diff -r be41f435c3f3 -r 6785d7aa7c49 talks/20091108-Riverside/riverside1.tex --- a/talks/20091108-Riverside/riverside1.tex Tue Nov 03 16:02:37 2009 +0000 +++ b/talks/20091108-Riverside/riverside1.tex Tue Nov 03 19:54:46 2009 +0000 @@ -2,7 +2,7 @@ % '[beamer]' for a digital projector % '[trans]' for an overhead projector % '[handout]' for 4-up printed notes -\documentclass[beamer]{beamer} +\documentclass[beamer, compress]{beamer} % change talk_preamble if you want to modify the slide theme, colours, and settings for trans and handout modes. \newcommand{\pathtotrunk}{../../} @@ -19,11 +19,6 @@ \frame{\titlepage} -\begin{frame} - \frametitle{Outline} - \tableofcontents -\end{frame} - \beamertemplatetransparentcovered \mode{\setbeamercolor{block title}{bg=green!40!black}} @@ -36,13 +31,14 @@ \section{Overview} -\AtBeginSection[] -{ \begin{frame} - \frametitle{Outline} - \tableofcontents[currentsection] - \end{frame} -} + \frametitle{Blob homology} + \begin{quote} + ... homotopical topology and TQFT have grown so close that I have started thinking that they are turning into the language of new foundations. + \end{quote} + \flushright{--- \href{http://www.ams.org/notices/200910/rtx091001268p.pdf}{Yuri Manin, September 2008}} + \tableofcontents +\end{frame} \begin{frame}{What is \emph{blob homology}?} \begin{block}{} @@ -83,9 +79,12 @@ I'm not going to go into details; I'll draw $2$-dimensional pictures, and rely on your intuition for pivotal $2$-categories. \end{block} \begin{block}{} +Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'. +\end{block} + +\begin{block}{} \begin{itemize} \item -Kevin's talk (part $\mathbb{II}$) will explain the notions of `topological $n$-categories' and `$A_\infty$ $n$-categories'.\item Defining $n$-categories: a choice of `shape' for morphisms. \item We allow all shapes! A vector space for every ball. @@ -95,9 +94,9 @@ \end{block} \end{frame} -\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} +\newcommand{\roundframe}[1]{\begin{tikzpicture}[baseline=-2pt]\node[rectangle,inner sep=1pt,rounded corners,fill=white] {#1};\end{tikzpicture}} - +\section{Definition} \begin{frame}{Fields and pasting diagrams} \begin{block}{Pasting diagrams} Fix an $n$-category with strong duality $\cC$. A \emph{field} on $\cM$ is a pasting diagram drawn on $\cM$, with cells labelled by morphisms from $\cC$. @@ -106,7 +105,7 @@ $$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$ \end{example} \begin{block}{} -Given a field on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. +Given a pasting diagram on a ball, we can evaluate it to a morphism. We call the kernel the \emph{null fields}. \vspace{-3mm} $$\text{ev}\Bigg(\roundframe{\mathfig{0.12}{definition/evaluation1}} - \frac{1}{d}\roundframe{\mathfig{0.12}{definition/evaluation2}}\Bigg) = 0$$ \end{block} @@ -119,12 +118,12 @@ \begin{block}{} \center -$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary fields on $\cM$. +$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$. \end{block} \begin{block}{} \vspace{-1mm} -$$\bc_1(\cM; \cC) = \setc{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ +$$\bc_1(\cM; \cC) = \setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$ \end{block} \vspace{-3.5mm} $$\mathfig{.5}{definition/single-blob}$$ @@ -145,7 +144,7 @@ \begin{block}{} \vspace{-5mm} \begin{align*} -\bc_2^{\text{disjoint}} & = \roundframe{\mathfig{0.5}{definition/disjoint-blobs}} & u_i \in \ker{\text{ev}_{B_i}} +\bc_2^{\text{disjoint}} & = \setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0} \end{align*} \vspace{-4mm} $$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$ @@ -153,13 +152,55 @@ \begin{block}{} \vspace{-5mm} \begin{align*} -\bc_2^{\text{nested}} & = \roundframe{\mathfig{0.5}{definition/nested-blobs}} & u \in \ker{\text{ev}_{B_1}} +\bc_2^{\text{nested}} & = \setcl{\roundframe{\mathfig{0.5}{definition/nested-blobs}}}{\text{ev}_{B_1}(u)=0} \end{align*} \vspace{-4mm} $$d_2 : (B_1, B_2, u, r', r) \mapsto (B_2, u \circ r', r) - (B_1, u, r \circ r')$$ \end{block} \end{frame} +\begin{frame}{Definition, general case} +\begin{block}{} +$$\bc_k = \set{\mathfig{0.4}{tempkw/blobkdiagram}}$$ +$k$ blobs, properly nested or disjoint, with ``innermost'' blobs labelled by pasting diagrams that evaluate to zero. +\end{block} +\begin{block}{} +\vspace{-2mm} +$$d_k : \bc_k \to \bc_{k-1} = {\textstyle \sum_i} (-1)^i (\text{erase blob $i$})$$ +\end{block} +\end{frame} + +\section{Properties} +\begin{frame}{An action of $\CH{\cM}$} +\begin{thm} +There's a chain map +$$\CH{\cM} \tensor \bc_*(\cM) \to \bc_*(\cM).$$ +which is associative up to homotopy, and compatible with gluing. +\end{thm} +\begin{block}{} +Taking $H_0$, this is the mapping class group acting on a TQFT skein module. +\end{block} +\end{frame} + +\begin{frame}{Gluing} +\begin{block}{$\bc_*(Y \times [0,1])$ is naturally an $A_\infty$ category} +\begin{itemize} +\item[$m_2$:] gluing $[0,1] \simeq [0,1] \cup [0,1]$ +\item[$m_k$:] reparametrising $[0,1]$ +\end{itemize} +\end{block} +\begin{block}{} +If $Y \subset \bdy X$ then $\bc_*(X)$ is an $A_\infty$ module over $\bc_*(Y)$. +\end{block} +\begin{thm}[Gluing formula] +When $Y \sqcup Y^{\text{op}} \subset \bdy X$, +\vspace{-5mm} +\[ + \bc_*(X \bigcup_Y \selfarrow) \iso \bc_*(X) \bigotimes_{\bc_*(Y)}^{A_\infty} \selfarrow. +\] +\end{thm} +In principle, we can compute blob homology from a handle decomposition, by iterated Hochschild homology. +\end{frame} \end{document} % ----------------------------------------------------------------