Commit 7296a2a8 authored by Quentin Aristote's avatar Quentin Aristote
Browse files

continued diapo

parent 4ffe164a
No preview for this file type
......@@ -21,6 +21,8 @@
\usepackage{caption}
\usepackage{float}
\usepackage{mwe}
\usepackage{array}
\usepackage{pst-solides3d}
\usepackage{hyperref}
\usepackage[sorting = none, backend = biber]{biblatex}
......@@ -43,6 +45,11 @@
% ------------------------------ Commands ------------------------------
\newcommand{\PreserveBackslash}[1]{\let\temp=\\#1\let\\=\temp}
\newcolumntype{C}[1]{>{\PreserveBackslash\centering}p{#1}}
\newcolumntype{R}[1]{>{\PreserveBackslash\raggedleft}p{#1}}
\newcolumntype{L}[1]{>{\PreserveBackslash\raggedright}p{#1}}
\newcommand{\PM}[2]{\emph{$#1$--to--$#2$ Pachner move}}
\newcommand{\PMs}[2]{\emph{$#1$--to--$#2$ Pachner moves}}
......@@ -92,7 +99,7 @@
\begin{frame}
\title{Marche quantique sur un réseau triangulaire sujet à des \textit{Pachner moves}}
\author{Quentin Aristote}
\author{Quentin Aristote \\ Giuseppe Di Molfetta (LiS Lab, équipe CaNa)}
\date{}
\titlepage
\end{frame}
......@@ -149,56 +156,31 @@
% premiere partie stage : travail bibliographique -> approprier
\begin{frame}{Marche quantique sur le réseau triangulaire}
\pause
\begin{figure}[H]
\centering
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\foreach \j in {-1,...,6} {
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\draw (0,0) grid (11,-5) ;
\end{tikzpicture}
\nolabelcaption{Le réseau triangulaire}
\includetikzpicture{triangular_lattice}
\pause
\nolabelcaption{$\mathbb{P}(\mathrm{particule\:à\:l'instant\:}t\mathrm{\:sur\:l'arête\:}e) = \norm{\psi(t,e)}^2$}
\end{figure}
\pause
\begin{figure}[H]
\centering
\begin{tikzpicture}[xscale = 1.5, yscale = 1.5]
\node at (-.5, .866) {étape 1 :} ;
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\begin{subfigure}[t]{.4\textwidth}
\centering
\includetikzpicture{triangle_rotation}
\nolabelcaption{étape 1}
\end{subfigure}
\qquad
\pause
\begin{tikzpicture}[xscale = 1.5, yscale = 1.5]
\node at (-.5, .866) {étape 2 :} ;
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\draw[->] (\i, \j - \s * .1) -- (\i, \j + \s * .1) ;
}
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\draw[<->, red, thick] (\i + \di, \j + \dj) -- (\i - \di, \j - \dj) ;
}
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\end{tikzpicture}
\begin{subfigure}[t]{.5\textwidth}
\centering
\includetikzpicture{edge_exchange}
\pause
\nolabelcaption{étape 2 : $\psi(t+\epsilon,e) = W\begin{pmatrix}\psi^\uparrow(t,e) \\ \psi^\downarrow(t,e)\end{pmatrix}$}
\end{subfigure}
\end{figure}
\end{frame}
% première théorie : marche ...
......@@ -212,5 +194,81 @@
% bonne matrice -> converge vers eq dirac (propagation particule matiere surface euclidienne)
% s'adapte pour dirac courbé
\begin{frame}{\textit{Pachner moves}}
\pause
\begin{tabular}{L{.5\textwidth} L{.5\textwidth}}
Triangulation & Graphe \\ \hline
\end{tabular}
\pause
\begin{figure}[H]
\centering
\begin{subfigure}[t]{.3\textwidth}
\centering
\includetikzpicture[width = .75\linewidth]{tetrahedre}
\end{subfigure}
\qquad\qquad
\pause
\begin{subfigure}[t]{.3\textwidth}
\centering
\includetikzpicture[width = .75\linewidth]{discrete_3_dim_sphere}
\end{subfigure}
\end{figure}
\pause
\begin{figure}[H]
\centering
\begin{subfigure}[t]{.4\textwidth}
\centering
\includetikzpicture[width = 1.2\linewidth]{triangle_1-to-3_pachner_move}
\end{subfigure}
\qquad\qquad
\pause
\begin{subfigure}[t]{.4\textwidth}
\centering
\includetikzpicture[width = 1.2\linewidth]{1-to-3_pachner_move}
\end{subfigure}
\end{figure}
\end{frame}
% objectif : étendre cette marche quantique à réseau triangulaire dynamique
% 2nde théorie : introduit dynamisme réseau : pachner moves
% pq : ne change pas la topologie
% voir de 2 façons : triangulation / graphe associé (sommet = triangle, arête = arête)
% pm : prendre complementaire d'un ensemble de triangles dans tetrahedre
% ex : 1-3 (un triangle -> 3 triangles)
\section{Couplage}
% 2 théories -> couplées, s'interprète bien theorie relativité générale
\begin{frame}{Action des \textit{Pachner moves} sur l'étiquettage}
\pause
\includetikzpicture[width = .4\textwidth]{binary_label}
\qquad
\pause
\includetikzpicture[width = .5\textwidth]{labeled_discrete_3_dim_sphere}
\vspace{30pt}
\pause
\includetikzpicture[width = \textwidth]{labeled_1-to-3_pachner_move}
\end{frame}
% 1ere chose à définir : comment étiquetter les nouveaux triangles
% etiquettage binaire impossible (3-cycles)
% 4 étiquettes, étiquettage du tétrahèdre
% correspondance kème caractère étiquette <-> coordonnée du kème côté portée par le triangle
% pm = prendre complémentaire étiquetté puis inversion des flèches
\begin{frame}{1-to-3 \& 2-to-2 \textit{Pachner moves}}
\pause
\begin{figure}[H]
\centering
\includetikzpicture[width = .8\textwidth]{walker_1-to-3}
\end{figure}
\vspace{20pt}
\pause
\begin{figure}[H]
\centering
\includetikzpicture[width = \textwidth]{walker_2-to-2}
\end{figure}
\end{frame}
\end{document}
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\ No newline at end of file
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\usepackage{tikz}
\usepackage{amsmath}
\usepackage[graphics,tightpage,active]{preview}
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