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latex/chapters/chap3.tex

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-% chapters/chap2.tex
+% chapters/chap3.tex
 \documentclass[../main.tex]{subfiles}
 
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 \begin{document}
 
-\chapter{Seigel-Bargmann Space}
-
-In this chapter, we will collect ideas and other perspective we have understanding the concentration of measure phenomenon. Especially with symmetric product of $\C P^1$ and see how it relates to Riemman surfaces and Seigel-Bargmann spaces.
+\chapter{Segal-Bargmann Space}
+
+In this chapter, we collect ideas and another perspective on the concentration of measure phenomenon. In particular, we look at symmetric products of $\C P^1$ and how they relate to Riemann surfaces and Segal-Bargmann spaces.
 
 \begin{figure}[h]
     \centering
@@ -37,21 +37,23 @@ Basically, there is a bijection between the complex projective space $\mathbb{C}
 
 We can use a symmetric group of permutations of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is, $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.
 
-One might be interested in the random sampling over the $\operatorname{Sym}_n(\mathbb{C}P^1)$ and the concentration of measure phenomenon on that.
-
-\section{Majorana stellar representation of the quantum state}
-
-\begin{defn}
-    Let $n$ be a positive integer. The Majorana stellar representation of the quantum state is the set of all roots of a polynomial of degree $n$ in $\mathbb{C}$.
+One might be interested in the random sampling over the $\operatorname{Sym}_n(\mathbb{C}P^1)$ and the concentration of measure phenomenon on that.
+
+\section{Majorana stellar representation of the quantum state}
+
+This branch continues the projective-space viewpoint from the previous chapters, but now through symmetric states and polynomial data. The advantage of this model is that it converts projective quantum states into configurations of points, making it plausible to compare concentration questions with geometry on symmetric products and holomorphic function spaces.
+
+\begin{defn}
+    Let $n$ be a positive integer. The Majorana stellar representation of the quantum state is the set of all roots of a polynomial of degree $n$ in $\mathbb{C}$.
 
     
 \end{defn}
 \section{Space of complex valued functions and pure states}
 
 
-
-\ifSubfilesClassLoaded{
-  \printbibliography[title={References for Chapter 2}]
-}
+
+\ifSubfilesClassLoaded{
+  \printbibliography[title={References for Chapter 3}]
+}
 
 \end{document}