# Outside in

Given a group $G$, embed $G$ in a larger group $H$ such that any automorphism of $G$ is the restriction of an inner automorphism of $H$.

# Too complex to be true but too symmetrical to be false

For $i\in\{1,2\}$, let $D_i$ be a complex domain and let $z_i \in D_i$. Let $B_i$ be a complex Brownian motion starting from $z_i$. Set $T_i = \inf \{t > 0 : B_i(t) \notin D_i\}$.

Suppose that there exists a conformal isomorphism $\varphi : D_1 \to D_2$ such that $\varphi (z_1) = z_2$ . Set $\tilde{T} = \int_0^T |\varphi' (B_1(t))|dt$ and define for $t< \tilde{T}$

$\tau(t)= \inf \{s>0 : \int_0^s|\varphi' (B_1(r))|dr=t\}$ and $\tilde{B}(t) = \varphi(B_1(\tau(t)))$.

Then $(\tilde{T}, (\tilde{B}(t))_{t<\tilde{T}})$ and $(T_2, (B_2(t))_{t have the same distribution.

# Squaring the circle

Is there a bijection of the plane sending every circle to a square?

# Spooky algebra II

Exhibit a group $G$, a set of generators $X$ for $G$ and a subgroup $H$ such that $xHx^{-1} \subseteq H$ for all $x\in X$ and yet $H$ is not normal in $G$.

# It’s counting time

What is the number of idempotent functions $f:\{1,\dots,n\}\to\{1,\dots,n\}$?

# [Some punny title about determinants and fixed points]

Prove the following identity

$\sum_\sigma (-1)^\sigma t^{\mathrm{Fix}\;\sigma} = \det\begin{pmatrix} t & 1 &\cdots& 1& 1\\ 1 & t &\cdots& 1& 1 \\ \vdots & \vdots &\ddots & \vdots & \vdots \\ 1 & 1 &\cdots& t& 1\\ 1 & 1 &\cdots& 1& t \end{pmatrix}$

the sum running through $S_n$ and the matrix being of size $n\times n$.

# A two-liner

Is there a ring $A$ such that there are two different ring injections $\Bbb{R}\to A$?