\documentclass[11pt,notitlepage]{article} \usepackage{amsmath, graphicx, amssymb, stmaryrd, datetime, amscd, multicol, txfonts, ../picins} \usepackage{jumplines, xcolor} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks, linkcolor={blue!50!black}, citecolor={blue!50!black}, urlcolor=blue } \usepackage[all]{xy} \usepackage{soul} % strikeout text. \usepackage[margin=0.25in,foot=0.2in]{geometry} \def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/Projects/Agenda/}}} \def\conf#1{{ \vskip 0pt % maximal at 12pt \par$\bullet$ \bf #1 % maximal with \Large\bf }} \def\fnoc{} \def\rconf#1{{ \vskip 2pt % maximal at 12pt \par$\bullet$ \bf #1 % maximal with \Large\bf }} \def\fnocr{} %\paperwidth 8in %\paperheight 10.5in %\textwidth 8in %\textheight 10.5in %\oddsidemargin -0.75in %\evensidemargin \oddsidemargin %\topmargin -0.75in %\headheight 0in %\headsep 0in %\footskip 0in \parindent 0in \setlength{\topsep}{0pt} \pagestyle{empty} \dmyydate \newcounter{linecounter} \newcommand{\cheatline}{\vskip 1mm\refstepcounter{linecounter}\thelinecounter. } % Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top: \def\imagetop#1{\vtop{\null\hbox{#1}}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\ad{\operatorname{ad}} \def\Ad{\operatorname{Ad}} \def\arXiv#1{{\href{http://arxiv.org/abs/#1}{arXiv:\linebreak[0]#1}}} \def\attr{\operatorname{\mathfrak{tr}}} \def\End{\operatorname{End}} \def\Hom{\operatorname{Hom}} \def\Mat{\operatorname{Mat}} \def\qed{{\hfill\text{$\Box$}}} \def\remove{\!\setminus\!} \def\FA{{F\!A}} \def\FG{{F\!G}} \def\sder{\operatorname{\mathfrak{sder}}} \def\tder{\operatorname{\mathfrak{tder}}} \def\bbK{{\mathbb K}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calC{{\mathcal C}} \def\calD{{\mathcal D}} \def\calG{{\mathcal G}} \def\calK{{\mathcal K}} \def\calO{{\mathcal O}} \def\calW{{\mathcal W}} \def\frakg{{\mathfrak g}} \def\RP{\bbR{\mathrm P}} \def\red{\color{red}} \begin{document} %\setlength{\jot}{0ex} \setlength{\abovedisplayskip}{0.5ex} \setlength{\belowdisplayskip}{0.5ex} \setlength{\abovedisplayshortskip}{0ex} \setlength{\belowdisplayshortskip}{0ex} \parbox[b]{3.5in}{ {\LARGE\bf Dror Bar-Natan: Confessions} } \hfill\parbox[b]{2.5in}{\tiny \null\hfill\sheeturl \newline\null\hfill initiated 9/9/13; modified \today, \ampmtime } \vskip -3mm \rule{\linewidth}{1pt} \vspace{-8mm} \begin{multicols}{2} \conf{I've never understood ``resolution of singularities''.\fnoc} \conf{I don't understand the Koszul condition.\fnoc} \conf{I don't yet appreciate infinity-algebras.\fnoc} \conf{I don't really understand Poisson structures: Why do they automatically arise from action principles? Why do they necessarily emerge in computing path integrals? Why should I care about their deformation quantizations?\fnoc} \conf{I don't understand Tamarkin's work on formality.\fnoc} \conf{Spectral sequences never became me.\fnoc} \conf{I don't understand homotopy theory, loop spaces, spectra, etc.\fnoc} \conf{I don't understand minimal models.\fnoc} Books on rational homotopy theory: F\'elix-Helperin-Thomas, Griffiths-Morgan. \conf{I don't understand thermal physics - energy, entropy, enthalpy, and all that. Such basic things these are that it is really embarrassing that I don't understand the constraints my air-conditioner is bound by.\fnoc} --- From Feynman's {\em Lectures on Physics}: $\bullet$~``equal volumes of gases, at the same pressure and temperature, contain the same number of molecules''; $N_0=6.022\times10^{23}$ as in (1 mole)$=$12g of $^{12}C$. $\bullet$~$P=F/A$. $\bullet$~$dW=-PdV$. $\bullet$~$PV=\frac23N\langle\frac12mv^2\rangle=\frac23U\ (\ldots=NkT)$. $\bullet$~With $\gamma-1=\frac23$, $PV^\gamma=C$. $\bullet$~In gas mixtures, $\frac12m_1v_1^2=\frac12m_2v_2^2$ (messy!). $\bullet$~$\frac12mv^2=:\frac32kT$, with $k=1.38\times10^{-23}$ $J$/degree ($J$ = joule = newton metre = watt second). --- From Bamberg-Sternberg: $\bullet$ First law of thermodynamics: $\alpha+\omega=dU$, with $\alpha$: heat 1-form, $\omega$: work 1-form, $U$: internal energy. $\bullet$ Second law of thermodynamics: $\alpha=TdS$, with $T$: temperature, $S$: entropy. --- From Schroeder: $\bullet$ 1cal $=$ $10^{-3}$ food calorie $\coloneqq$ $4.186J$ $\sim$ heat to raise $1g$ of water by $1^\circ C$. --- See also Lieb-Yngvason. \conf{I don't understand supersymmetry.\fnoc} \conf{I don't understand renormalization theory.\fnoc} Minor point: it would be great if I could present the renormalization of associators/vertices as a special case. \conf{I don't understand the Mostow rigidity theorem.\fnoc} \conf{I'm not as comfortable with special relativity as I want to be.\fnoc} \conf{I don't really understand general relativity.\fnoc} \conf{I don't know how to put figures in \LaTeX\ efficiently.\fnoc} \conf{I don't fully understand the $h$-cobordism theorem.\fnoc} Perhaps follow Milnor's lecture notes? {\bf Def.} An $h$-cobordism is a cobordism in which the boundary inclusions are deformation retracts. {\bf Thm.} In {\em Diff}, {\em PL}, or {\em Top}, a simply-connected $h$-cobordism between simply-connected $(n\geq 5)$-manifolds is trivial. \conf{I haven't internalized the distinction between continuous, smooth, and triangulated.\fnoc} \conf{I don't really understand Faddeev-Popov and/or BRST.\fnoc} \conf{I don't understand the Batalin-Vilkovisky formalism.\fnoc} --- Mnev's example. ``Space of fields'' $M=R^3_{txy}\times S^1_z$; ``classical action'' $S_{\!cl}\coloneqq\frac12t^2$; ``Gauge symmetry'' $E\coloneqq\operatorname{span}\left(\partial_y, \partial_x+ty\partial_z\right)$, integrable on $EL=[t=0]$ surface but not on $M$, $S_{\!cl}$ is invariant. $M/E$ is not $T_2$ and $\int_{M/E}e^{-S}$ makes no sense. BV space of fields $F=T^\ast[-1](\bbR^2[1]\times M)$ with coords $c_{1,2}$ (ghost number $1$), $t,x,y,z$ (g.n.\ $0$), $t^\dagger,x^\dagger,y^\dagger,z^\dagger$ (g.n.\ $-1$) and $c_{1,2}^\dagger$ (g.n.\ $-2$). The BV action is $S=\frac12t^2+c_1y^\dagger+c_2(x^\dagger+tyz^\dagger)+c_1c_2t^\dagger z^\dagger$; satisfies QME \& consistent with $S_{\!cl}$ and $E$. Gauge fixing Lagrangian $L=[x=y=t^\dagger=z^\dagger=c_{1,2}^\dagger=0]\subset F$ gives \hfill$\displaystyle \int_L e^{-S} = \int dtdzdc_1dc_2dx^\dagger dy^\dagger e^{-S_{cl}} c_1c_2x^\dagger y^\dagger = \sqrt{2\pi}T. $\hfill \vskip -2mm \parpic[r]{$\displaystyle \int_{[f=0]}\omega=\int_{TM\oplus\bbR^{1|1}_{l|\lambda}}\omega e^{-d(f\lambda)}.$} \picskip{2} --- Losev: For $\omega\in\Omega^{n-1}(M^n)$, $f\colon M\to\bbR$, --- Further: old paper by Schwarz; \arXiv{0812.0464} by Albert, Bleile, Fr\"ohlich; notes by Kazhdan; thesis by Gwilliam; notes by Ens. \conf{I still don't understand the BF TQFT.\fnoc} From Cattaneo-Rossi's \arXiv{math-ph/0210037} {\em Wilson Surfaces}: $A\in\Omega^1(\bbR^4,\frakg)$ a connection, $B\in\Omega^2(\bbR^4,\frakg^*)$, \hfill$\displaystyle S(A,B)\coloneqq\int_{\bbR^4}\langle B,F_A\rangle.$\hfill\null $\calG\coloneqq\exp\Omega^0(\bbR^4,\frakg)$ is (u-)gauge transformations, $(g,\sigma)\in\tilde{\calG}\coloneqq\calG\ltimes\Omega^1(\bbR^4,\frakg^*)$ acts by \[ A\mapsto A^g \qquad B\mapsto B^{(g,\sigma)}\coloneqq\Ad^*_{g^{-1}}B+d_{A^g}\sigma. \] With $f\colon\bbR^2\to\bbR^4$, $\xi\in\Omega^0(\bbR^2,\frakg)$, $\beta\in\Omega^1(\bbR^2,\frakg^*)$, set \[ \calO(A,B,f)\coloneqq\int\calD\xi\calD\beta\exp\left( \frac{i}{\hbar}\int_{\bbR^2}\left\langle\xi,d_{f^*A}\beta+f^*B\right\rangle \right). \] \conf{I forgot too much of what I used to know about Lie theory.\fnoc} From Humphreys: Weyl's formula: For $\lambda\in\Lambda^+$, \[ ch_\lambda*\sum_{\sigma\in\calW}(-)^\sigma\epsilon_{\sigma\delta} = \sum_{\sigma\in\calW}(-)^\sigma\epsilon_{\sigma(\lambda+\delta)}. \] \conf{I know nothing about $\theta$ functions.\fnoc} \conf{I don't understand Witten's exact solution of Chern-Simons theory (what he understood in 1988).\fnoc} \conf{I'm \href{http://drorbn.net/AcademicPensieve/2017-09/one/A_Conspiracy_Theory.pdf}{uncomfortable} with quantum groups.\fnoc} Is there a diagrammatic perspective? On a philosophical level, quantum groups as they appear in topology are ``constructions'' or ``images''. I wish I understood them as associated with ``kernels''. Rotational virtual tangles explain quantum groups as associated with a kernel of an extension, but I don't have an explanation within that context for why clean formulas arise. What is the relationship between quantum groups and expansions? %Failure to understand over a period of 25 years (and counting) slowly turned into fear and mistrust. \conf{I don't understand the first thing about Heegaard-Floer homology.\fnoc} Maybe Juh\'asz' \arXiv{1310.3418}, Manolescu's \arXiv{1401.7107}, or Lipshitz' \arXiv{1411.4540}. \conf{If it has the word K\"ahler in it, I shy away.\fnoc} \conf{I don't understand projective and injective resolutions, Ext and Tor, the universal coefficients theorem, etc.\fnoc} \conf{I am yet to internalize ``sheafs''.\fnoc} \conf{I've never figured ``derived''.\fnoc} Perhaps Yekutieli's \arXiv{1501.06731}? \conf{I've never figured ``perverse''.\fnoc} \conf{I don't understand the K\"unneth and Eilenberg-Zilber theorems.\fnoc} \conf{I don't understand the relationship between $gr$ and $H$, as it appears, for example, in braid theory.\fnoc} --- Perhaps Berglund's {\em Koszul Spaces}? \conf{I have no clue what are ``motives''.\fnoc} \conf{I don't understand Tannakian reconstruction principles, and I wish I did.\fnoc} --- Given an algebra $A$ let $\calD\coloneqq A-\text{Mod}$ (projective (?) left $A$-modules), let $\calC\coloneqq{\text{Vect}}$ and $G\colon\calD\to\calC$ be the forGetful functor. Then $A\simeq\End(G)$ by \[ a\in A\mapsto(\text{the action of $a$ on any $X\in\calD$}), \] \[ \{a_X\colon G(X)\to G(X)\}_{X\in\calD} \mapsto a_A(1)\in A. \] --- Given a monoidal $\calD$ and an exact $G\colon\calD\to\calC\eqqcolon\text{Vect}$ with a natural isomorphism $\alpha_{X,Y}\colon G(X)G(Y)\to G(XY)$, there is a Hopf algebra structure on $H\coloneqq\End(G)$: product is composition, coproduct $\Delta\colon H\to H^2 = \End(G^2\colon\calD\times\calD\to\calC)$ by \[ (h_X)_{X\in\calD}\mapsto\left( (X,Y)\mapsto \alpha_{X,Y}\act h_{XY}\act \alpha_{X,Y}^{-1} \in \End(G(X)G(Y)) \right). \] \conf{I don't understand Pfaffians (though of all my troubles, this is perhaps the least).\fnoc} --- See Wikipedia, Parameswaran, Ledermann. Concisely, if $\lambda_{\{ij\}}=0$, then \[ (\lambda_{ij}dx^i\wedge dx^j)^{n/2} = \sqrt{\det(\lambda_{ij})}\bigwedge_idx^i\] (common in symplectic geometry), so $\sqrt{\det(\lambda_{ij})}$ is a polynomial in the $\lambda_{ij}$'s. Itai/Yael: with $\omega=\lambda_{ij}dx^i\wedge dx^j$, need \[ \det(\omega(u_i,v_j)) = \omega^{n/2}(u_1,\ldots,u_n)\omega^{n/2}(v_1,\ldots,v_n). \] Easy from multi-linearity and anti-symmetry if $(u_i)$ and $(v_j)$ are in a symplectic basis for $\omega$. \conf{I don't understand the Goussarov-Polyak-Viro theorem.\fnoc} \conf{I don't understand knot signatures (and signatures in general).\fnoc} \conf{I don't fully understand the Goussarov-Habiro theory of claspers.\fnoc} \conf{I don't understand Gr\"obner bases.\fnoc} \conf{I still don't know a proof of the Milnor-Moore theorem.\fnoc} --- Maybe ``Spencer Bloch's course on Hopf Algebras'' \st{or Kreimer's thesis}. Maybe search inside? \conf{I still don't understand Vogel's construction.\fnoc} \conf{I'm missing the key to equivariant cohomology, $EG$, $BG$, and all that.\fnoc} --- I need a framework for $X_G\coloneqq (X\times EG)/G$. \conf{I don't understand fusion categories and subfactors.\fnoc} --- Morrison's \href{http://drorbn.net/dbnvp/Morrison-140220}{drorbn.net/dbnvp/Morrison-140220}? \end{multicols} \vskip -6mm \rule{\textwidth}{0.4pt} \vskip -1mm \conf{I don't understand group cohomology.\fnoc} --- \href{../../2013-02/index.html}{Pensieve: 2013-02}: $G$ group; $M$ a $G$-module; $C^n(G,M)\coloneqq\{\varphi\colon G^n\to M\}$;\hfill``derived from $M\to M^G$'' \[ (d\varphi)(g_1,\dots,g_{n+1}) \coloneqq g_1\varphi(g_2,\dots,g_{n+1}) + \sum_{i=1}^n (-)^i\varphi(\dots,g_ig_{i+1},\dots) + (-)^{n+1}\varphi(g_1,\dots,g_n). \] \[ (\varphi\cup\psi)(g_1,\ldots,g_{n+m}) \coloneqq \hspace{-15mm}\sum_{\text{$\sigma$ monotone on $1..n$ \& on $(n+1)..(n+m)$}}\hspace{-15mm} (-)^\sigma \varphi(g_{\sigma 1},\dots,g_{\sigma n}) \psi(g_{\sigma(n+1)},\dots,g_{\sigma(n+m)}) \] At $M=\bbK$: $\bullet$ $H^*=H^*(K(G,1))$. $\bullet$ $H^1=\Hom(G,\bbK)$. $\bullet$ $H^2\leftrightarrow$ central extensions by $\bbK$. \hfill $H^3(G,\bbK^\times)\leftrightarrow$ categorifications of $\bbZ G$. \rule{\textwidth}{0.4pt} \vskip -1mm \conf{I don't understand the basics of three-dimensional topology: the loop and sphere theorems, JSJ decompositions, etc.\fnoc} \hfill{\tiny Continuing \href{../../2013-11/CheatSheet3DTopology.pdf}{2013-11}: CheatSheet3DTopology.pdf} \vskip -3mm \rule{\textwidth}{0.2pt}\vspace{-5mm} \begin{multicols}{2} From Hatcher's notes: {\bf Definition.} $M$ prime: $M=P\#Q\Rightarrow(P=S^3)\vee(Q=S^3)$. M Irreducible: an embedded 2-sphere in $M$ bounds a 3-ball. (Irreducible $\Rightarrow$ Prime). {\bf Theorem} (Alexander, 1920s). $S^3$ is irreducible. {\bf Proof.} Study the change to the ``canonical closure'' of a cropped embedded $S^2$ under the following cases: \[ \includegraphics[width=3in]{figs/Slices.png} \] {\bf Theorem.} Orientable, prime, not irreducible $\Rightarrow\ S^2\times S^1$. Nonorientable? Also $S^2\widetilde{\times}S^1$ (Klein 3D). {\bf Theorem.} Compact connected orientable 3-manifolds have unique decomposition into primes. {\bf Proof.} $\bullet$ Given a system of splitting spheres (sss) and a $\theta$-partition of one member, at least one part will make an sss. $\bullet$ An sss can be simplified relative to a fixed triangulation $\tau$: only disk intersections with simplices; circle and single-edge-arc intersections with faces of $\tau$ can be eliminated. $\bullet$ The size of an sss is bounded by $4|\tau|+\operatorname{rank}H_1(M;\bbZ/2)$ and hence prime-decompositions exist. $\bullet$ Uniqueness. \qed Nonorientable $M$? Same but $M\#(S^2\times S^1)=M\#(S^2\widetilde{\times}S^1)$. {\bf Theorem.} If a covering is irreducible, so is the base. ([Ha] proof is fishy). {\bf Examples.} Lens spaces, surface bundles $F\to M\to S^1$ with $F\neq S^2,\RP^2$. Yet $S^1\times S^2/(x,y)\sim(\bar{x},-y)=\RP^3\#\RP^3$, a prime covers a sum. {\bf Definition.} $S\subset M^3$ a 2-sided surface, $S\neq S^2$, $S\neq D^2$. {\em Compressing disk for $S$} is a disk $D\subset M$ with $D\cap S=\partial D$. If for every compressing $D$ there's a disk $D'\subset S$ with $\partial D'=\partial D$, $S$ is {\em incompressible}. {\bf Claims.} $\bullet$ $\pi_1(S)\hookrightarrow\pi_1(M)$ $\Rightarrow$ $S$ incompressible. $\bullet$ No incompressibles in $\bbR^3$/$S^3$. $\bullet$ In irreducible $M^3$, $T^2$ is 2-sided incompressible iff $T$ bounds a $D^2\times S^1$ or $T$ is contained in a $B^3$. $\bullet$ A $T^2$ in $S^3$ bounds a $D^2\times S^1$ on at least one side. $\bullet$ $S\subset M$ incompressible $\Rightarrow$ ($M$ irreducible iff $M|S$ irreducible). $\bullet$ $S$ a collection of disjoint incompressibles or disks or spheres in $M$, $T\subset M|S$. Then $T$ is incompressible in $M$ iff in $M|S$. \end{multicols} \vskip -6mm \rule{\textwidth}{0.2pt}\vspace{-5mm} \begin{multicols}{2} From Hempel's book: {\bf Dehn's Lemma} (Dehn 1910 (wrong), Papakyriakopoulos 1950s). $M$ a 3-manifold, $f\colon B^2\to M$ s.t.\ for some neighborhood $A$ of $\partial B^2$ in $B^2$ the restriction $F|_A$ is an embedding and $f^{-1}(f(A))=A$. Then $f|_{\partial B^2}$ extends to an embedding $g\colon B^2\to M$. {\bf The Loop Theorem} (Stallings 1960, implies Dehn's lemma). $M$ a 3-manifold, $F$ a connected 2-manifold in $\partial M$, $\ker(\pi_1(F)\to\pi_1(M)\not\subset N\triangleleft\pi_1(F)$. Then there is a proper embedding $g\colon(B^2,\partial B^2)\to(M,F)$ s.t.\ $[g|_{\partial B^2}]\not\in N$. {\bf The Sphere Theorem.} $M$ orientable 3-manifold, $N$ a $\pi_1(M)$-invariant proper subgroup of $\pi_2(M)$. Then there is an embedding $g\colon S^2\to M$ s.t.\ $[g]\not\in N$. \end{multicols} \rule{\textwidth}{0.4pt} \vspace{-5mm} \begin{multicols}{2} {\Large\bf Redeemed Confessions.} \rconf{I don't understand Galois theory, for real. Abstractness is fun, but Galois surely understood everything in very concrete terms. I wish I did too.\fnocr} --- \href{http://youtu.be/RhpVSV6iCko}{youtu.be/RhpVSV6iCko} and then \href{http://drorbn.net/dbnvp/AKT-140314.php}{drorbn.net/dbnvp/AKT-140314.php} and \url{http://www.math.toronto.edu/~drorbn/Talks/CMU-1504/} do the job! \end{multicols} \end{document} \endinput