standard MW structure (left) and different symplectic structures (middle and right), exhibiting distinct behaviors. 
    }\label{fig:different_Ham_flows}
\end{figure}

\subsection{Prequantum geometry for polygons on the sphere}

Symplectic and prequantum geometry are frameworks for studying dynamics, but they can also be useful for purely geometric problems. This project is such an instance. Unlike the above projects 
which concerned dynamics and infinite-dimensional manifolds, 
the following project deals with a purely geometric, finite-dimensional problem.

\subsection*{Project 4: Area formula for spherical polygons via prequantization}
This part is based on the article \cite{bauerMichorIshida2024symplectic}.

In this project, we derive a new formula for the area of spherical polygons via prequantization. A spherical polygon is a finite sequence of ordered points on $\SS^ 
2$, connected by geodesics. Calculating the solid angle of the region enclosed by such a polygon has practical applications in fields like fluid dynamics, light transport theory, and geography (\autoref{fig:austria_on_earth}).

The areas of spherical polygons are often computed using a formula based on the Gauss-Bonnet theorem. However, this formula requires the non-degeneracy assumption, meaning that no two consecutive points can lie on the same location. This constraint renders the formula unusable or numerically unstable in certain situations.

Our new formula mimics Green's formula, which is not directly applicable on $\SS^2$
  because the standard area form is not exact. We circumvent this issue using a  prequantum bundle. We lift the area form onto the bundle, where the resulting 2-form becomes exact. This converts an area integral on $\SS^2$ into a line integral on the bundle.

Through this procedure, we derive a formula which is applicable to a wider range of degenerate spherical curves and polygons.

\begin{figure}[h]
\centering
    \begin{minipage}[b]{.45\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 0cm 0cm 0cm},clip]{istaustriathesis-master/images_spherical/austria_sphere_greyBG.jpeg}
    \end{minipage}
    \begin{minipage}[b]{.45\linewidth}
        \centering
        \includegraphics[width=0.99\linewidth, trim={0cm 0cm 0cm 0cm},clip]{istaustriathesis-master/images_spherical/austria_clip_greyBG.jpg}
    \end{minipage}
    \caption{A spherical polygon representing the shape of Austria (left), and a zoomed-in view (right). Our formula can robustly compute the area of such an intricate shape.}\label{fig:austria_on_earth}
\end{figure}

\subsection{Organization of the thesis}
Chapters 2--4 are based on the article \cite{chernIshida2025prequantum}.

\autoref{ch:preliminary} reviews the symplectic geometry of codimension-2 shape spaces using the language of currents from geometric measure theory. This allows a unified description of results for both the classical setting with explicit representations and the implicit representations developed in the subsequent chapters.
% \autoref{ch:implicit_representations} and \autoref{ch:prequantum_bundle}.
% As a byproduct, we extend the previous result on the existence of a Liouville form for the MW form on the space of space curves to a generic codimension-2 submanifolds in arbitrary dimension.

\autoref{ch:implicit_representations} introduces implicit representations of codimension-2 shapes using a class of complex-valued functions. We study the geometry of the space of such representations as a fiber bundle over the space of explicit representations. In particular, we show that over each codimension-2 shape, the fiber may have multiple connected components indexed by $H^1_{dR}(M, \mathbb{Z})$, the first integral de Rham cohomology group of the ambient manifold.

\autoref{ch:prequantum_bundle} constructs a prequantum structure over the codimension-2 shape space. We show that a quotient bundle of the space of implicit representations forms a prequantum \( G \)-bundle with structure group \( G = \mathbb{S}^1 \times H^1_{dR}(M, \mathbb{Z}) \). As a corollary, we obtain a new geometric interpretation of the Marsden--Weinstein structure.
% as the curvature of the connection form of this bundle, which measures the average swept volume of the phase hypersurfaces of the implicit representation representing the codimension-2 shape.

\autoref{ch:hidden_degrees} is the content of the article \cite{iwc2022implicit_filaments}. 
% Exploiting the redundancy of implicit representations, we develop a numerical method for simulating space curve dynamics, with emphasis on its applications to vortex filaments in hydrodynamics. To do so, we leverage the infinite degrees of freedom, hidden in implicit representations of both the configuration and the dynamics.
% Among possible representations, we choose specifically \emph{untwisted Clebsch variables} and \emph{non-swirling dynamics}.
% 
Using implicit representations, we develop a numerical method for simulating space curve dynamics. We leverage the infinite degrees of freedom, hidden in implicit representations of both the configuration and the dynamics. This results in a simulation method that can handle otherwise numerically instable scenarios like vortex filaments from hydrodynamics, while automatically processing topological changes.

\autoref{ch:more_symplectic} is the content of the article \cite{bauerMichorIshida2024symplectic}. We introduce new symplectic structures on the space of space curves by modifying the Liouville 1-form of the Marsden--Weinstein structure, inspired by a recent trend in shape analysis. We also derive the corresponding Hamiltonian vector fields for several Hamiltonian functions with respect to these new symplectic structures, and numerically illustrate a few examples.

\autoref{ch:spherical_polygon} is the content of the article \cite{chernIshida2024sphericalPolygons}. We derive a formula for the signed area of a spherical polygon, which is a piecewise geodesic on $\SS^2$. For this purpose, we use a prequantum bundle. This gives rise to a version of Green's theorem that translates the area integral on the base manifold into a line integral along a lifted perimeter in the bundle.

\autoref{ch:postface} is independent of the rest of the thesis. I present my thoughts on mathematics, physics, and computer graphics, which I have developed over the past few years.

\paragraph{Dependencies of chapters}
\autoref{ch:preliminary} is a prerequisite for \autoref{ch:implicit_representations} and \autoref{ch:prequantum_bundle}.  The remaining chapters \autoref{ch:hidden_degrees}, \autoref{ch:more_symplectic}, and \autoref{ch:spherical_polygon} are written in a self-contained manner, so readers can  jump directly to any of them. For a deeper understanding, note that \autoref{ch:hidden_degrees} is an application of implicit representations introduced in \autoref{ch:implicit_representations}, and \autoref{ch:more_symplectic} is a generalization of the Marsden--Weinstein structure explained in \autoref{ch:preliminary}.

% \begin{tikzpicture}[
%   chapter/.style={rectangle, draw, rounded corners, minimum width=3cm, minimum height=1cm, fill=blue!10},
%   ->, >=Stealth, node distance=1.5cm and 2cm
% ]

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% \node[chapter, right=of prelim] (theory) {Chapter 3\\Main Theory};
% \node[chapter, below=of theory] (applications) {Chapter 4\\Applications};
% \node[chapter, left=of applications] (appendix) {Chapter 5\\Appendix};

% % Define arrows
% \draw (intro) -> (prelim);
% \draw (prelim) -> (theory);
% \draw (theory) -> (applications);
% \draw (appendix) -> (applications);
% \draw (prelim) -> (appendix);

% \end{tikzpicture}
