\documentclass[12pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[full]{textcomp}
\usepackage[osf]{newtxtext}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{graphicx}
\usepackage{mhequ}

\def\can{{\mathrm{can}}}
\def\coker{\operatorname{coker}\nolimits}
\def\End{\operatorname{End}\nolimits}
\def\eps{\varepsilon}
\def\gr{\operatorname{gr}\nolimits}
\def\Hom{\operatorname{Hom}\nolimits}
\def\id{\operatorname{id}\nolimits}
\def\idun{\mathbf{1}}
\def\iso{\buildrel \sim\over\to}
\def\opp{{\operatorname{opp}\nolimits}}
\def\rev{\operatorname{rev}}
\def\CC{{\mathcal{C}}}
\def\CD{{\mathcal{D}}}
\def\CE{{\mathcal{E}}}
\def\CL{{\mathcal{L}}}
\def\GS{{\mathfrak{S}}}

\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\Ad}{\mrm{Ad}}
\newcommand{\liegroup}{{\mathbf{G}}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\cov}{\mathrm{inv}}
\newcommand{\covd}{\textup{D}}
\newcommand{\PP}{\ensuremath{\mathbb{P}}}

\DeclareMathOperator{\slim}{slim}
\let\div\undefined
\DeclareMathOperator{\div}{div}

\makeatletter
\newcommand*{\bigcdot}{}
\DeclareRobustCommand*{\act}{%
  \mathbin{\mathpalette\bigcdot@{}}%
}
\DeclareRobustCommand*{\place}{%
  \mathord{\kern0.1em\mathpalette\bigcdot@{}\kern0.1em}%
}
\newcommand*{\bigcdot@scalefactor}{.5}
\newcommand*{\bigcdot@widthfactor}{1.15}
\newcommand*{\bigcdot@}[2]{%
  \sbox0{$#1\vcenter{}$}%
  \sbox2{$#1\cdot\m@th$}%
  \hbox to \bigcdot@widthfactor\wd2{%
    \hfil
    \raise\ht0\hbox{%
      \scalebox{\bigcdot@scalefactor}{%
        \lower\ht0\hbox{$#1\bullet\m@th$}%
      }%
    }%
    \hfil
  }%
}
\makeatother

\newtheorem*{problem}{Problem}
\newtheorem{theorem}{Theorem}

\title{Second Batch Community Experiment --- Week \#2 Problems}
\date{\today}
\author{First Proof}
\begin{document}

\maketitle

\section{Problem C3}

Fix a compact Lie group $\liegroup$ with Lie algebra $\fg$, endowed with an
$\Ad$-invariant norm.
We define invariant $\CC^{-\kappa}_\cov$ ``norms'' on smooth $\liegroup$-equivariant 
connections $B_0$ on the trivial $\liegroup$-bundle over the torus $\T^2$ by
\begin{equ}[e:covariantNorm]
\|B_0\|_{\cov,-\kappa} = \sup_{s \in (0,1]} s^{\frac{\kappa}2}\bigl(s^{\frac12} \|F(s)\|_{L^\infty} + s\|\covd F(s)\|_{L^\infty}\bigr)\;.
\end{equ}
Here $F(s) = F_{B(s)}$ denotes the curvature tensor for the solution $B(s)$ at time $s$ 
to the Yang--Mills gradient flow with
initial condition $B_0$ and $\covd F(s)$ denotes the covariant derivative of $F(s)$ with 
respect to the connection $B(s)$.
The quantity given by \eqref{e:covariantNorm} is of course not a genuine norm since the relation 
$B_0 \mapsto F$ is nonlinear. Recall also that the group of smooth functions $g \colon \T^2 \to \liegroup$
acts on such connections $B_0$ by 
\begin{equ}
g\act B_0(x) = \Ad_{g(x)}B_0(x) - dg(x)\,g(x)^{-1}\;.
\end{equ}
Give a proof of the following theorem.

\begin{theorem}\label{theo:cov2norm}
Given $\kappa \in (0,\frac12)$, there exist constants $c, C>0$ such that the following is true.
For every smooth connection $B_0$ as above, there exists a gauge transformation $g$ such that
\begin{equ}[e:apriori]
\|g\act B_0\|_{\CC^{-\kappa}} + \sup_{s \in (0,1]}s^{\kappa} \lVert\div(g\act  B(s))\rVert_{L^\infty}  \le C (1 + \|B_0\|_{\cov,-\kappa})^c\;.
\end{equ}
\end{theorem}

\section{Problem C4}
Give a proof of the following theorem.
\begin{theorem}
    Given $c > 1$ and $G_n \sim G(n,c/n)$, let $C_{max}$ denote the giant component. Conditional on $x,y,z \in C_{max}$, sample $[x,y], [y,z]$ and $[z,x]$ uniformly at random (from the set of shortest paths between the corresponding vertex pairs) to form $\Delta(x,y,z)$. 

    For every fixed $\epsilon > 0$, there exists a constant $M(\epsilon) > 0$ such that for the scale
    \[ \delta_n := \left\lfloor{ \frac{1}{\log c} \left( \frac{1}{2}\log n - 9 \log \log n\right) } \right\rfloor - M(\epsilon), \]
    there exists $N$ such that for all $n \geq N$, conditioned on the uniformly randomly chosen vertices $x,y,z$ lie in the giant component of $G_n$,
\[ \PP_{G_n}(\PP(\slim(\Delta(x,y,z)) > \delta_n) > 1 - \epsilon) > 1 - \epsilon \]
    where $\Delta(x,y,z)$ is a geodesic triangle.
\end{theorem}

\end{document}
