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+% use options
+% '[beamer]' for a digital projector
+% '[trans]' for an overhead projector
+% '[handout]' for 4-up printed notes
+\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}{../../}
+\input{\pathtotrunk talks/talk_preamble.tex}
+
+%\setbeameroption{previous slide on second screen=right}
+
+\author[Scott Morrison]{Scott Morrison \\ \texttt{http://tqft.net/} \\ joint work with Kevin Walker}
+\institute{UC Berkeley / Miller Institute for Basic Research}
+\title{Blob homology, part $\mathbb{I}$}
+\date{Homotopy Theory and Higher Algebraic Structures, UC Riverside, November 10 2009 \\ \begin{description}\item[slides, part $\mathbb{I}$:]\url{http://tqft.net/UCR-blobs1} \item[slides, part $\mathbb{II}$:]\url{http://tqft.net/UCR-blobs2} \item[draft:]\url{http://tqft.net/blobs}\end{description}}
+
+\begin{document}
+
+\frame{\titlepage}
+
+\beamertemplatetransparentcovered
+
+\mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}}
+
+\beamersetuncovermixins
+{\opaqueness<1->{60}}
+{}
+
+
+
+\section{Overview}
+
+ \begin{frame}<beamer>
+ \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}{}
+The blob complex takes an $n$-manifold $\cM$ and an `$n$-category with strong duality' $\cC$ and produces a chain complex, $\bc_*(\cM; \cC)$.
+\end{block}
+\tikzstyle{description}=[gray, font=\tiny, text centered, text width=2cm]
+\begin{tikzpicture}[]
+\setbeamercovered{%
+ transparent=5,
+% still covered={\opaqueness<1>{15}\opaqueness<2>{10}\opaqueness<3>{5}\opaqueness<4->{2}},
+ again covered={\opaqueness<1->{50}}
+}
+
+\node[red] (blobs) at (0,0) {$H(\bc_*(\cM; \cC))$};
+\uncover<1>{
+\node[blue] (skein) at (4,0) {$\cA(\cM; \cC)$};
+\node[below=5pt, description] (skein-label) at (skein) {(the usual TQFT Hilbert space)};
+\path[->](blobs) edge node[above] {$*= 0$} (skein);
+}
+
+\uncover<2>{
+ \node[blue] (hoch) at (0,3) {$HH_*(\cC)$};
+ \node[right=20pt, description] (hoch-label) at (hoch) {(the Hochschild homology)};
+ \path[->](blobs) edge node[right] {$\cM = S^1$} (hoch);
+}
+
+\uncover<3>{
+ \node[blue] (comm) at (-2.4, -1.8) {$H_*(\Delta^\infty(\cM), k)$};
+ \node[description, below=5pt] (comm-label) at (comm) {(singular homology of the infinite configuration space)};
+ \path[->](blobs) edge node[right=5pt] {$\cC = k[t]$} (comm);
+}
+
+\end{tikzpicture}
+\end{frame}
+
+\begin{frame}{$n$-categories}
+\begin{block}{Defining $n$-categories is fraught with difficulties}
+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
+Defining $n$-categories: a choice of `shape' for morphisms.
+\item
+We allow all shapes! A vector space for every ball.
+\item
+`Strong duality' is integral in our definition.
+\end{itemize}
+\end{block}
+\end{frame}
+
+\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 $\cC$.
+\end{block}
+\begin{example}[$\cC = \text{TL}_d$ the Temperley-Lieb category]
+$$\roundframe{\mathfig{0.35}{definition/example-pasting-diagram}} \in \cF^{\text{TL}_d}\left(T^2\right)$$
+\end{example}
+\begin{block}{}
+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}
+\end{frame}
+
+\begin{frame}{Background: TQFT invariants}
+\begin{defn}
+A decapitated $n+1$-dimensional TQFT associates a vector space $\cA(\cM)$ to each $n$-manifold $\cM$.
+\end{defn}
+(`decapitated': no numerical invariants of $n+1$-manifolds.)
+
+\begin{block}{}
+If the manifold has boundary, we get a category. Objects are boundary data, $\Hom{\cA(\cM)}{x}{y} = \cA(\cM; x,y)$.
+\end{block}
+
+\begin{block}{}
+We want to extend `all the way down'. The $k$-category associated to the $n-k$-manifold $\cY$ is $\cA(\cY \times B^k)$.
+\end{block}
+
+\begin{defn}
+Given an $n$-category $\cC$, the associated TQFT is
+\vspace{-3mm}
+$$\cA(\cM) = \cF(M) / \ker{ev},$$
+
+\vspace{-3mm}
+fields modulo fields which evaluate to zero inside some ball.
+\end{defn}
+\end{frame}
+
+\begin{frame}{\emph{Definition} of the blob complex, $k=0,1$}
+\begin{block}{Motivation}
+A \emph{local} construction, such that when $\cM$ is a ball, $\bc_*(\cM; \cC)$ is a resolution of $\cA(\cM,; \cC)$.
+\end{block}
+
+\begin{block}{}
+\center
+$\bc_0(\cM; \cC) = \cF(\cM)$, arbitrary pasting diagrams on $\cM$.
+\end{block}
+
+\begin{block}{}
+\vspace{-1mm}
+$$\bc_1(\cM; \cC) = \Complex\setcr{(B, u, r)}{\begin{array}{c}\text{$B$ an embedded ball}\\\text{$u \in \cF(B)$ in the kernel}\\ r \in \cF(\cM \setminus B)\end{array}}.$$
+\end{block}
+\vspace{-3.5mm}
+$$\mathfig{.5}{definition/single-blob}$$
+\vspace{-3mm}
+\begin{block}{}
+\vspace{-6mm}
+\begin{align*}
+d_1 : (B, u, r) & \mapsto u \circ r & \bc_0 / \im(d_1) \iso A(\cM; \cC)
+\end{align*}
+\end{block}
+\end{frame}
+
+\begin{frame}{Definition, $k=2$}
+\begin{block}{}
+\vspace{-1mm}
+$$\bc_2 = \bc_2^{\text{disjoint}} \oplus \bc_2^{\text{nested}}$$
+\end{block}
+\begin{block}{}
+\vspace{-5mm}
+\begin{align*}
+\bc_2^{\text{disjoint}} & = \Complex\setcl{\roundframe{\mathfig{0.5}{definition/disjoint-blobs}}}{\text{ev}_{B_i}(u_i) = 0}
+\end{align*}
+\vspace{-4mm}
+$$d_2 : (B_1, B_2, u_1, u_2, r) \mapsto (B_2, u_2, r \circ u_1) - (B_1, u_1, r \circ u_2)$$
+\end{block}
+\begin{block}{}
+\vspace{-5mm}
+\begin{align*}
+\bc_2^{\text{nested}} & = \Complex\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}{}
+$$\bc_k = \Complex\set{\roundframe{\mathfig{0.7}{definition/k-blobs}}}$$
+$k$ 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}{Hochschild homology}
+\begin{block}{TQFT on $S^1$ is `coinvariants'}
+\vspace{-3mm}
+$$\cA(S^1, A) = \Complex\set{\roundframe{\mathfig{0.1}{hochschild/m-a-b}}}\scalebox{2}{$/$}\set{\roundframe{\mathfig{0.065}{hochschild/ma}} - \roundframe{\mathfig{0.12}{hochschild/m-a}}} = A/(ab-ba)$$
+\end{block}
+\begin{block}{}
+The Hochschild complex is `coinvariants of the bar resolution'
+\vspace{-2mm}
+$$ \cdots \to A \tensor A \tensor A \to A \tensor A \xrightarrow{m \tensor a \mapsto ma-am} A$$
+\end{block}
+\begin{thm}[$ \HC_*(A) \iso \bc_*(S^1; A)$]
+$$m \tensor a \mapsto
+\roundframe{\mathfig{0.35}{hochschild/1-chains}}
+$$
+\vspace{-5mm}
+\begin{align*}
+u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 &= \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2}
+\end{align*}
+\end{thm}
+\end{frame}
+
+\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}{Higher Deligne conjecture}
+\begin{block}{Deligne conjecture}
+Chains on the little discs operad acts on Hochschild cohomology.
+\end{block}
+
+\begin{block}{}
+Call $\Hom{A_\infty}{\bc_*(\cM)}{\bc_*(\cM)}$ `blob cochains on $\cM$'.
+\end{block}
+
+\begin{block}{Theorem* (Higher Deligne conjecture)}
+\scalebox{0.96}{Chains on the $n$-dimensional fat graph operad acts on blob cochains.}
+\vspace{-3mm}
+$$\mathfig{.85}{deligne/manifolds}$$
+\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}
+% ----------------------------------------------------------------
+