updates

presentation/ZheyuanWu_HonorThesis_Presentation.tex

@@ -538,26 +538,56 @@
 
 \section{Geometry of State Space}
 
-
+\begin{frame}{Observable diameter: the inner definition}
-\begin{frame}{Observable Diameter}
+    \begin{block}{Partial diameter on $\mathbb{R}$}
-    \begin{block}{Definition}
+        Let $\nu$ be a Borel probability measure on $\mathbb{R}$ and let $\alpha \in (0,1]$.
-        For a metric-measure space $X$ and $\kappa>0$,
+        The \textbf{partial diameter} of $\nu$ at mass level $\alpha$ is
         $$
-        \obdiam_{\mathbb{R}}(X;-\kappa)
+        \diameter(\nu;\alpha):=
-        =
+        \{\diameter(A):A \subseteq \mathcal{B}(\mathbb{R}),
-        \sup_{f\in \operatorname{Lip}_1(X,\mathbb{R})}
+             \nu(A)\ge \alpha
-        \diameter(f_*\mu_X;1-\kappa).
+        \},
+        $$
+        where
+        $$
+        \diameter(A):=\sup_{x,y\in A}|x-y|.
         $$
     \end{block}
 
+    \vspace{0.4em}
     \begin{itemize}
-        \item It asks how concentrated every $1$-Lipschitz real observable must be.
+        \item This asks for the shortest interval-like region containing at least $\alpha$ of the total mass.
-        \item In the thesis, entropy is used as a concrete observable-diameter proxy.
+        \item So $\diameter(\nu;1-\kappa)$ measures how tightly we can capture \emph{most} of the distribution, allowing us to discard a set of mass at most $\kappa$.
-        \item Hopf fibration lets us compare $\mathbb{C}P^n$ with spheres.
     \end{itemize}
+
+\end{frame}
+
+\begin{frame}{Observable diameter of a metric-measure space}
+    \begin{block}{Definition}
+        Let $X=(X,d_X,\mu_X)$ be a metric-measure space and let $\kappa>0$.
+        The \textbf{observable diameter} of $X$ is
+        $$
+        \obdiam_{\mathbb{R}}(X;-\kappa)
+        :=
+        \sup_{f\in \operatorname{Lip}_1(X,\mathbb{R})}
+        \diameter(f_*\mu_X;1-\kappa),
+        $$
+        where $\operatorname{Lip}_1(X,\mathbb{R})$ is the set of all $1$-Lipschitz functions
+        $f:X\to\mathbb{R}$, and $f_*\mu_X$ is the pushforward measure on $\mathbb{R}$.
+    \end{block}
+
+    \vspace{0.4em}
+    \begin{itemize}
+        \item Each $1$-Lipschitz function $f$ is viewed as an \textbf{observable} on $X$.
+        \item The pushforward measure $f_*\mu_X$ is the distribution of the values of that observable.
+        \item If $\obdiam_{\mathbb{R}}(X;-\kappa)$ is small, then \emph{every} $1$-Lipschitz observable is strongly concentrated.
+    \end{itemize}
+
 \end{frame}
 
 \begin{frame}{A Geometric Consequence}
+    In this thesis, entropy functions are used as concrete observables to estimate observable diameter, and the Hopf fibration helps transfer information between $S^{2n+1}$ and $\mathbb{C}P^n$.
+    \vspace{0.4em}
     \begin{block}{Projective-space estimate}
         For $0<\kappa<1$,
         $$