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\begin{document}
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\parbox[b]{3.5in}{
  {\LARGE\bf Dror Bar-Natan: Confessions}
  }
\hfill\parbox[b]{4.5in}{\small
  \null\hfill\sheeturl
  \newline\null\hfill initiated 9/9/13; modified \today, \ampmtime
}

\rule{\textwidth}{1pt}

\conf{I don't understand the Koszul condition.\fnoc}
\conf{I don't understand 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{I don't understand homotopy theory, loop spaces, spectra, etc.\fnoc}
\conf{I don't understand minimal models.\fnoc}

\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}

\vspace{-3mm}
\begin{multicols}{2}
--- 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 $k=1.38\times10^{-23}$ joule/degree (joule=newton metre).

--- 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.
\end{multicols}
\vskip -3mm

\conf{I don't understand supersymmetry.\fnoc}

\conf{I don't understand renormalization theory.\fnoc}

\vskip 1mm
--- 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 understand love and sex.\fnoc}

\conf{I don't understand the $h$-cobordism theorem.\fnoc}

\vspace{-3mm}\begin{multicols}{2}

--- 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.

\end{multicols}\vspace{-3mm}

\conf{I don't understand the basics of three-dimensional topology: the loop and sphere theorems, JSJ decompositions, etc.\fnoc} Progress: \href{../../2013-09/CheatSheet3DTopology.pdf}{Pensieve: 2013-09: CheatSheet3DTopology.pdf}.

\conf{I don't understand the Batalin-Vilkovisky formalism.\fnoc}

\vspace{-3mm}
\begin{multicols}{2}

--- Mnev's example. ``Space of fields'' $M=R^3_{txy}\times S^1_z$;
``classical action'' $S_{cl}:=\frac12t^2$;
``Gauge symmetry'' $E:=\operatorname{span}\left(\partial_y, \partial_x+ty\partial_z\right)$, 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
\[ \int_L e^{-S}
  = \int dtdzdc_1dc_2dx^\dagger dy^\dagger e^{-S_{cl}} c_1c_2x^\dagger y^\dagger
  = \sqrt{2\pi}T.
\]

--- Losev: For $\omega\in\Omega^{n-1}(M^n)$, $f\colon M\to\bbR$,
\[ \int_{[f=0]}\omega=\int_{TM\oplus\bbR^{1|1}_{l|\lambda}}\omega e^{-d(f\lambda)}. \]

--- Further references: old paper by Schwarz, \arXiv{0812.0464} by Albert, Bleile, Fr\"ohlich; notes by Kazhdan.

\end{multicols}
\vskip -3mm

\conf{I forgot too much of what I used to know about Lie theory.\fnoc}
\conf{I don't understand Witten's exact solution of Chern-Simons theory (what he understood in 1988).\fnoc}
\conf{I don't feel comfortable with quantum groups.\fnoc}
\conf{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.\fnoc}

\conf{I don't understand Heegaard-Floer homology.\fnoc}
\vskip 1mm
--- Maybe Juh\'asz' \arXiv{1310.3418}?

\conf{If it has the word K\"ahler in it, I shy away.\fnoc}

\conf{I don't understand group cohomology.\fnoc}

\vskip 1mm
--- From \href{../../2013-02/index.html}{Pensieve: 2013-02}: $G$ group; $M$ a $G$-module; $C^n(G,M):=\{\varphi\colon G^n\to M\}$;\hfill``derived from $M\to M^G$''
\[ (d\varphi)(g_1,\dots,g_{n+1}) :=
  g_1\varphi(g_2,\dots,g_{n+1})
  + \sum_{i=1}^n (-)^i\varphi(\dots,g_ig_{i+1},\dots)
  + (-)^{n+1}\varphi(g_1,\dots,g_n).
\]
\[
  (\varphi\cup\psi)(g_1,\ldots,g_{n+m}) :=
  \hspace{-15mm}\sum_{\text{$\sigma$ monotone on $1..n$ \& on $(n+1)..(n+m)$}}\hspace{-15mm}
    (-)^\sigma
    \varphi(g_{\sigma 1},\dots,g_{\sigma n})
    \psi(g_{\sigma(n+1)},\dots,g_{\sigma(n+m)})
\]

For $M=\bbK$:
$\bullet$ $H^*=H^*(K(G,1))$.
$\bullet$ $H^1=\Hom(G,\bbK)$.
$\bullet$ $H^2$ counts central extensions by $\bbK$.

\conf{I don't understand projective and injective resolutions, Ext and Tor, the universal coefficients theorem, etc.\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}
\conf{I have no clue what are ``motives''.\fnoc}
\conf{I don't understand Tannakian reconstruction principles, and I wish I did.\fnoc}

\conf{I don't understand Pfaffians (though of all my troubles, this is perhaps the least).\fnoc}

--- See Wikipedia, Parameswaran, Ledermann.

\conf{I don't fully understand Habiro's 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}

\rule{\textwidth}{1pt}
\textbf{Mathematical Monoblog Experiment.} (Initiated October 7, 2013)
\vspace{-3mm}
\begin{multicols}{2}

{\bf\tiny (131104)} The infinitesimal braid relations on a surface: (pensieve/2013-04)
\[
  [v_i,w_j]=\langle v, w\rangle t_{ij}
  \quad [v_i, t_{jk}]=0
  \quad [x_i,y_i]=-\sum_{j\neq i}t_{ij}
\]
\vspace{-5mm}

{\bf\tiny (131103)} The Yoshikawa moves:
\[ \includegraphics[width=3.8in]{YoshikawaMoves.png} \]
\vspace{-5mm}

{\bf\tiny (131027)} Cattaneo: ``BV is the `right' de-Rham differential on super-manifolds.''

{\bf\tiny (131027)} The Hilbert basis theorem: An ideal in the ring of multivariable polynomials over a Noetherian ring is finitely generated.

{\em Pf.} Enough, $R$ Noetherian $\Rightarrow$ any $I\subset R[x]$ is finitely generated. Let $p_n\in I\remove\langle p_1,\ldots,p_{n-1}\rangle$ be of minimal degree. As $R$ is Noetherian, for large $N$ the leading coefficient of $p_N$ is a combination of previous leading coefficients, so it can be killed off contradicting the minimality of $P_N$. \qed

Can be made constructive using Gr\"obner bases.

{\bf\tiny (131023)} Markl: ``like a bottle under a waterfall''.

{\bf\tiny (131020)} Artin-Wedderburn: A semi-simple ring is uniquely (up to a permutation) isomorphic to a product of finitely many finite matrix rings over division rings.

{\bf\tiny (131017)} If $\lambda_{\{ij\}}=0$,
$(\lambda_{ij}dx^i\wedge dx^j)^{n/2} = \sqrt{\det(\lambda_{ij})}\bigwedge_idx^i$.

{\bf\tiny (131014)} Problem with the projectivization paradigm: No room for negative degrees and for degree-decreasing ops.

{\bf\tiny (131009)} Let $\Gamma_{1,2,3}$ be thickened surfaces. Is there an expansion for the composition map
\[ \calK(\Gamma_1\hookrightarrow\Gamma_2)\times\calK(\Gamma_2\hookrightarrow\Gamma_3)
  \overset{\sslash}{\rightarrow} \calK(\Gamma_1\hookrightarrow\Gamma_3)?
\]

{\bf\tiny (131007)} I don't understand Severa's
\[ \xymatrix{
  (A\otimes A)\otimes A \ar[rr]^\Phi \ar[d]^{m\otimes 1}
  & & A\otimes(A\otimes A) \ar[d]_{1\otimes m} \\
  A\otimes A \ar[r]^m
  & A
  & A\otimes A \ar[l]_m
} \]

\end{multicols}

\rule{\textwidth}{1pt}
\vspace{-3mm}
\begin{multicols}{2}

\textbf{Archived Items.}

{\bf\tiny (131026)} Time to make an ``agenda browser''.

\end{multicols}

\end{document}

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