\documentclass{beamer}
\mode
{
\usetheme{default}
% or ...
\setbeamercovered{transparent}
% or whatever (possibly just delete it)
}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\title[Short Paper Title] % (optional, use only with long paper titles)
{My Beautiful Beamer Presentation}
\subtitle
{it's really gorgeous}
\author[] % (optional, use only with lots of authors)
{Karl Gauss}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
% affiliation.
\institute[Southern Illinois University Edwardsville] % (optional, but mostly needed)
{
%\inst{1}%
Department of Mathematics and Statistics\\
Southern Illinois University Edwardsville}
%\and
%\inst{2}%
%Department of Theoretical Philosophy\\
%University of Elsewhere}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.
\date[] % (optional, should be abbreviation of conference name)
{Conference on Important Mathematical Objects}
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
% yourself) who are reading the slides online
%\subject{Theoretical Computer Science}
% This is only inserted into the PDF information catalog. Can be left
% out.
% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:
% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSection[]
{
\begin{frame}
\frametitle{Outline}
\tableofcontents[currentsection]%,currentsubsection]
\end{frame}
}
% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:
%\beamerdefaultoverlayspecification{<+->}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{Outline}
\tableofcontents
% You might wish to add the option [pausesections]
\end{frame}
% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
% 15 and 30 frames, all told.
% - A conference audience is likely to know very little of what you
% are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
% enough. Leave out details, even if it means being less precise than
% you think necessary.
% - If you omit details that are vital to the proof/implementation,
% just say so once. Everybody will be happy with that.
\section{Basic definitions}
\subsection{Definitions and examples}
\begin{frame}
\frametitle{Definition of a Minimal Surface}
\begin{definition}
A \alert{minimal surface} is a 2-dimensional surface in $\R^{3}$ with mean curvature $H \equiv 0$.
\end{definition} %can justify intutition with pg. 144 of thick doCarmo
\pause
\begin{block}{Where does the name minimal come from?}
Let $F: U \subset \C \to \R^{3}$ parameterize a minimal surface; let $d: U \to \R$ be smooth with compact support. Define a deformation of $M$ by $F_{\varepsilon}: p \mapsto F(p) + \varepsilon d(p) N(p)$.
\[ \frac{d}{d\varepsilon} Area(F_{\varepsilon}(U))\bigg|_{\varepsilon = 0}\!\! = 0 \iff H \equiv 0 \]
Thus, ``minimal surfaces'' may really only be \alert{critical points} for the area functional (but the name has stuck).
\end{block}
\end{frame}
\begin{frame}
\frametitle{Definition of Triply Periodic Minimal Surface}
\begin{definition}
A \alert{triply periodic minimal surface} M is a minimal surface in $\R^3$ that is invariant under the action of a lattice $\Lambda$. The quotient surface $M / \Lambda \subset \R^3 / \Lambda$ is compact and minimal.
\end{definition}
\bigskip
\pause
Physical scientists are interested in these surfaces:
\begin{itemize}
\item Interface in polymers
\item Physical assembly during chemical reactions
\item Microcelluar membrane structures
\end{itemize}
\end{frame}
\subsection{Goal and motivation}
\begin{frame} \frametitle{Classification of TPMS}
Rough classification by the genus of $M / \Lambda$:
\begin{theorem}
(Meeks, 1975) Let $M$ be a triply periodic minimal surface of genus $g$. The Gauss map of $M / \Lambda$ is a conformal branched covering map of the sphere of degree $g-1$.
\end{theorem}
\pause
\begin{proof}
Since $M$ is minimal, $G$ is holomorphic (Weierstra\ss). Then $M / \Lambda$ is a conformal branched cover of $S^{2}$. By Gauss-Bonnet:
\[ \uncover<5->{-degree(G) 4 \pi = }\uncover<4->{-\int |K| dA = }\int K dA = 2 \pi \chi (M)\uncover<3->{ = 4\pi (1-g)} \]
\end{proof}
\begin{corollary}<6->
The smallest possible genus of $M / \Lambda$ is 3.
\end{corollary}
\end{frame}
\begin{frame} \frametitle{Other classifications?}
Many triply periodic surfaces are known to come in a continuous family (or deformation).
\bigskip
\begin{theorem}(Meeks, 1975)
There is a \alert{five-dimensional continuous family} of embedded triply periodic minimal surfaces of genus 3.
\end{theorem}
\hyperlink{psurface}{\beamergotobutton{Picture}}
\pause
\bigskip
All proven examples of genus 3 triply periodic surfaces are in the Meeks' family, with \alert{two exceptions}, the gyroid and the lidinoid.
\end{frame}
\begin{frame}\frametitle{The Gyroid}
\begin{columns}
\column{.7\textwidth}
%\includegraphics[width=4in]{gyroid.png}
\column{.3\textwidth}
\begin{itemize}
\item Schoen, 1970
\item Triply periodic surface
\item Contains no straight lines or planar symmetry curves
\end{itemize}
\end{columns}
\end{frame}
\section{Sketch of the gyroid family}
\begin{frame}\frametitle{Philosophy of the Problem}
\begin{block}{From $H \equiv 0$ to Complex Analysis}Using Weierstra\ss{}~Representation construct surfaces by finding a Riemann surface $X$, a meromorphic function $G$ on $X$, and a holomorphic 1-form $dh$ on the $X$ so that:
\begin{itemize}
\item The period problem is solved
\item Certain mild compatibility conditions are satisfied
\end{itemize}
\end{block}
\pause
\begin{block}{From Complex Analysis to Euclidean Polygons}
The period problem is typically \alert{hard}. Using flat structures, transfer the period problem to one involving Euclidean polygons and compute explicitly (algebraically!) the periods. To achieve this we:
\begin{itemize}
\item Assume (fix) some symmetries of the surface to reduce the number of parameters (and the number of conditions)
\item Find a suitable class of polygons to study
\end{itemize}
\end{block}
\end{frame}
\end{document}