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zdq92uk_!lh`->(HVT(5OD~^EzG5_RG9KQ*T04+R!mXkn1_GAMM3oXSPXcEwp_$Q5k z1o286X;3uJMjGZXeMS=SV4JkD909~({YeAysUYTSBMpkT+(^SiYccfquO1*JK%CwN z8Va=d7Y&Lr+*l4z*f?IW=lc7*iiS`D8|tF*knw~5{^bP?Ch0F4;V&Hm5YNA4kHX*q z+Id5J7y|UGLVxywMgH9$76oD!H(o%^M$KfSl{QN4qs<8hB-oRGK literal 0 HcmV?d00001 diff --git a/thesis.tex b/thesis.tex index 3e0c285..cba4794 100644 --- a/thesis.tex +++ b/thesis.tex @@ -59,7 +59,7 @@ \newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}} \renewcommand{\cite}[1]{{\tiny{\fcolorbox{red}{red!10}{[#1]}}}} -\newcommand{\firstlogo}{img/unito.pdf} +\newcommand{\firstlogo}{img/unito.png} \newcommand{\thefirstlogo}{% \begin{figure} \centering @@ -237,12 +237,6 @@ $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$ \\ \end{tabular} - - \pause - - \begin{center} - \highlight{Conformal properties fixed by \textbf{OPE}s.} - \end{center} \end{column} \end{columns} \end{frame} @@ -311,7 +305,7 @@ \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} + \item contains algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ \end{itemize} \end{column} @@ -326,8 +320,8 @@ \vfill \begin{columns}[T, totalwidth=0.95\linewidth] - \begin{column}{0.475\linewidth} - \textbf{Kähler manifolds} $\qty( M,\, g )$ such that: + \begin{column}{0.55\linewidth} + \textbf{Calabi--Yau manifolds} $\qty( M,\, g )$ such that: \begin{itemize} \item $\dim\limits_{\mathds{C}} M = m$ @@ -335,23 +329,25 @@ \item $\mathrm{Ric}( g ) \equiv 0$ (equiv.\ $c_1\qty( M ) \equiv 0$) \end{itemize} + + \hspace{1em}\cite{Calabi (1957), Yau (1977), Candelas \emph{et al.} (1985)} \end{column} \hfill \pause - \begin{column}{0.475\linewidth} + \begin{column}{0.4\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. + (no.\ of harmonic $(r,s)$-forms). \end{column} \end{columns} \end{frame} \begin{frame}{D-branes and Open Strings} - Polyakov's action naturally introduces \textbf{Neumann b.c.} for \textbf{open strings}: + Polyakov's action naturally introduces \textbf{Neumann b.c.}: \begin{equation*} \eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0 \end{equation*} @@ -441,46 +437,51 @@ \centering \resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}} - \cite{Chan, Paton (1969)} + \hfill\cite{Chan, Paton (1969)} \begin{equation*} \ket{n;\, r} = \sum\limits_{i,\, j = 1}^N \ket{n;\, i,\, j}\, \tensor{\uplambda}{^r_{ij}} + \quad + \Rightarrow + \quad + \highlight{\mathrm{U}(N)} \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.} + \begin{column}{0.5\linewidth} + \centering + \resizebox{\columnwidth}{!}{\import{img}{smbranes.pgf}} + + % \begin{block}{Symmetry Enhancement} + % 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} \end{column} \end{columns} \end{frame} - \begin{frame}{Standard Model-like Scenarios} - \centering - \resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}} - \hfill\cite{Zwiebach (2009)} - \end{frame} + % \begin{frame}{Standard Model-like Scenarios} + % \centering + % \resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}} + % \hfill\cite{Zwiebach (2009)} + % \end{frame} \subsection[D-branes at Angles]{D-branes Intersecting at Angles} \begin{frame}{Intersecting D-branes} - Consider \highlight{$N$ intersecting $D6$-branes} filling $\mathscr{M}^{1,3}$ and \textbf{embedded in} $\mathds{R}^6$ + Consider \highlight{$3$ intersecting $D6$-branes} filling $\mathscr{M}^{1,3}$ and \textbf{embedded in} $\mathds{R}^6$ \begin{equationblock}{Twist Fields Correlators} \begin{equation*} @@ -512,7 +513,7 @@ \item \textbf{relative rotations} are $\mathrm{SO}(2) \simeq \mathrm{U}(1)$ elements - \item $S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) \sim \text{Area}\qty( \qty{ f_{(t)},\, \mathrm{R}_{(t)} }_{1 \le t \le N_B} )$ + \item $S_{E}^{(\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) \sim \text{Area}\qty( \qty{ f_{(t)},\, \mathrm{R}_{(t)} }_{1 \le t \le N_B} )$ \end{itemize} \hfill\cite{Cremades, Ibanez, Marchesano (2003); Pesando (2012)} @@ -523,8 +524,6 @@ \begin{frame}{$\mathrm{SO}(4)$ Rotations} Consider \highlight{$\mathds{R}^4 \times \mathds{R}^2$} (focus on $\mathds{R}^4$): - \pause - \begin{columns} \begin{column}{0.4\linewidth} \centering @@ -538,12 +537,10 @@ \tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I \in \mathds{R}^4 \end{equation*} - \pause where \begin{equation*} R_{(t)} \in \frac{\mathrm{SO}(4)}{\mathrm{S}\qty( \mathrm{O}(2) \times \mathrm{O}(2) )} \end{equation*} - \pause that is \begin{equation*} \qty[ R_{(t)} ] @@ -554,11 +551,7 @@ \end{columns} \end{frame} - \begin{frame}{Boundary Conditions} - What are the consequences for \highlight{open strings?} - - \pause - + \begin{frame}{Boundary Conditions and Open Strings} \begin{columns} \begin{column}{0.6\linewidth} \begin{itemize} @@ -566,13 +559,13 @@ \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} +% \item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet} \end{itemize} \end{column} - + \hfill \begin{column}{0.4\linewidth} \centering - \resizebox{0.9\columnwidth}{!}{\import{img}{branchcuts.pgf}} + \resizebox{\columnwidth}{!}{\import{img}{branchcuts.pgf}} \end{column} \end{columns} @@ -668,27 +661,31 @@ \end{frame} \begin{frame}{Hypergeometric Basis} - \begin{columns} - \begin{column}{0.3\linewidth} + \begin{columns}[totalwidth=0.95\linewidth] + \begin{column}{0.4\linewidth} \centering - \resizebox{0.9\columnwidth}{!}{\import{img}{threebranes_plane.pgf}} + \resizebox{\columnwidth}{!}{\import{img}{threebranes_plane.pgf}} \end{column} \hfill - \begin{column}{0.7\linewidth} + \begin{column}{0.6\linewidth} Sum over \highlight{all contributions:} \begin{equation*} \begin{split} \partial_z \mathcal{X}( z ) & = + \pdv{\omega_z}{z}\, \sum\limits_{l,\, r = -\infty}^{+\infty} c_{lr}\, \qty( - \upomega_z )^{A_{lr}}\, \qty( 1 - \upomega_z )^{B_{lr}}\, + \\ + & \times B_{0,\, l}^{(L)}( \omega_z )\, \qty( B_{0,\, r}^{(R)}( \omega_z ) )^T \end{split} \end{equation*} \end{column} \end{columns} + \vfill \pause @@ -717,23 +714,17 @@ \begin{enumerate} \item rotation matrix $=$ monodromy matrix - \pause - \item contiguity relations $\Rightarrow$ independent hypergeometrics - \pause - \item finite action $\Rightarrow$ $2$ solutions (no.\ of d.o.f.\ is correctly saturated) - \pause - \item boundary conditions $\Rightarrow$ fix free constants $c_{lr}$ \end{enumerate} \pause \begin{block}{Physical Interpretation} - \only<5>{% + \only<2>{% \begin{columns} \begin{column}{0.4\linewidth} \centering @@ -756,7 +747,7 @@ \end{columns} \vfill } - \only<6->{% + \only<3->{% \begin{columns} \begin{column}{0.35\linewidth} \centering @@ -771,7 +762,7 @@ \item classical action \textbf{larger} than factorised case \end{itemize} - \hspace{0.65\columnwidth}\cite{RF, Pesando (2019)} + \hspace{0.65\columnwidth}\cite{\textbf{RF}, Pesando (2019)} \end{column} \end{columns} } @@ -809,46 +800,45 @@ \end{equationblock} \end{column} \end{columns} + \vfill \pause - \begin{equationblock}{Stress-energy Tensor} - \begin{equation*} - \mathcal{T}_{\pm\pm}( \upxi_{\pm} ) + \begin{equation*} + \mathcal{T}_{\pm\pm}( \upxi_{\pm} ) + = + -i\, \frac{T}{4}\, + \uppsi^*_{\pm,\, I}( \upxi_{\pm} )\, + \overset{\leftrightarrow}{\partial} \uppsi^I_{\pm}( \upxi_{\pm} ) + \quad + \Rightarrow + \quad + \begin{cases} + \dot{\mathrm{H}}( \uptau ) + & + % = + % \partial_{\uptau} + % \qty(% + % \int\limits_0^{\uppi} \dd{\upsigma} + % \mathcal{T}_{\uptau\uptau}( \uptau, \upsigma ) + % ) = - -i\, \frac{T}{4}\, - \uppsi^*_{\pm,\, I}( \upxi_{\pm} )\, - \overset{\leftrightarrow}{\partial} \uppsi^I_{\pm}( \upxi_{\pm} ) - \quad - \Rightarrow - \quad - \begin{cases} - \dot{\mathrm{H}}( \uptau ) - & - % = - % \partial_{\uptau} - % \qty(% - % \int\limits_0^{\uppi} \dd{\upsigma} - % \mathcal{T}_{\uptau\uptau}( \uptau, \upsigma ) - % ) - = - 0 - \quad \Leftrightarrow \quad - \uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} ) - \\ - \dot{\mathrm{P}}( \uptau ) - & - % = - % \partial_{\uptau} - % \qty(% - % \int\limits_0^{\uppi} \dd{\upsigma} - % \mathcal{T}_{\uptau\upsigma}( \uptau, \upsigma ) - % ) - \neq - 0 - \end{cases} - \end{equation*} - \end{equationblock} + 0 + \quad \Leftrightarrow \quad + \uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} ) + \\ + \dot{\mathrm{P}}( \uptau ) + & + % = + % \partial_{\uptau} + % \qty(% + % \int\limits_0^{\uppi} \dd{\upsigma} + % \mathcal{T}_{\uptau\upsigma}( \uptau, \upsigma ) + % ) + \neq + 0 + \end{cases} + \end{equation*} \end{frame} \begin{frame}{Conserved Product and Operators} @@ -912,7 +902,7 @@ \end{frame} \begin{frame}{Twisted Complex Fermions} - Consider the case $R_{(t)} = e^{i \uppi \upalpha_{(t)}} \in \mathrm{U}( 1 )$: + Consider $R_{(t)} = e^{i \uppi \upalpha_{(t)}} \in \mathrm{U}( 1 )$: \begin{equation*} \Uppsi( x_{(t)} + e^{2\uppi i} \updelta ) = @@ -969,6 +959,10 @@ \begin{columns} \begin{column}{0.6\linewidth} \begin{equation*} + \braket{\qty{x_{(t)}}} = 1 + \quad + \Rightarrow + \quad \mathrm{L} = n_{(t)} + \widetilde{n}_{(t)} @@ -986,7 +980,7 @@ \end{frame} \begin{frame}{Stress-energy Tensor and CFT Approach} - Compute the OPEs leading to the \textbf{stress-energy tensor:} + Compute the OPEs leading to the \highlight{time dependent} \textbf{stress-energy tensor:} \begin{equation*} \mathcal{T}( z ) = @@ -1002,7 +996,7 @@ + \frac{1}{2} \qty( \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}} )^2 \end{equation*} - \hfill\cite{RF, Pesando (2019)} + \hfill\cite{\textbf{RF}, Pesando (2019)} \pause @@ -1021,7 +1015,7 @@ \begin{equationblock}{Equivalence with Bosonization} \begin{equation*} \begin{split} - \partial_{x_{(t)}} \braket{\qty{x_{(t)}}} + \partial_{x_{(t)}} \ln \braket{\qty{x_{(t)}}} & = \oint\limits_{x_{(t)}} \frac{\dd{z}}{2\uppi i} \frac{% @@ -1046,21 +1040,11 @@ \begin{itemize} \item (semi-)phenomenological models involve \textbf{twist and spin} fields and \textbf{open strings} - \pause + \item framework for \textbf{bosonic} open strings with \textbf{intersecting D-branes} - \item general framework for \textbf{bosonic} open strings with \textbf{intersecting D-branes} + \item \textbf{spin fields} as \textbf{boundary changing operators} (hidden in \textbf{defects}) - \pause - - \item leading contribution for \textbf{twist fields} - - \pause - - \item \textbf{spin fields} as \textbf{boundary changing operators} on \textbf{defects} - - \pause - - \item alternative framework for amplitudes (extension to (non) Abelian twist/spin fields?) + \item framework for amplitudes (extension to (non) Abelian twist/spin fields?) \end{itemize} \end{frame} @@ -1072,7 +1056,7 @@ \begin{frame}{A Few Words on a Theory of Everything} \begin{center} - string theory = theory of everything = nuclear forces + gravity + \textbf{string theory} = theory of everything = \textbf{nuclear forces + gravity} \end{center} \pause @@ -1109,64 +1093,64 @@ \end{columns} \end{frame} - \begin{frame}{Orbifolds} - \begin{columns}[T] - \begin{column}{0.475\linewidth} - \begin{tabular}{@{}p{0.975\columnwidth}@{}} - \textbf{Mathematics} - \\ - \toprule - \begin{itemize} - \item manifold $M$ + % \begin{frame}{Orbifolds} + % \begin{columns}[T] + % \begin{column}{0.475\linewidth} + % \begin{tabular}{@{}p{0.975\columnwidth}@{}} + % \textbf{Mathematics} + % \\ + % \toprule + % \begin{itemize} + % \item manifold $M$ - \item (Lie) group $G$ + % \item (Lie) group $G$ - \item \emph{stabilizer} $G_p = \qty{g \in G \mid gp = p \in M}$ + % \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 \emph{orbit} $Gp = \qty{gp \in M \mid g \in G}$ - \item charts $\upphi = \uppi \circ \mathscr{P}$ where: + % \item charts $\upphi = \uppi \circ \mathscr{P}$ where: - \begin{itemize} - \item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$ + % \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{tabular} - \end{column} - \hfill - \begin{column}{0.475\linewidth} - \begin{tabular}{@{}p{0.975\columnwidth}@{}} - \textbf{Physics} - \\ - \toprule - \begin{itemize} - \item global orbit space $M / G$ + % \item $\uppi\colon U / G \to M$ + % \end{itemize} + % \end{itemize} + % \end{tabular} + % \end{column} + % \hfill + % \begin{column}{0.475\linewidth} + % \begin{tabular}{@{}p{0.975\columnwidth}@{}} + % \textbf{Physics} + % \\ + % \toprule + % \begin{itemize} + % \item global orbit space $M / G$ - \item $G$ group of isometries + % \item $G$ group of isometries - \item fixed points + % \item fixed points - \item additional d.o.f.\ (\emph{twisted states}) + % \item additional d.o.f.\ (\emph{twisted states}) - \item singular limits of CY manifolds - \end{itemize} - \end{tabular} - \end{column} - \end{columns} + % \item singular limits of CY manifolds + % \end{itemize} + % \end{tabular} + % \end{column} + % \end{columns} - \pause + % \pause - \vspace{2em} - \begin{center} - Use \textbf{time-dependent orbifolds} to model singularities in time - \end{center} + % \vspace{2em} + % \begin{center} + % Use \textbf{time-dependent orbifolds} to model singularities in time + % \end{center} - \begin{tikzpicture}[remember picture, overlay] - \draw[line width=4pt, red] (5em,3.5em) rectangle (35em, 1em); - \end{tikzpicture} - \end{frame} + % \begin{tikzpicture}[remember picture, overlay] + % \draw[line width=4pt, red] (5em,3.5em) rectangle (35em, 1em); + % \end{tikzpicture} + % \end{frame} \begin{frame}{Cosmological Singularities} Use \textbf{time-dependent orbifolds} to model \textbf{space-like singularities}: @@ -1216,10 +1200,12 @@ \pause - \begin{equationblock}{Killing Vector and Null Boost Oribfold} + \begin{equationblock}{Killing Vector and Null Boost Orbifold} \begin{equation*} \upkappa = -i \qty( 2 \uppi \Updelta ) J_{+2} = 2 \uppi \partial_z + \quad \Rightarrow + \quad z \sim z + 2 \uppi n \end{equation*} \end{equationblock} @@ -1276,7 +1262,7 @@ \begin{center} \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 + most terms \textbf{do not converge} due to \textbf{isolated zeros} \emph{($l_{(*)} \equiv 0$)} and cannot be recovered even with a \textbf{distributional interpretion} due to the term $\propto u^{-1}$ in the exponential } \end{center} \end{frame} @@ -1284,20 +1270,15 @@ \begin{frame}{String and Field Theory} So far: \begin{itemize} - \item field theory presents \textbf{divergences} (even sQED $\rightarrow$ eikonal?) - - \pause + \item field theory presents \textbf{divergences} (see sQED) \item obvious ways to regularise (Wilson lines, higher derivative couplings, etc.) \textbf{do not work} - - \pause \item divergences are \textbf{not (only) gravitational} - \pause - - \item \textbf{vanishing volume} in phase space of the compact direction is responsible for the divergence + \item \textbf{vanishing volume} in phase space responsible for the divergence \end{itemize} + \vfill \pause @@ -1356,7 +1337,7 @@ \pause - \begin{equationblock}{Distributional Interpretation} + \begin{equationblock}{No isolated zeros $\Rightarrow$ distributional Interpretation} \begin{equation*} \widetilde{\upphi}_{\qty{ k_+,\, p,\, l,\, \vec{k},\, r}}\qty( u ) = @@ -1371,36 +1352,28 @@ \begin{itemize} \item divergences are present in sQED and \textbf{open string} sector - \pause - \item singularities $\Rightarrow$ \textbf{massive states} are no longer spectators - \pause - \item vanishing volume (\textbf{compact orbifold directions}) $\Rightarrow$ particles ``cannot escape'' - \pause - \item \textbf{non compact} orbifold directions $\Rightarrow$ interpretation of \textbf{amplitudes as distributions} - \pause - \item issue not restricted to NBO/GNBO but also BO, null brane, etc. (it is a \textbf{general issues} connected to the geometry of the underlying space) \end{itemize} + \vfill \pause - \vspace{2em} \begin{center} \it - divergences are \textbf{hidden into contact terms} and interactions with \textbf{massive states} (the gravitational eikonal deals with massless interactions) + divergences are \textbf{hidden into EFT contact terms} and interactions with \textbf{string massive states}: gravity is not the only cause as the same problems are present also in gauge theories. \end{center} \begin{tikzpicture}[remember picture, overlay] \draw[line width=4pt, red] (0em, 4.5em) rectangle (40em, 1em); \end{tikzpicture} - \hfill\cite{Arduino, RF, Pesando (2020)} + \hfill\cite{Arduino, \textbf{RF}, Pesando (2020)} \end{frame} @@ -1439,9 +1412,9 @@ \mathds{P}^n\colon \qquad \begin{cases} - p_a\qty( Z^0,\, \dots,\, Z^n ) & = P_{I_1 \dots I_a} Z^{I_1} \dots Z^{I_a} = 0 + p_i\qty( Z^0,\, \dots,\, Z^n ) & = P_{a_1 \dots a_i} Z^{a_1} \dots Z^{a_i} = 0 \\ - p_a\qty( \uplambda Z^0,\, \dots,\, \uplambda Z^n ) & = \uplambda^a p_a\qty( Z^0,\, \dots,\, Z^n ) + p_i\qty( \uplambda Z^0,\, \dots,\, \uplambda Z^n ) & = \uplambda^i p_i\qty( Z^0,\, \dots,\, Z^n ) \end{cases} \end{equation*} \hfill\cite{Green, Hübsch (1987); Hübsch (1992)} @@ -1489,11 +1462,11 @@ \begin{block}{Machine Learning Approach} What is $\mathscr{R}$ in \textbf{machine learning} approach? \begin{equation*} - \mathscr{R}( M ) \longrightarrow \mathscr{R}_n( M;\, w ) + \mathscr{R}( M ) \rightarrow \mathscr{R}_n( M;\, w ) \rightarrow \widehat{h}^{p,\,q} \qquad \text{s.t.} \qquad - \exists n > M > 0 \mid \mathcal{L}\qty(\mathscr{R}( M ),\, \mathscr{R}_n( M;\, w )) < \upepsilon + \exists n > M > 0 \quad \mid \quad \mathcal{L}_n\qty(\widehat{h}^{p,\,q},\, h^{p,\,q}) < \upepsilon \quad \forall \upepsilon > 0 \end{equation*} @@ -1502,19 +1475,15 @@ \begin{frame}{Machine Learning} \begin{itemize} - \item exchange \textbf{analytical solution} with \textbf{optimisation problem} + \item \textbf{optimisation problem} $\Rightarrow$ \highlight{gradient descent} (or similar) \pause - \item use \textbf{various algorithms} and exploit \textbf{large datasets} + \item use \textbf{various algorithms} and exploit \textbf{large datasets} (more training) \pause - \item learn a \textbf{representation} rather than a \textbf{solution} - - \pause - - \item knowledge from \textbf{computer science, mathematics and physics} to solve problems + \item intersection of \textbf{computer science, mathematics and physics} \pause @@ -1523,6 +1492,8 @@ \begin{center} \includegraphics[width=0.7\linewidth]{img/label-distribution_orig.pdf} + + \hfill\cite{Green \emph{et al.} (1987)} \end{center} \end{frame} @@ -1533,21 +1504,13 @@ \begin{frame}{Dataset} \begin{itemize} \item $7890$ CICY manifolds (full dataset) - - \pause \item \textbf{dataset pruning}: no product spaces, no ``very far'' outliers (reduction of $0.49\%$) - \pause - \item $h^{1,\, 1} \in \qty[ 1,\, 16 ]$ and $h^{2,\, 1} \in \qty[ 15,\, 86 ]$ - \pause - \item $80\%$ training, $10\%$ validation, $10\%$ test - \pause - \item choose \textbf{regression}, but evaluate using \textbf{accuracy} (round the result) \end{itemize} @@ -1559,7 +1522,6 @@ \end{frame} \begin{frame}{Exploratory Data Analysis} - Machine Learning \highlight{pipeline:} \begin{center} \textbf{exploratory} data analysis $\rightarrow$ @@ -1568,19 +1530,25 @@ Hodge numbers \end{center} - \hfill\cite{Ruehle (2020); Erbin, RF (2020)} + \hfill\cite{Ruehle (2020); Erbin, \textbf{RF} (2020)} + \vfill \pause \begin{columns} - \begin{column}{0.5\linewidth} + \begin{column}{0.33\linewidth} \centering - \includegraphics[width=0.9\columnwidth]{img/corr-matrix_orig.pdf} + \includegraphics[width=\columnwidth]{img/corr-matrix_orig.pdf} \end{column} \hfill - \begin{column}{0.5\linewidth} + \begin{column}{0.33\linewidth} \centering - \includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig.pdf} + \includegraphics[width=\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig.pdf} + \end{column} + \hfill + \begin{column}{0.33\linewidth} + \centering + \includegraphics[width=\columnwidth, trim={0 0.5in 6in 0}, clip]{img/vector-tensor-features_orig.pdf} \end{column} \end{columns} \end{frame} @@ -1590,7 +1558,7 @@ \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}}; + \node[anchor=base] at (16em,18em) {\cite{from \href{https://scikit-learn.org/stable/tutorial/machine_learning_map/index.html}{scikit-learn.org}}}; \end{tikzpicture} \pause @@ -1612,13 +1580,13 @@ \begin{column}{0.4\linewidth} What is PCA for a $X \in \mathds{R}^{n \times p}$? \begin{itemize} - \item project data onto a \textbf{lower dimensional} space where variance is maximised + \item project data such that \textbf{variance is maximised} - \item equivalently compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$ + \item \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$ - \item isolate \textbf{the signal} from the \textbf{background} + \item isolate \textbf{signal} from \textbf{background} - \item ease the machine learning job of finding a better representation of the input + \item ease the ML job of finding a better representation of the input \end{itemize} \end{column} \hfill @@ -1642,7 +1610,7 @@ \begin{column}{0.5\linewidth} \centering \textbf{Best Training Set} - \cite{Erbin, RF (2020)} + \cite{Erbin, \textbf{RF} (2020)} \includegraphics[width=0.75\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf} \end{column} \end{columns} @@ -1666,13 +1634,13 @@ \begin{block}{Layers} \vspace{0.5em} \begin{tabular}{@{}ll@{}} - \textbf{fully connected}: + fully connected: & - $\upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$ + $\upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$ \\ - \textbf{convolutional}: + convolutional: & - $\upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$ + $\upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$ \end{tabular} \end{block} @@ -1694,14 +1662,14 @@ Why convolutional? \begin{columns} \begin{column}{0.4\linewidth} - \begin{itemize}[<+->] + \begin{itemize} \item retain \textbf{spacial awareness} \item smaller \textbf{no.\ of parameters} ($\approx 2 \times 10^5$ vs.\ $\approx 2 \times 10^6$) \item weights are \textbf{shared} - \item CNN can isolate \textbf{``defining features''} + \item CNNs isolate \textbf{``defining features''} \item find patterns as in \textbf{computer vision} \end{itemize} @@ -1709,21 +1677,21 @@ \hfill \begin{column}{0.6\linewidth} \centering - \only<1-3>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}} - \only<4>{\animategraphics[autoplay,loop,controls={play,stop},width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}} - \only<5>{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}} + \only<1>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}} + \only<2>{\animategraphics[autoplay,loop,controls={play,stop},width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}} + \only<3>{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}} \end{column} \end{columns} \end{frame} \begin{frame}{Inception Neural Networks} - Recent development by Google's deep learning teams led to: + Recent development by deep learning research at \highlight{Google} led to: \begin{itemize} - \item neural networks with \textbf{better generalisation properties} + \item neural networks with better \textbf{generalisation properties} - \item smaller networks (both parameters and depth) + \item \textbf{smaller} networks (both parameters and depth) - \item different \textbf{concurrent kernels} (e.g.\ one over \textbf{equations} one over \textbf{coordinates}) + \item different \textbf{concurrent kernels} \end{itemize} \pause @@ -1738,64 +1706,50 @@ \begin{column}{0.5\linewidth} \centering \textbf{Best Training Set} - \cite{Erbin, RF (2020)} + \cite{Erbin, \textbf{RF} (2020)} \only<1>{\includegraphics[width=0.8\columnwidth, trim={0 0 1.65in 0}, clip]{img/cicy_best_plots.pdf}} \only<2->{\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.pdf} - \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve.pdf} - \end{column} + \visible<2->{ + \begin{column}{0.5\linewidth} + \centering + \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} \begin{frame}{A Few Comments and Future Directions} - Why \highlight{deep learning in physics?} - \begin{itemize} - \item reliable \textbf{predictive method} \pause (provided good data analysis) - - \pause - - \item reliable \textbf{source of inspiration} \pause (provided good data analysis) - - \pause - - \item reliable \textbf{generalisation method} \pause (provided good data analysis) - - \pause - - \item \textbf{CNNs are powerful tools} (this is the \emph{first time in physics!}) - - \pause - - \item interdisciplinary approach $=$ win-win situation! - \end{itemize} - - \pause - - What now? - - \begin{itemize} - \item representation learning $\Rightarrow$ what is the best way to represent CICYs? - - \pause - - \item study invariances $\Rightarrow$ invariances should not influence the result (graph representations?) - - \pause - - \item higher dimensions $\Rightarrow$ what about CICY 4-folds? - - \pause - - \item geometric deep learning $\Rightarrow$ explain the geometry of the ``AI'' behind deep learning! - - \pause - - \item reinforcement learning $\Rightarrow$ give the rules, not the result! - \end{itemize} + \begin{tabular}{@{}l@{}} + Why \highlight{deep learning in physics?} + \\ + \toprule + $\circ$ reliable \textbf{predictive method} \pause (provided good data analysis) + \\ + $\circ$ reliable \textbf{source of inspiration} \pause (provided good data analysis) + \\ + $\circ$ reliable \textbf{generalisation method} \pause (provided good data analysis) + \\ + $\circ$ \textbf{CNNs are powerful tools} (this is the \emph{first time in physics!}) + \\ + $\circ$ interdisciplinary approach $=$ win-win situation! + \\[1em] + \pause + What now? + \\ + \toprule + $\circ$ representation learning $\Rightarrow$ what is the best way to represent CICYs? + \\ + $\circ$ study invariances $\Rightarrow$ invariances should not influence the result (graph representations?) + \\ + $\circ$ higher dimensions $\Rightarrow$ what about CICY 4-folds? + \\ + $\circ$ geometric deep learning $\Rightarrow$ explain the geometry of the ``AI'' behind deep learning! + \\ + $\circ$ reinforcement learning $\Rightarrow$ give the rules, not the result! + \end{tabular} \end{frame} {% @@ -1806,6 +1760,24 @@ } \addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{} \begin{frame}[noframenumbering]{The End?} + \begin{columns}[T, totalwidth=\linewidth] + \begin{column}{0.7\linewidth} + \begin{itemize} + \item general framework for \textbf{D-branes at angles} + + \item alternative computations of \textbf{correlators of spin fields} + + \item strings and divergences in \textbf{time dependent orbifolds} + + \item string compactifications and \textbf{deep learning techniques} + \end{itemize} + \end{column} + \begin{column}{0.3\linewidth} + \centering + \includegraphics[width=0.5\columnwidth]{\firstlogo} + \end{column} + \end{columns} + \vfill \begin{center} \Huge THANK YOU