diff --git a/.gitignore b/.gitignore index b71074f..c2694d4 100644 --- a/.gitignore +++ b/.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 diff --git a/img/animation/conv.pdf b/img/animation/conv.pdf new file mode 100644 index 0000000..ba31872 Binary files /dev/null and b/img/animation/conv.pdf differ diff --git a/img/animation/conv.tex b/img/animation/conv.tex new file mode 100644 index 0000000..e3f39b8 --- /dev/null +++ b/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} diff --git a/img/animation/sequence/conv-0.png b/img/animation/sequence/conv-0.png new file mode 100644 index 0000000..8ed0889 Binary files /dev/null and b/img/animation/sequence/conv-0.png differ diff --git a/img/animation/sequence/conv-1.png b/img/animation/sequence/conv-1.png new file mode 100644 index 0000000..0f99107 Binary files /dev/null and b/img/animation/sequence/conv-1.png differ diff --git a/img/animation/sequence/conv-10.png b/img/animation/sequence/conv-10.png new file mode 100644 index 0000000..b4e8513 Binary files /dev/null and b/img/animation/sequence/conv-10.png differ diff --git a/img/animation/sequence/conv-11.png b/img/animation/sequence/conv-11.png new file mode 100644 index 0000000..0432bb4 Binary files /dev/null and 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-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{column}[t]{0.5\linewidth} - \highlight{Symmetries:} - \begin{itemize} - \item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$ - - \item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$ - - \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$ - \end{itemize} + \begin{columns}[T, totalwidth=0.935\linewidth] + \begin{column}{0.45\linewidth} + \begin{tabular}{@{}ll@{}} + Symmetries: & + \\ + \toprule + \textbf{Poincaré transf.}: & + $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$ + \\ + \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} - \highlight{Conformal symmetry:} - \begin{itemize} - \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} + \begin{center} + \highlight{Conformal properties fixed by \textbf{OPE}s.} + \end{center} \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{column}{0.7\linewidth} + \begin{columns}[T, totalwidth=0.95\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 (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}}; + \node[anchor=base] at (-2em,-6em) {\cite{code in Hanson (1994)}}; + \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} - \begin{column}[t]{0.5\linewidth} - \highlight{Kähler manifolds} $\qty( M,\, g )$ such that + \vfill + \begin{columns}[T, totalwidth=0.95\linewidth] + \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} - Characterised by \highlight{Hodge numbers} + \begin{column}{0.475\linewidth} + 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} ) - = - X( z ) + \overline{X}( \overline{z} ) + \overline{X}( z ) \mapsto - \overline{X}( z ) \quad - \stackrel{T}{\Rightarrow} + \Rightarrow \quad - X( z ) - \overline{X}( \overline{z} ) - = - Y( z, \overline{z} ) - = - Y( z ) + \overline{Y}( \overline{z} ) + \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0 + \quad + \stackrel{T-duality}{\longrightarrow} + \quad + \eval{\partial_{\uptau} \widetilde{X}^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0 \end{equation*} - \end{equationblock} - - \pause - - 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)} + thus \textbf{open strings} can be \textbf{constrained} to $D(D - p - 1)$-branes. + \hfill + \cite{Polchinski (1995, 1996)} + \end{block} \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} $\mathrm{SO}( D - 2 )$)} - \begin{equation*} - \mathcal{A}^{\upmu} - \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} + \vspace{2em} + \begin{columns}[T, totalwidth=0.95\linewidth] + \begin{column}{0.475\linewidth} \centering - \resizebox{0.55\columnwidth}{!}{\import{img}{chanpaton.pgf}} - \end{column} + \resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}} - \begin{column}{0.5\linewidth} - Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}: - \hfill\cite{Chan, Paton (1969)} + \cite{Chan, Paton (1969)} \begin{equation*} - \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1) - \quad - \longrightarrow - \quad - \mathrm{U}( N ) + \ket{n;\, r} + = + \sum\limits_{i,\, j = 1}^N + \ket{n;\, i,\, j}\, + \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 - \resizebox{0.25\columnwidth}{!}{\import{img}{brane3d.pgf}} + \begin{columns} + \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} - \item manifold $M$ - - \item (Lie) group $G$ - - \item \emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$ - - \item \emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$ - - \item charts $\upphi = \uppi \circ \mathscr{P}$ where: - - \begin{itemize} - \item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$ - - \item $\uppi\colon U / G \to M$ - \end{itemize} - \end{itemize} - \end{center} - \end{column} - \begin{column}{0.05\linewidth} - \centering - $\Rightarrow$ + \\ + \toprule + manifold $M$ + \\ + (Lie) group $G$ + \\ + \emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$ + \\ + \emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$ + \\ + charts $\upphi = \uppi \circ \mathscr{P}$ where: + \\ + \vspace{1em}$\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$ + \\ + \vspace{1em}$\uppi\colon U / G \to M$ + \\ + \end{tabular} \end{column} + \hfill \begin{column}{0.475\linewidth} - \begin{center} + \begin{tabular}{@{}l@{}} \textbf{Physics} - - \begin{itemize} - \item global orbit space $M / G$ - - \item $G$ group of isometries - - \item fixed points - - \item additional d.o.f.\ (\emph{twisted states}) - - \item singular limits of CY manifolds - \end{itemize} - \end{center} + \\ + \toprule + global orbit space $M / G$ + \\ + $G$ group of isometries + \\ + fixed points + \\ + additional d.o.f.\ (\emph{twisted states}) + \\ + singular limits of CY manifolds + \\ + \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 - 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 + \emph{ + 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=\columnwidth]{img/svd_orig} + \includegraphics[width=0.5\columnwidth]{img/marchenko-pastur.pdf} + \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@{}} - fully connected: + \begin{tabular}{@{}ll@{}} + \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<2>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat}} - \only<3>{\includegraphics[width=0.75\columnwidth, trim={12in 0 0 10in}, clip]{img/input_mat}} - \only<4->{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}} + \only<1-3>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}} + \only<4>{\animategraphics[autoplay,loop,controls,width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}} + \only<5>{\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} + \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11.pdf} + \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve.pdf} \end{column} \end{columns} \end{frame}