Sample document

1 Basic typesetting

Of course you can type paragraphs containing some mathematics, such as the following one.

If \(G=\operatorname{GL}_2\) then \({G_{\operatorname{ad}}}=\operatorname{PGL}_2\) and so, as above, \(G_1=\operatorname{SL}_2\). Now \(G_2\) is the set of \((g,h)\in \operatorname{SL}_2\times \operatorname{GL}_2\) with \(g=h\) in \(\operatorname{PGL}_2\), so \(h=\lambda g\) for some unique \(\lambda \in \mathbf{G}_m\) and \(G_2=\operatorname{SL}_2\times \mathbf{G}_m\), with the obvious map to \(\operatorname{GL}_2\) sending \(\mathbf{G}_m\) into the centre (or perhaps its inverse depending on how one is thinking about things, but this doesn’t matter). The subgroup \(\mu _2\) is embedded diagonally of course, because it’s the kernel of the map \(G_2\to G\). Finally we push out via \(\mu _2\to \mathbf{G}_m\) and this gives us \(\operatorname{SL}_2\times \mathbf{G}_m\times \mathbf{G}_m\) modulo the subgroup of order 2 with non-trivial element \((-1,-1,-1)\). But there’s an automorphism of \(\mathbf{G}_m\times \mathbf{G}_m\) sending \((-1,-1)\) to \((-1,1)\) (namely, send \((x,y)\) to \((x,xy)\)) so again \({\tilde{G}}\) is just \(G\times \mathbf{G}_m\).

You can also use displayed formulas such as:

\[ \int _{I \times \Sigma } \Phi ^*\omega = \int _I\left(\int _{\Phi _t(\Sigma )} \iota _X \omega \right)dt. \]

and refer to displayed formulas such as Equation 1 below.

\begin{equation} \label{eq:stokes} \int _M d\omega = \int _{\partial M} \omega \end{equation}

Commutative diagrams using tikz-cd are supported as well.

\begin{tikzcd} [row sep=1cm]
  	{}   & \ker \pi' = T_mM \times \{0_p\} \ar[d, hookrightarrow] \ar["T_m F_p", dr] &  &  \\
    \ker \pi = T_\sigma\Sigma \ar[r] \ar[r] \ar[r] \ar[r, hookrightarrow] &
  	T_m M \times T_p P \ar["\pi", bend right, swap, rr] \ar["\pi'", d]  \ar["T_\sigma F",
  	r, swap]& T_{F(\sigma)} N \ar["\rho", r, twoheadrightarrow]& \nu_{F(\sigma)} A \\
     & T_p P &  &