Add animation of the convolution network

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

.gitignore

@@ -9,4 +9,8 @@ img/*.xcf
 *.synctex.gz
 *.thm
 *.toc
-
+img/animation/*.aux
+img/animation/*.fdb_latexmk
+img/animation/*.fls
+img/animation/*.log
+img/animation/*.synctex.gz

img/animation/conv.tex

@@ -0,0 +1,54 @@
+\documentclass[10pt,tikz]{standalone}
+
+\usepackage{tikz}
+\usepackage{ifthen}
+
+\newlength{\cell}
+\setlength{\cell}{1cm}
+
+\newlength{\separation}
+\setlength{\separation}{3cm}
+
+\begin{document}
+\foreach \m in {0,...,9}
+{
+  \foreach \n in {0,...,7}
+  {
+    \begin{tikzpicture}
+      
+      % draw main grid
+      \draw[thick] (0,0) rectangle (10 * \cell, 12 * \cell);
+
+      \foreach \x in {0,...,9}
+      {
+        \foreach \y in {0,...,11}
+        {
+          \draw (\x * \cell, \y * \cell) rectangle (\x * \cell + \cell, \y * \cell + \cell);
+        }
+      }
+
+      \node[anchor=base] at (5 * \cell, -\cell) {INPUT};
+
+      % draw convolution
+      \draw[thick, fill=red!20, opacity=0.5] (\n * \cell, 12 * \cell - \m * \cell) rectangle (\n * \cell + 3 * \cell, 12 * \cell - 3 * \cell - \m * \cell);
+
+      \draw[fill=red!20, opacity=0.5] (\n * \cell + 10 * \cell + \separation, 9 * \cell - \m * \cell) rectangle (\n * \cell + 10 * \cell + \separation + \cell, 9 * \cell + \cell - \m * \cell);
+
+      \draw[thick, red] (\n * \cell + 1.5 * \cell, 12 * \cell - \m * \cell) -- (\n * \cell + 10 * \cell + \separation + 0.5 * \cell, 10 * \cell - 0.5 * \cell - \m * \cell) -- (\n * \cell + 1.5 * \cell, 9 * \cell - \m * \cell);
+
+      % draw filter
+      \draw[thick] (10 * \cell + \separation, 0) rectangle (18 * \cell + \separation, 10 * \cell);
+
+      \foreach \x in {0,...,7}
+      {
+        \foreach \y in {0,...,9}
+        {
+          \draw (10 * \cell + \separation + \x * \cell, \y * \cell) rectangle (10 * \cell + \separation + \x * \cell + \cell, \y * \cell + \cell);
+        }
+      }
+
+      \node[anchor=base] at (10 * \cell + \separation + 4 * \cell, -\cell) {OUTPUT};
+    \end{tikzpicture}
+  }
+}
+\end{document}

thesis.tex

@@ -12,6 +12,7 @@
 \usepackage{upgreek}
 \usepackage{physics}
 \usepackage{tensor}
+\usepackage{animate}
 \usepackage{graphicx}
 \usepackage{transparent}
 \usepackage{tikz}
@@ -26,7 +27,7 @@
 \usecolortheme{crane}
 \usefonttheme{structurebold}
 \setbeamertemplate{navigation symbols}{}
-\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
+\addtobeamertemplate{background canvas}{\transfade[duration=0.15]}{}
 
 \author[Finotello]{Riccardo Finotello}
 \title[D-branes and Deep Learning]{D-branes and Deep Learning}
@@ -58,7 +59,7 @@
 \newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}}
 \renewcommand{\cite}[1]{{\tiny{\fcolorbox{red}{red!10}{[#1]}}}}
 
-\newcommand{\firstlogo}{img/unito}
+\newcommand{\firstlogo}{img/unito.pdf}
 \newcommand{\thefirstlogo}{%
   \begin{figure}
     \centering
@@ -66,7 +67,7 @@
   \end{figure}
 }
 
-\newcommand{\secondlogo}{img/infn}
+\newcommand{\secondlogo}{img/infn.pdf}
 \newcommand{\thesecondlogo}{%
   \begin{figure}
     \centering
@@ -160,7 +161,7 @@
   }
 
   {%
-    % \setbeamertemplate{footline}{}
+    \setbeamertemplate{footline}{}
     \usebackgroundtemplate{%
       \transparent{0.1}
       \includegraphics[width=\paperwidth]{img/torino.png}
@@ -203,79 +204,52 @@
 
     \pause
 
-    \begin{columns}
+    \begin{columns}[T, totalwidth=0.935\linewidth]
-      \begin{column}[t]{0.5\linewidth}
+      \begin{column}{0.45\linewidth}
-        \highlight{Symmetries:}
+        \begin{tabular}{@{}ll@{}}
-        \begin{itemize}
+          Symmetries: &
-          \item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
+          \\
-
+          \toprule
-          \item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
+          \textbf{Poincaré transf.}: & 
-
+          $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
-          \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
+          \\
-        \end{itemize}
+          \textbf{2D diff.}: &
+          $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
+          \\
+          \textbf{Weyl transf.}: &
+          $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
+          \\
+        \end{tabular}
       \end{column}
+      \hfill
+      \begin{column}{0.45\linewidth}
+        \begin{tabular}{@{}ll@{}}
+          Conformal symmetry: &
+          \\
+          \toprule
+          \textbf{vanishing} stress-energy tensor: &
+          $\mathcal{T}_{\upalpha \upbeta} = 0$
+          \\
+          \textbf{traceless} stress-energy tensor: &
+          $\trace{\mathcal{T}} = 0$
+          \\
+          \textbf{conformal gauge}: &
+          $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
+          \\
+        \end{tabular}
 
-      \pause
+        \pause
 
-      \begin{column}[t]{0.5\linewidth}
+        \begin{center}
-        \highlight{Conformal symmetry:}
+          \highlight{Conformal properties fixed by \textbf{OPE}s.}
-        \begin{itemize}
+        \end{center}
-          \item \textbf{vanishing} stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$
-          
-          \item \textbf{traceless} stress-energy tensor: $\trace{\mathcal{T}} = 0$
-
-          \item \textbf{conformal gauge} $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
-        \end{itemize}
       \end{column}
     \end{columns}
   \end{frame}
 
 
-  % \begin{frame}{Action Principle and Conformal Symmetry}
-  %   \begin{columns}
-  %     \begin{column}{0.6\linewidth}
-  %       \highlight{%
-  %         Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
-  %       }
-  %       \begin{equation*}
-  %         \mathcal{T}( z )\, \Upphi_h( w )
-  %         \stackrel{z \to w}{\sim}
-  %         \frac{h}{(z - w)^2} \Upphi_h( w )
-  %         +
-  %         \frac{1}{z - w} \partial_w \Upphi_h( w )
-  %       \end{equation*}
-  %       \begin{equation*}
-  %         \mathcal{T}( z )\, \mathcal{T}( w )
-  %         \stackrel{z \to w}{\sim}
-  %         \frac{\frac{c}{2}}{(z - w)^4}
-  %         +
-  %         \order{(z - w)^{-2}}
-  %       \end{equation*}
-
-  %       \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
-  %         \begin{eqnarray*}
-  %           \qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
-  %           & = &
-  %           (n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
-  %           \\
-  %           \qty[ L_n,\, \overline{L}_m ]
-  %           & = &
-  %           0
-  %         \end{eqnarray*}
-  %       \end{equationblock}
-  %     \end{column}
-
-  %     \begin{column}{0.4\linewidth}
-  %       \begin{figure}[h]
-  %         \centering
-  %         \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
-  %       \end{figure}
-  %     \end{column}
-  %   \end{columns}
-  % \end{frame}
-
   \begin{frame}{Action Principle and Conformal Symmetry}
-    \highlight{Superstrings in $D$ dimensions:}
+    Superstrings in $D$ dimensions $\longrightarrow$ \emph{Virasoro algebra} (central extension of de Witt's algebra):
     \begin{equation*}
       \mathcal{T}( z )
       =
@@ -311,7 +285,7 @@
 
     \pause
 
-    \highlight{Consequence:}
+    Consequence:
     \begin{equation*}
       c_{\text{full}} = c + c_{\text{ghost}} = 0
       \quad
@@ -324,152 +298,145 @@
 
   \begin{frame}{Extra Dimensions and Compactification}
     \begin{block}{Compactification}
-      \begin{columns}
+      \begin{columns}[T, totalwidth=0.95\linewidth]
-        \begin{column}{0.7\linewidth}
+        \begin{column}{0.8\linewidth}
           \begin{equation*}
             \mathscr{M}^{1,\, 9}
             =
             \mathscr{M}^{1,\, 3} \otimes \mathscr{X}_6
           \end{equation*}
+          \vspace{-1em}
           \begin{itemize}
             \item $\mathscr{X}_6$ is a \textbf{compact} manifold
 
-            \item $N = 1$ \textbf{supersymmetry} is preserved in 4D
+            \item $N = 1$ \textbf{supersymmetry} preserved in 4D
 
             \item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ in arising \textbf{gauge group}
           \end{itemize}
         \end{column}
 
         \begin{tikzpicture}[remember picture, overlay]
-          \node[anchor=base] at (8em,-3.3em) {\cite{code in Hanson (1994)}};
+          \node[anchor=base] at (-2em,-6em) {\cite{code in Hanson (1994)}};
-          \node[anchor=base] at (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}};
+          \node[anchor=base] at (-7em,-6em) {\includegraphics[width=0.25\linewidth]{img/cy.png}};
         \end{tikzpicture}
-        \begin{column}{0.3\linewidth}
-        %   \centering
-        %   \includegraphics[width=0.9\columnwidth]{img/cy}
-        \end{column}
       \end{columns}
     \end{block}
 
     \pause
 
-    \begin{columns}
+    \vfill
-      \begin{column}[t]{0.5\linewidth}
+    \begin{columns}[T, totalwidth=0.95\linewidth]
-        \highlight{Kähler manifolds} $\qty( M,\, g )$ such that
+      \begin{column}{0.475\linewidth}
+        \textbf{Kähler manifolds} $\qty( M,\, g )$ such that:
         \begin{itemize}
           \item $\dim\limits_{\mathds{C}} M = m$
 
           \item $\mathrm{Hol}( g ) \subseteq \mathrm{SU}( m )$
 
-          \item $g$ is Ricci-flat (equiv.\ $c_1\qty( M )$ vanishes)
+          \item $\mathrm{Ric}( g ) \equiv 0$ (equiv.\ $c_1\qty( M ) \equiv 0$)
         \end{itemize}
       \end{column}
+      \hfill
 
       \pause
 
-      \begin{column}[t]{0.5\linewidth}
+      \begin{column}{0.475\linewidth}
-        Characterised by \highlight{Hodge numbers}
+        Characterised by \textbf{Hodge numbers}
         \begin{equation*}
           h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} )
         \end{equation*}
-        counting the no.\ of harmonic $(r,\,s)$-forms.
+        counting the no.\ of harmonic $(r,s)$-forms.
       \end{column}
     \end{columns}
   \end{frame}
 
   \begin{frame}{D-branes and Open Strings}
-    Polyakov's action naturally introduces \highlight{Neumann b.c.:}
+    Polyakov's action naturally introduces \textbf{Neumann b.c.} for \textbf{open strings}:
     \begin{equation*}
       \eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
     \end{equation*}
-    satisfied by \textbf{open and closed} strings living in $D$ dimensions s.t.\ $\square X = 0$.
+    satisfied by \highlight{\textbf{open and closed strings} in $D$ dim.} s.t.\ $\square X = 0 \Rightarrow X( z, \overline{z} ) = X( z ) + \overline{X}( \overline{z} )$.
 
     \pause
 
-    \begin{equationblock}{T-duality}
+    % \begin{equationblock}{Equivalent Theories of Closed String Compactification}
+    %   \begin{equation*}
+    %     X( z, \overline{z} )
+    %     =
+    %     X( z ) + \overline{X}( \overline{z} )
+    %     \quad
+    %     \stackrel{T-dual}{\Rightarrow}
+    %     \quad
+    %     X( z ) - \overline{X}( \overline{z} )
+    %     =
+    %     Y( z, \overline{z} )
+    %     =
+    %     Y( z ) + \overline{Y}( \overline{z} )
+    %   \end{equation*}
+    % \end{equationblock}
+
+    % \pause
+
+    \begin{block}{T-duality}
+      \textbf{Dirichlet b.c.} consequence of \textbf{T-duality} on $p$ directions:
       \begin{equation*}
-        X( z, \overline{z} )
+        \overline{X}( z ) \mapsto - \overline{X}( z )
-        =
-        X( z ) + \overline{X}( \overline{z} )
         \quad
-        \stackrel{T}{\Rightarrow}
+        \Rightarrow
         \quad
-        X( z ) - \overline{X}( \overline{z} )
+        \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
-        =
+        \quad
-        Y( z, \overline{z} )
+        \stackrel{T-duality}{\longrightarrow}
-        =
+        \quad
-        Y( z ) + \overline{Y}( \overline{z} )
+        \eval{\partial_{\uptau} \widetilde{X}^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
       \end{equation*}
-    \end{equationblock}
+      thus \textbf{open strings} can be \textbf{constrained} to $D(D - p - 1)$-branes.
-
+      \hfill
-    \pause
+      \cite{Polchinski (1995, 1996)}
-
+    \end{block}
-    Resulting effect (repeated $p \le D - 1$ times) leads to \highlight{Dirichlet b.c.:}
-    \begin{equation*}
-      \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
-      \quad
-      \stackrel{T}{\Rightarrow}
-      \quad
-      \eval{\partial_{\uptau} Y^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
-      \quad
-      \forall i = 1, 2,\, \dots,\, p
-    \end{equation*}
-    thus \textbf{open strings} can be \textbf{constrained} to \highlight{$D(D - p - 1)$-branes.}
-    \hfill
-    \cite{Polchinski (1995, 1996)}
   \end{frame}
 
   \begin{frame}{D-branes and Open Strings}
-    Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$.}
+    Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$}:
+    \begin{equation*}
+      \mathcal{A}^{\upmu} \rightarrow \qty( \mathcal{A}^A,\, \mathcal{A}^a )
+      \quad
+      \Rightarrow
+      \quad
+      \mathrm{U}( 1 )~\text{theory~in}~p+1~\text{dimensions~(and~scalars)}
+    \end{equation*}
 
     \pause
 
-    \begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$)}
+    \vspace{2em}
-      \begin{equation*}
+    \begin{columns}[T, totalwidth=0.95\linewidth]
-        \mathcal{A}^{\upmu}
+      \begin{column}{0.475\linewidth}
-        \quad \leftrightarrow \quad
-        \upalpha_{-1}^{\upmu} \ket{0}
-        \qquad
-        \longrightarrow
-        \qquad
-        \begin{tabular}{@{}llll@{}}
-          $\mathcal{A}^A$
-          &
-          $\leftrightarrow$
-          &
-          $\upalpha_{-1}^A \ket{0},$
-          &
-          $A = 0,\, 1,\, \dots,\, p$
-          \\
-          $\mathcal{A}^a$
-          &
-          $\leftrightarrow$
-          &
-          $\upalpha_{-1}^a \ket{0},$
-          &
-          $a = 1,\, 2,\, \dots,\, D - p - 1$
-        \end{tabular}
-      \end{equation*}
-    \end{equationblock}
-
-    \pause
-
-    \begin{columns}
-      \begin{column}{0.5\linewidth}
         \centering
-        \resizebox{0.55\columnwidth}{!}{\import{img}{chanpaton.pgf}}
+        \resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}}
-      \end{column}
 
-      \begin{column}{0.5\linewidth}
+        \cite{Chan, Paton (1969)}
-        Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}:
-        \hfill\cite{Chan, Paton (1969)}
         \begin{equation*}
-          \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
+          \ket{n;\, r}
-          \quad
+          =
-          \longrightarrow
+          \sum\limits_{i,\, j = 1}^N
-          \quad
+          \ket{n;\, i,\, j}\,
-          \mathrm{U}( N )
+          \tensor{\uplambda}{^r_{ij}}
         \end{equation*}
+      \end{column}
+      \hfill
+      \pause
+
+      \begin{column}{0.475\linewidth}
+        \begin{block}{Chan--Paton Factors}
+          When branes are \textbf{coincident}:
+          \begin{equation*}
+            \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
+            \quad
+            \longrightarrow
+            \quad
+            \mathrm{U}( N )
+          \end{equation*}
+        \end{block}
         \pause
         \highlight{Build gauge bosons, fermions and scalars.}
       \end{column}
@@ -542,6 +509,7 @@
           \qty( X_{(t)} )^I
           =
           \tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
+          \in \mathds{R}^4
         \end{equation*}
         \pause
         where
@@ -567,9 +535,9 @@
     \begin{columns}
       \begin{column}{0.6\linewidth}
         \begin{itemize}
-          \item consider $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
+          \item $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
 
-          \item let $x_{(t)} < x_{(t-1)}$ be the \textbf{worldsheet intersection points} on \textbf{real axis}
+          \item $x_{(t)} < x_{(t-1)}$ \textbf{worldsheet intersection points}
 
           \item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet}
         \end{itemize}
@@ -685,7 +653,7 @@
           \begin{split}
             \partial_z \mathcal{X}( z )
             & =
-            \sum\limits_{l,\, r} c_{lr}\,
+            \sum\limits_{l,\, r = -\infty}^{+\infty} c_{lr}\,
             \qty( - \upomega_z )^{A_{lr}}\,
             \qty( 1 - \upomega_z )^{B_{lr}}\,
             B_{0,\, l}^{(L)}( \omega_z )\,
@@ -718,7 +686,7 @@
   \end{frame}
 
   \begin{frame}{The Solution}
-    \highlight{Operations sequence:}
+    Sequence of the operations:
     \begin{enumerate}
       \item rotation matrix $=$ monodromy matrix
 
@@ -742,20 +710,19 @@
         \begin{columns}
           \begin{column}{0.4\linewidth}
             \centering
-            \resizebox{0.607\columnwidth}{!}{\import{img}{branesangles.pgf}}
+            \resizebox{0.6\columnwidth}{!}{\import{img}{branesangles.pgf}}
           \end{column}
           \hfill
           \begin{column}{0.6\linewidth}
             \begin{equation*}
               \begin{split}
-                \eval{S_{\mathds{R}^4}}_{\text{on-shell}}
+                2 \uppi \upalpha' \eval{S_{\mathds{R}^4}}_{\text{on-shell}}
                 & =
-                \frac{1}{2\uppi \upalpha'}
                 \sum\limits_{t = 1}^3
                 \qty( \frac{1}{2} \abs{g_{(t)}^{\perp}} \abs{f_{(t-1)} - f_{(t)}} )
                 \\
                 & =
-                \text{Area}\qty( \qty{ f_{(t)} } )
+                \text{Area}\qty( \qty{ f_{(t)} }_{1 \le t \le N_B} )
               \end{split}
             \end{equation*}
           \end{column}
@@ -763,8 +730,23 @@
         \vfill
       }
       \only<6->{%
-        \centering
+        \begin{columns}
-        \resizebox{0.25\columnwidth}{!}{\import{img}{brane3d.pgf}}
+          \begin{column}{0.35\linewidth}
+            \centering
+            \resizebox{0.8\columnwidth}{!}{\import{img}{brane3d.pgf}}
+          \end{column}
+          \hfill
+          \begin{column}{0.6\linewidth}
+            \begin{itemize}
+              \item strings no longer confined to plane
+
+              \item strings form a \emph{small bump} from the D-brane
+
+              \item classical action \textbf{larger} than factorised case
+            \end{itemize}
+          \end{column}
+        \end{columns}
+        \vfill
       }
     \end{block}
   \end{frame}
@@ -773,7 +755,7 @@
   \subsection[Fermions]{Fermions and Point-like Defect CFT}
 
   \begin{frame}{Fermions on the Strip}
-    \begin{columns}
+    \begin{columns}[totalwidth=0.95\linewidth]
       \begin{column}{0.4\linewidth}
         \centering
         \resizebox{0.9\columnwidth}{!}{\import{img}{defects.pgf}}
@@ -803,7 +785,7 @@
 
     \pause
 
-    \begin{block}{Stress-energy Tensor}
+    \begin{equationblock}{Stress-energy Tensor}
       \begin{equation*}
         \mathcal{T}_{\pm\pm}( \upxi_{\pm} )
         =
@@ -839,11 +821,11 @@
           0
         \end{cases}
       \end{equation*}
-    \end{block}
+    \end{equationblock}
   \end{frame}
 
   \begin{frame}{Conserved Product and Operators}
-    Expand on a \highlight{basis of solutions}
+    Expand on a \textbf{basis of solutions}
     \begin{equation*}
       \uppsi_{\pm}( \upxi_{\pm} )
       =
@@ -890,7 +872,7 @@
 
     \pause
 
-    Derive the \highlight{algebra of operators:}
+    Derive the \textbf{algebra of operators:}
     \begin{equation*}
       \qty[ b_n,\, b_m^{\dagger} ]_+
       =
@@ -956,7 +938,7 @@
 
     \pause
 
-    Theories are subject to \highlight{consistency conditions:}
+    Theories are subject to \textbf{consistency conditions:}
     \begin{columns}
       \begin{column}{0.6\linewidth}
         \begin{equation*}
@@ -977,7 +959,7 @@
   \end{frame}
 
   \begin{frame}{Stress-energy Tensor and CFT Approach}
-    Compute the OPEs leading to the \highlight{stress-energy tensor:}
+    Compute the OPEs leading to the \textbf{stress-energy tensor:}
     \begin{equation*}
       \mathcal{T}( z )
       =
@@ -1070,7 +1052,7 @@
     \begin{columns}
       \begin{column}{0.5\linewidth}
         \centering
-        \includegraphics[width=0.9\columnwidth]{img/cone}
+        \includegraphics[width=0.9\columnwidth]{img/cone.pdf}
       \end{column}
       \hfill
       \begin{column}{0.5\linewidth}
@@ -1100,50 +1082,45 @@
   \end{frame}
 
   \begin{frame}{Orbifolds}
-    \begin{columns}[c]
+    \begin{columns}[T]
       \begin{column}{0.475\linewidth}
-        \begin{center}
+        \begin{tabular}{@{}l@{}}
           \textbf{Mathematics}
-
+          \\
-          \begin{itemize}
+          \toprule
-            \item manifold $M$
+          manifold $M$
-
+          \\
-            \item (Lie) group $G$
+          (Lie) group $G$
-
+          \\
-            \item \emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$
+          \emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$
-
+          \\
-            \item \emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$
+          \emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$
-
+          \\
-            \item charts $\upphi = \uppi \circ \mathscr{P}$ where:
+          charts $\upphi = \uppi \circ \mathscr{P}$ where:
-
+          \\
-            \begin{itemize}
+          \vspace{1em}$\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
-              \item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
+          \\
-
+          \vspace{1em}$\uppi\colon U / G \to M$
-              \item $\uppi\colon U / G \to M$
+          \\
-            \end{itemize}
+        \end{tabular}
-          \end{itemize}
-        \end{center}
-      \end{column}
-      \begin{column}{0.05\linewidth}
-        \centering
-        $\Rightarrow$
       \end{column}
+      \hfill
       \begin{column}{0.475\linewidth}
-        \begin{center}
+        \begin{tabular}{@{}l@{}}
           \textbf{Physics}
-
+          \\
-          \begin{itemize}
+          \toprule
-            \item global orbit space $M / G$
+          global orbit space $M / G$
-
+          \\
-            \item $G$ group of isometries
+          $G$ group of isometries
-
+          \\
-            \item fixed points
+          fixed points
-
+          \\
-            \item additional d.o.f.\ (\emph{twisted states})
+          additional d.o.f.\ (\emph{twisted states})
-
+          \\
-            \item singular limits of CY manifolds
+          singular limits of CY manifolds
-          \end{itemize}
+          \\
-        \end{center}
+        \end{tabular}
       \end{column}
     \end{columns}
 
@@ -1217,7 +1194,7 @@
 
     \pause
 
-    Consider \highlight{scalar QED:}
+    Scalars on NBO:
     \begin{equation*}
       \upphi_{\qty{ k_+,\, l,\, \vec{k},\, r}}\qty( u,\, v,\, z,\, \vec{x} )
       =
@@ -1266,8 +1243,9 @@
     \pause
 
     \begin{center}
-      \it
+      \emph{
-      most terms \textbf{do not converge} due to \textbf{isolated zeros} ($l_{(*)} \equiv 0$) and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial
+        most terms \textbf{do not converge} due to \textbf{isolated zeros} \emph{($l_{(*)} \equiv 0$)} and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial
+      }
     \end{center}
   \end{frame}
 
@@ -1464,6 +1442,12 @@
 
     \pause
 
+    \begin{tikzpicture}[remember picture, overlay]
+      \draw[line width=4pt, red] (13em, 5.5em) rectangle (22em, 0em);
+    \end{tikzpicture}
+
+    \pause
+
     \begin{block}{Machine Learning Approach}
       What is $\mathscr{R}$?
       \begin{equation*}
@@ -1490,13 +1474,15 @@
 
         \pause
 
-      \item effectively use knowledge from \textbf{computer science, mathematics and physics} to solve problems
+      \item knowledge from \textbf{computer science, mathematics and physics} to solve problems
+
+        \pause
+
+      \item provide in-depth \textbf{data analysis} of the datasets
     \end{itemize}
 
-    \pause
-
     \begin{center}
-      \includegraphics[width=0.7\linewidth]{img/label-distribution_orig}
+      \includegraphics[width=0.7\linewidth]{img/label-distribution_orig.pdf}
     \end{center}
   \end{frame}
 
@@ -1504,31 +1490,6 @@
   \subsection[Machine Learning]{Machine Learning for String Theory}
 
 
-  \begin{frame}{Exploratory Data Analysis}
-    Machine Learning \highlight{pipeline:}
-    \begin{center}
-      \textbf{exploratory} data analysis
-      $\rightarrow$
-      feature \textbf{selection}
-      $\rightarrow$
-      Hodge numbers
-    \end{center}
-
-    \pause
-
-    \begin{columns}
-      \begin{column}{0.5\linewidth}
-        \centering
-        \includegraphics[width=0.9\columnwidth]{img/corr-matrix_orig}
-      \end{column}
-      \hfill
-      \begin{column}{0.5\linewidth}
-        \centering
-        \includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig}
-      \end{column}
-    \end{columns}
-  \end{frame}
-
   \begin{frame}{Dataset}
     \begin{itemize}
       \item $7890$ CICY manifolds (full dataset)
@@ -1553,13 +1514,38 @@
     \pause
 
     \begin{center}
-      \includegraphics[width=0.7\linewidth]{img/label-distribution-compare_orig}
+      \includegraphics[width=0.7\linewidth]{img/label-distribution-compare_orig.pdf}
     \end{center}
   \end{frame}
 
+  \begin{frame}{Exploratory Data Analysis}
+    Machine Learning \highlight{pipeline:}
+    \begin{center}
+      \textbf{exploratory} data analysis
+      $\rightarrow$
+      feature \textbf{selection}
+      $\rightarrow$
+      Hodge numbers
+    \end{center}
+
+    \pause
+
+    \begin{columns}
+      \begin{column}{0.5\linewidth}
+        \centering
+        \includegraphics[width=0.9\columnwidth]{img/corr-matrix_orig.pdf}
+      \end{column}
+      \hfill
+      \begin{column}{0.5\linewidth}
+        \centering
+        \includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig.pdf}
+      \end{column}
+    \end{columns}
+  \end{frame}
+
   \begin{frame}{Machine Learning}
     \centering
-    \includegraphics[width=0.85\linewidth]{img/ml_map}
+    \includegraphics[width=0.85\linewidth]{img/ml_map.png}
     
     \begin{tikzpicture}[remember picture, overlay]
       \node[anchor=base] at (16em,18em) {\cite{from scikit-learn.org}};
@@ -1584,9 +1570,9 @@
       \begin{column}{0.4\linewidth}
         What is PCA for a $X \in \mathds{R}^{n \times p}$?
         \begin{itemize}
-          \item find new coordinates to \textbf{``put the variance in order''}
+          \item project data onto a \textbf{lower dimensional} space where variance is maximised
 
-          \item \highlight{equivalently} compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$
+          \item equivalently compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$
 
           \item isolate \textbf{the signal} from the \textbf{background}
 
@@ -1594,10 +1580,11 @@
         \end{itemize}
       \end{column}
       \hfill
+      \pause
       \begin{column}{0.6\linewidth}
         \centering
-        \includegraphics[width=0.5\columnwidth]{img/marchenko-pastur}
+        \includegraphics[width=0.5\columnwidth]{img/marchenko-pastur.pdf}
-        \includegraphics[width=\columnwidth]{img/svd_orig}
+        \includegraphics[width=\columnwidth]{img/svd_orig.pdf}
       \end{column}
     \end{columns}
   \end{frame}
@@ -1607,13 +1594,13 @@
       \begin{column}{0.5\linewidth}
         \centering
         \textbf{Configuration Matrix Only}
-        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots}
+        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots.pdf}
       \end{column}
       \hfill\pause
       \begin{column}{0.5\linewidth}
         \centering
         \textbf{Best Training Set}
-        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots}
+        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf}
       \end{column}
     \end{columns}
   \end{frame}
@@ -1635,14 +1622,14 @@
 
         \begin{block}{Neural Networks}
           \vspace{0.5em}
-          \begin{tabular}{@{}lc@{}}
+          \begin{tabular}{@{}ll@{}}
-            fully connected:
+            \textbf{fully connected}:
             &
-            $a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
+            $a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$
             \\
-            convolutional:
+            \textbf{convolutional}:
             &
-            $a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
+            $a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$
           \end{tabular}
         \end{block}
 
@@ -1679,10 +1666,9 @@
       \hfill
       \begin{column}{0.6\linewidth}
         \centering
-        \only<1>{\includegraphics[width=0.75\columnwidth, trim={12in 10in 0 0}, clip]{img/input_mat}}
+        \only<1-3>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}}
-        \only<2>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat}}
+        \only<4>{\animategraphics[autoplay,loop,controls,width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}}
-        \only<3>{\includegraphics[width=0.75\columnwidth, trim={12in 0 0 10in}, clip]{img/input_mat}}
+        \only<5>{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}}
-        \only<4->{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}}
       \end{column}
     \end{columns}
   \end{frame}
@@ -1709,13 +1695,13 @@
       \begin{column}{0.5\linewidth}
         \centering
         \textbf{Best Training Set}
-        \includegraphics[width=\columnwidth]{img/cicy_best_plots}
+        \includegraphics[width=\columnwidth]{img/cicy_best_plots.pdf}
       \end{column}
       \hfill
       \begin{column}{0.5\linewidth}
         \centering
-        \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11}
+        \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11.pdf}
-        \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve}
+        \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve.pdf}
       \end{column}
     \end{columns}
   \end{frame}