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%----------------- TITLE AND INTRODUCTION -----------------------------
\title{Summary of changes in the revised thesis}
\author{Sadashige Ishida}
\date{\today}

\begin{document}
\maketitle

\vspace{.5cm}
I thank the thesis committee members for their careful reading of the draft and constructive feedback.  This is the summary of the revision with the changes described below together with the corresponding comments in the reviews. In the supplementary file ``thesis\_revisions\_highlighted\_size\_reduced.pdf'', 
 changes in the manuscript are highlighted in light blue.


\section{Changes and responses to comments}
\begin{itemize}
    \item I resolved "TODO" parts, corrected errors and broken references, and improved the exposition in several chapters. 
    \item Chapter 6 \emph{Symplectic structures on the space of space curves} is based on a preprint. After the defense, we received reviews of the article and revised the manuscript. I have reflected these revisions in the thesis. A major addition is the discussion of (locally) conformal symplectic structures in Chapter 6.7 "Appendix: Approach via an almost complex
structure". 
\end{itemize}
%----------------- REVIEWER 1 -------------------------------------------

\begin{comment*}
    In Figure 3.3 is probably meant the Hopf link instead of the Hopf ring.  In the display at page 70 the variable t should be an s.
\end{comment*}
\begin{answer*}
    Thank you for catching them. I fixed these mistakes.
\end{answer*}

\begin{comment*}
In Proposition 4.3.5 why is the definition of a principal connection 1-form weakened by allowing to reproduce infinitesimal generators up to a constant factor?
\end{comment*}
\begin{answer*}
This was mainly for expositional simplicity. As it was mentioned in the footnote, we can always make the 1-form satisfy the classical definition of vertical reproducibility by scaling the volume form $\mu$ of the ambient manifold $M$, which does not change any results in the manuscript. But I acknowledge that this was confusing to readers. 

 I discussed this issue with Albert and introduced a normalization factor so that the normalized $\Theta$ is indeed a connection form with scaling factor $1$. In Remark 4.3.7, I explained our motivation in doing so from viewpoints of physical units, rather than phrasing "$\omega$ is prequantizable if the volume form $\mu$ is of integral class (times $2\pi$)".
\end{answer*}

\begin{comment*}
    In section 2.1.2 it is not clear that the $\operatorname{Diff}(M)$ orbit $\mathcal O$ (of a current) is a differentiable manifold. If this is not the case, then the tangent space in the formula (2.1.2) has to be seen in a formal way.
\end{comment*}
\begin{answer*}
    Indeed, it is apriori not clear whether $\mathcal O$ carries a manifold structure or not. I added explanation that, if it is not the case, the tangent space should be interpreted formally as the image of fundamental vector field mapping of $\operatorname{Diff}(M)$ action.
\end{answer*}


\begin{comment*}
    The diffeomorphism group defined in the first lines of page 29 seems to be the identity component of the isotropy subgroup of $\gamma$.
\end{comment*}
\begin{answer*}
    Yes, indeed. I updated the text accordingly. 
\end{answer*}
\begin{comment*}
Proposition 3.3.3 I think could be rephrased like this: $\Pi$ over the orbit $\mathcal O$ of $\gamma$ is a principal bundle with structure group the semidirect product  $\operatorname{DC}_\gamma \ltimes H^1(M,\mathbb Z)$.
\end{comment*} 

\begin{answer*}
We may phrase $\Pi:\mathcal F_{\mathcal O}\to \mathcal O$ as a principal bundle as suggested, but there are some obstacles as follows. 

The fiber bundle $\Pi$ can be indeed formally regarded as a principal bundle with structure group $\operatorname{Diff}_{\gamma_0}(M)\ltimes C^\infty(M,\mathbb C^\times)$, but the setting is a little complicated without groupoid framework. We first need to fix some $\gamma_0$ and then define the action of $\Diff_{\gamma_0}(M)$ on each fiber $\pi^{-1}\gamma$ where $\gamma=h\circ \gamma_0 \in \mathcal O$ with some $h\in \Diff_0(M)$ by 
\begin{equation}
    f \rhd \psi := \psi\circ h^{-1} \circ f^{-1} \circ h.
\end{equation}
Note that the entire action of $\operatorname{Diff}_{\gamma_0}(M)$ on the fiber does not depend on the choice of $h$ i.e., if $h_1\circ \gamma_0= h_2\circ \gamma_0$, the entire action of $\operatorname{Diff}_{\gamma_0}(M)$ using either $h_1$ or $h_2$ are identical.

However, the action of $\operatorname{Diff}_{\gamma_0}(M)\ltimes C^\infty(M,\mathbb C^\times)$ may still not be free. For example, some  action of $f\in \operatorname{Diff}_{\gamma_0}(M)$ alone can perform a constant phase shift $\psi \mapsto e^{ic}\psi$ for some $c\in \mathbb R$. Then $(f, e^{-ic}) \in \operatorname{Diff}_{\gamma_0}(M)\ltimes C^\infty(M,\mathbb C^\times)$ is in the stabilizer of $\psi$.

Due to such technical complicatedness, I chose not to phrase Proposition 3.3.3 as a principal bundle.
\end{answer*}


\begin{comment*}
 A prequantum bundle is always a circle bundle, so I find the name prequantum G-bundle misleading, especially when G is not a circle, but a collection of circles.
\end{comment*}
\begin{answer*}
Yes, we (Albert and I) are aware that a standard definition of prequantum bundle requires the structure group being either a circle or a (complex) line. In Chapter 4.1 and 4.3 I added clarification that our setting deviates from this standard definition. We still think our relaxation of prequantum bundle makes sense as the structure group $G$ being copies of a circle is a 1D group, in particular abelian, and we have $d\Theta=\pi^*\omega$ without extra term $\frac{1}{2}[\Theta \wedge \Theta]$.
\end{answer*}

\begin{comment*}
     In Corollary 4.3.8 what does it mean: bounds a "single" hypersurface.
\end{comment*}
\begin{answer*}
    Literally, just a hypersurface bounding the shape $\gamma$. But we agree that saying "single" hypersurface makes it confusing. So we just dropped the adjective in the statement of Corollary 4.3.8 (now Corollary 4.3.9).
\end{answer*}

\begin{comment*}
    The corollaries 4.3.7 and 4.3.8 require a proof.
\end{comment*}
\begin{answer*}
    I added a proof to Corollary 4.3.7 (which is now Corollary 4.3.8). I also newly wrote Chapter 4.5. "Appendix: Curvature as monodromy" for a proof of the standard result: the integral of curvature form is the monodromy of parallel transport measured by the connection form.
\end{answer*}

\end{document}