preamble.sty

% NB Recommended compiler: LuaLaTeX. Pdflatex should work as well, but uncomment the first two lines with fontenc and inputenc. The alternative (currently commented) sets of fonts can be made to work but will likely cause wrong outputs.

% \usepackage[T1]{fontenc}
% \usepackage[utf8]{inputenc}
\usepackage{anyfontsize}
\usepackage{geometry}
\geometry{verbose,tmargin=2cm,bmargin=1.5cm,lmargin=1cm,rmargin=1cm,heightrounded}

% \setlength{\parskip}{3pt plus 1pt minus 1pt}
% \setlength{\parindent}{0pt}
\usepackage[skip=.3\baselineskip plus 2pt\relax, parfill]{parskip}
\raggedbottom
% \renewenvironment{quote}[1]%
%   {\list{}{\leftmargin=#1\rightmargin=#1}\item[]}%
%   {\endlist}

\usepackage{color}
\usepackage{babel}
\usepackage{amsmath}
\usepackage{amsthm}
%\usepackage[amsthm]{ntheorem}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{graphicx}
% \usepackage{mathabx}
\usepackage{ytableau}
\usepackage{wasysym}
\usepackage{cancel}
\usepackage{imakeidx}
\usepackage{booktabs}
\usepackage{wrapfig}
\usepackage{adjustbox}
\usepackage{enumerate}

\usepackage{import}
% \graphicspath{{figures/}}
% \newcommand{\svginput}[1]{\input{figures/#1}}
% \usepackage{breqn}

\usepackage{stmaryrd}
\expandafter\def\csname opt@stmaryrd.sty\endcsname{only,shortleftarrow,shortrightarrow} %fake options for stmaryd so that loading extpfeil later doesn't cause a clash
\usepackage{extpfeil} % lets define multiple long arrows
\newextarrow{\toto}{{20}{20}{20}{20}}{\bigRelbar\bigRelbar{\bigtwoarrowsleft\rightarrow\rightarrow}}

   
\usepackage[stable,bottom]{footmisc}
\interfootnotelinepenalty=10000
\makeindex
\usepackage[footnotesize]{caption}
\captionsetup{width=0.8\textwidth}
\DeclareCaptionLabelFormat{bf-parens}{\textbf{Fig.~#2}}
\captionsetup{labelformat=bf-parens}

\usepackage{comment}
% \includecomment{comment}
\excludecomment{comment}


% center env with no vertical space before
\newenvironment{nscenter}{\parskip=0pt\par\nopagebreak\centering}{\par\noindent\ignorespacesafterend} 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Force numbering of all equations
% \renewenvironment{equation*}{\equation}{\endequation}
% \renewenvironment{align*}{\align}{\endalign}
% \renewenvironment{alignat*}{\alignat}{\endalignat}
% \renewenvironment{multline*}{\multline}{\endmultline}
% \renewenvironment{gather*}{\gather}{\endgather}

% Redefine \[...\] to be \begin{equation}...\end{equation}
\newcommand{\myequation}{\begin{equation}}
  \newcommand{\myendequation}{\end{equation}}
  \let\[\myequation
  \let\]\myendequation



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%% FONTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{fontspec}
\usepackage{newtxtext}
\usepackage[varvw]{newtxmath}
%OR
% \usepackage{fontspec}
% \usepackage{newtxtext}
% \renewcommand{\rmdefault}{ptm}
% \renewcommand{\ttdefault}{pcr}
% \renewcommand{\sfdefault}{phv}
% \let\Bbbk\relax
% \let\circledS\relax
% \usepackage[subscriptcorrection,mtphrb,mtpcal,mtpscr,mtpfrak]{mtpro2}
%OR
%\usepackage{pscyr}
%\usepackage[libertine]{newtxmath}
%OR
% \usepackage{stix2}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% FONT-DEPENDENT DEFINITIONS OF WIDE ACCENTS, BARS %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

            %%%%%% USE THESE WITH MTPRO2 %%%%%%%

% \newcommand\wh[1]{\what{#1}} 
% \newcommand\wt[1]{\wtilde{#1}} 
% \newcommand\wb[1]{\wbar{#1}}


           %%%%%% USE THESE WITHOUT MTPRO2 %%%%%%%

\newcommand{\altI}{I}

\usepackage{scalerel}
\newcommand\wh[1]{\hstretch{1.42}{\hat{\hstretch{.7}{#1\mkern2mu}}}\mkern-2mu} %requires \usepackage{graphicx,scalerel}
\newcommand\wt[1]{\hstretch{1.42}{\tilde{\hstretch{.7}{#1\mkern2mu}}}\mkern-2mu}

    %%%% \newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}
    %%%% \newcommand\wb[1]{\hstretch{1.42}{\bar{\hstretch{.7}{#1\mkern2mu}}}\mkern-2mu}

% define a slightly wider bar \wb
\makeatletter
\renewcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
\newcommand*\wb[1]{%
  \begingroup
  \def\mathaccent##1##2{%
    \rel@kern{0.8}%
    \overline{\rel@kern{-0.8}\macc@nucleus\rel@kern{0.2}}%
    \rel@kern{-0.2}%
  }%
  \macc@depth\@ne
  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
  \macc@set@skewchar\relax
  \let\mathaccentV\macc@nested@a
  \macc@nested@a\relax111{#1}%
  \endgroup
}
\makeatother


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%   FORMATTING and ENVS  %%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%% THIS ALLOWS PAGE BREAKS IN DISPLAY MATH GLOBALLY %%%%%
\allowdisplaybreaks

% This fixes math in headings not becoming bold 
% From https://tex.stackexchange.com/questions/41379/automatically-typeset-math-in-section-headings-in-bold-face
\makeatletter
    \g@addto@macro\bfseries{\boldmath}
\makeatother


% This increases spacing inside matrices
\def\arraystretch{1.1} % row spacing
\setlength{\arraycolsep}{2pt} % column spacing
\usepackage[math]{cellspace}
\cellspacetoplimit 1pt
\cellspacebottomlimit 1pt



%%%%%%%%%%%%%%%%%%%%%%%%   tikz-cd and highlighting for matrices
\usepackage{tikz-cd}
% \usetikzlibrary{matrix,backgrounds}
% \pgfdeclarelayer{myback}
% \pgfsetlayers{myback,background,main}

% \tikzset{mycolor/.style = {line width=1bp,color=#1}}%
% \tikzset{myfillcolor/.style = {draw,fill=#1}}%

% \NewDocumentCommand{\highlight}{O{blue!40} m m}{%
% \draw[mycolor=#1] (#2.north west)rectangle (#3.south east);
% }

% \NewDocumentCommand{\fhighlight}{O{blue!40} m m}{%
% \draw[myfillcolor=#1] (#2.north west)rectangle (#3.south east);
% }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\usepackage{marginnote}


\usepackage[activate={true},step=1,tracking=true,spacing=false,factor=1100,stretch=20,shrink=20]{microtype}%text appearance improvements
% \usepackage[activate={true},step=1,tracking=true,kerning=true,spacing=false,factor=1100,stretch=20,shrink=20]{microtype}%text appearance improvements
\UseMicrotypeSet[protrusion]{basictext}
%\microtypecontext{spacing=nonfrench}
\DisableLigatures[N]{encoding=*, family =*}
\SetTracking{encoding=*, shape=sc}{0} %reduce spacing in small caps

\usepackage{titlesec} %change title spacing for sections
\titlespacing*{\section}
{0pt}{0.7\baselineskip}{0.3\baselineskip}
\titlespacing*{\subsection}
{0pt}{0.7\baselineskip}{0.3\baselineskip}
\newcommand{\Part}{Part}
\newcommand{\partt}{part}
\newcommand{\Chap}{Chapter}
\newcommand{\chap}{chapter}
\newcommand{\Sect}{Section}
\newcommand{\sect}{section}
\newcommand{\Subsect}{Subsection}
\newcommand{\subsect}{subsection}


%%%%%%%%%  TOC FORMATTING %%%%%%%%%%%%%

\makeatletter
  \renewcommand\@pnumwidth{3em} %give more space to page numbers in TOC
\makeatother 

\usepackage[nottoc]{tocbibind} % list bibiliography and other secondary sections in TOC
% \usepackage{tocloft} % needed for the next line
% \addtolength{\cftsubsecnumwidth}{1em} % increase the space given to numbers in TOC

% increase the space given to numbers in TOC
\makeatletter
  \renewcommand*{\numberline}[1]{\hb@xt@3em{#1\hfil}} 
\makeatother

% \makeatletter
%   \let\toc@pre\relax
%   \let\toc@post\relax
% \makeatother

%%%%%%%%%%%%%%%%% LISTS FORMATTING  %%%%%%%%%%%%%%%%%%

\usepackage{enumitem}
\setlist[itemize]{leftmargin=2em}
\setlist[enumerate]{leftmargin=2em,topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex}


%%%%%%%%%%%%%%%%% THEOREM ENVS %%%%%%%%%%%%%%%%%%%%%%%

\flushbottom
\renewcommand\qedsymbol{$\blacksquare$}
\providecommand{\definitionname}{Definition}
\providecommand{\examplename}{Example}
\providecommand{\exercisename}{Exercise}
\providecommand{\propositionname}{Proposition}
\providecommand{\remarkname}{Remark}
\providecommand{\intuitionname}{Intuition}
\providecommand{\theoremname}{Theorem}

% number equations within sections
\numberwithin{equation}{section}
\numberwithin{figure}{section}
\numberwithin{table}{section}

\theoremstyle{plain}
\newtheorem{prop}{\protect\propositionname}[section]
\newtheorem{thm}[prop]{\protect\theoremname}%thm 
\newtheorem{lem}[prop]{Lemma}%lem with counter inherited from prop
\newtheorem{cor}[prop]{Corollary}%cor with counter inherited from prop
\theoremstyle{definition}
\newtheorem{rem}[prop]{\protect\remarkname}
\newtheorem*{intu*}{\protect\intuitionname}
\newtheorem*{rem*}{\protect\remarkname}
\newtheorem*{hrem*}{Historical note}
\newtheorem{example}[prop]{\protect\examplename}
\newtheorem{defn}[prop]{\protect\definitionname}
\newtheorem{xca}{\protect\exercisename}[section]


\usepackage{tablefootnote}
\usepackage[many,breakable]{tcolorbox}
\ExplSyntaxOn
\NewDocumentCommand{\tcbwrap}{mm}
 {
  \clist_map_inline:nn{#1}{\tcolorboxenvironment{##1}{#2}}
 }
\ExplSyntaxOff
\tcbwrap{thm,lem,cor,prop}{
    grow to left by=5pt,grow to right by=5pt,
    boxsep=1pt,left=4pt,right=4pt,
    top=2pt,bottom=2pt, colback=gray!20, sharp corners,enhanced,
    leftrule=3pt, rightrule=0pt, toprule=0pt, bottomrule=0pt,
    breakable=true}
\tcbwrap{intu*}{
  grow to left by=5pt,grow to right by=5pt,
  boxsep=1pt,left=4pt,right=4pt,
  top=2pt,bottom=2pt, colback=gray!10, sharp corners,enhanced,
  leftrule=1pt, rightrule=1pt, toprule=1pt, bottomrule=1pt,
  breakable=true}
\tcbwrap{defn}{
    grow to left by=5pt,grow to right by=5pt,
    boxsep=1pt,left=4pt,right=4pt,
    top=0pt,bottom=0pt,colback=white,boxrule=0pt,frame hidden,sharp corners,enhanced,borderline west={1.5pt}{0pt}{black!80},
    breakable=true}
\tcbwrap{rem,rem*}{
    grow to left by=5pt,grow to right by=5pt,
    boxsep=1pt,left=4pt,right=4pt,
    top=0pt,bottom=0pt,colback=white,boxrule=0pt,frame hidden,sharp corners,enhanced,borderline west={1pt}{0pt}{black!80},
    breakable=true}
\tcbwrap{hrem*}{
    grow to left by=5pt,grow to right by=5pt,
    boxsep=1pt,left=4pt,right=4pt,
    top=0pt,bottom=0pt, colback=white,boxrule=1pt,frame hidden, enhanced,
    breakable=true,fontlower=\small}
\tcbwrap{xca}{
    grow to left by=5pt,grow to right by=5pt,
    boxsep=1pt,left=4pt,right=4pt,
    top=0pt,bottom=0pt,colback=white,boxrule=0pt,frame hidden,sharp corners,enhanced,
    borderline west={1.5pt}{0pt}{black!80,dashed},
    breakable=true}
\usetikzlibrary{decorations}
\usetikzlibrary{decorations.pathmorphing}
\tcbuselibrary{skins}
\tcbwrap{example}{
    grow to left by=5pt,grow to right by=5pt,
    boxsep=1pt,left=4pt,right=4pt,
    top=0pt,bottom=0pt,colback=white,boxrule=0pt,frame hidden,sharp corners,enhanced,
    borderline west={0.75pt}{0pt}{decoration={zigzag,amplitude=1pt,pre=moveto,pre length=12pt},decorate},
    breakable=true}

    % Workaround for a hyperref-related bug in tcolorbox's parbox=false option: pass the definition of parskip directly
    
\AddToHook{env/rem/begin}{\parindent=0pt \parskip=0.3\baselineskip plus 2pt\relax}
\AddToHook{env/hrem/begin}{\parindent=0pt \parskip=\baselineskip}
\AddToHook{env/thm/begin}{\parindent=0pt \parskip=\baselineskip}
\AddToHook{env/lem/begin}{\parindent=0pt \parskip=\baselineskip}
\AddToHook{env/cor/begin}{\parindent=0pt \parskip=\baselineskip}
\AddToHook{env/prop/begin}{\parindent=0pt \parskip=\baselineskip}
\AddToHook{env/defn/begin}{\parindent=0pt \parskip=\baselineskip}
\AddToHook{env/example/begin}{\parindent=0pt \parskip=\baselineskip}
\AddToHook{env/xca/begin}{\parindent=0pt \parskip=\baselineskip}



%%%%%%%%%%%%%%%%%%%%%%%%
%Configure tcolorbox so that footnotes are printed outside of it
%From https://tex.stackexchange.com/a/558922/44577
\makeatletter
% restore footnote internals to those in normal page, not minipage
\def\tcb@restore@footnote{%
  \def\@mpfn{footnote}%
  \def\thempfn{\arabic{footnote}}%
  \let\@footnotetext\tcb@footnote@collect
}
% collect footnote text
\long\def\tcb@footnote@collect#1{%
  % expand \@thefnmark before appending before app to \tcb@footnote@acc
  \expandafter\gappto\expandafter\tcb@footnote@acc\expandafter{%
    \expandafter\footnotetext\expandafter[\@thefnmark]{#1}%
  }%
}
\def\tcb@footnote@use{%
  \tcb@footnote@acc
  \global\let\tcb@footnote@acc\@empty
}
\global\let\tcb@footnote@acc\@empty
\tcbset{
  % restore for every box
  every box/.style={
    before title pre=\tcb@restore@footnote, % just in case
    before upper pre=\tcb@restore@footnote,
    before lower pre=\tcb@restore@footnote,
  },
  % use for layer 1 boxes only
  every box on layer 1/.append style={
    % use \footnotetext befere the default /tcb/after ends the current paragraph
    after pre=\tcb@footnote@use
  }
}
\makeatother

%%%%%%%%%%%%%%%% Under construction symbol %%%%%%%%%%%%%%%%%%%%%
\usepackage{fontawesome} % requires fontspec when used with XeTeX or LuaTeX
\newcommand{\ucmark}{\faWrench}

%%%%%%%%%%%%%%%%%%%% Colored text commands %%%%%%%%%%%%%%%%%%%%%
\usepackage{color}
\def\white{\color{white}}
\def\black{\color{black}}
\def\red{\color{red}}
\def\green{\color{green}}
\def\blue{\color{blue}}
\def\cyan{\color{cyan}}
\def\magenta{\color{magenta}}
\def\yellow{\color{yellow}}


%%%%%%%%%%%%%%%%  HYPERREF (load as late as possible) %%%%%%%%%%%
\usepackage[unicode=true,pdfusetitle,
 bookmarks=true,bookmarksnumbered=false,bookmarksopen=false,
 breaklinks=false,pdfborder={0 0 1},backref=false,colorlinks=true]
 {hyperref}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%   MACROS  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \usepackage{bm}
\renewcommand{\bf}[1]{\mathbf{#1}}
\newcommand{\bm}[1]{\boldsymbol{#1}}

% backslash for left side quotients
\newcommand{\bslash}{\mkern-.5mu\backslash\mkern-2mu}

\newcommand{\bbA}{\mathbb{A}}
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbN}{\mathbb{N}}
\newcommand{\bbC}{\mathbb{C}}
\newcommand{\bbE}{\mathbb{E}}
\newcommand{\bbF}{\mathbb{F}}
\newcommand{\bbZ}{\mathbb{Z}}
\newcommand{\bbH}{\mathbb{H}}
\newcommand{\bbO}{\mathbb{O}}
\newcommand{\bbK}{\mathbb{K}}
\newcommand{\bbP}{\mathbb{P}}
\newcommand{\bbT}{\mathbb{T}}
\newcommand{\bbQ}{\mathbb{Q}}
\newcommand{\bbD}{\mathbb{D}}
\newcommand{\bbS}{\mathbb{S}}

\newcommand{\rmA}{\mathrm{A}}
\newcommand{\rmB}{\mathrm{B}}
\newcommand{\rmC}{\mathrm{C}}
\newcommand{\rmD}{\mathrm{D}}
\newcommand{\rmE}{\mathrm{E}}
\newcommand{\rmF}{\mathrm{F}}
\newcommand{\rmG}{\mathrm{G}}
\newcommand{\rmH}{\mathrm{H}}
\newcommand{\rmI}{\mathrm{I}}
\newcommand{\rmII}{\mathrm{II}}
\newcommand{\rmJ}{\mathrm{J}}
\newcommand{\rmj}{\mathrm{j}}
\newcommand{\rmK}{\mathrm{K}}
\newcommand{\rmL}{\mathrm{L}}
\newcommand{\rmN}{\mathrm{N}}
\newcommand{\rmP}{\mathrm{P}}
\newcommand{\rmQ}{\mathrm{Q}}
\newcommand{\rmR}{\mathrm{R}}
\newcommand{\rmS}{\mathrm{S}}
\newcommand{\rmT}{\mathrm{T}}
\newcommand{\T}{\mathrm{T}}
\newcommand{\D}{\mathrm{D}}
\newcommand{\rmV}{\mathrm{V}}
\newcommand{\rmZ}{\mathrm{Z}}
\newcommand{\rmi}{\i}
\newcommand{\rme}{\mathrm{e}}

\newcommand{\calA}{\mathcal{A}}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\calD}{\mathcal{D}}
\newcommand{\calE}{\mathcal{E}}
\newcommand{\calF}{\mathcal{F}}
\newcommand{\calG}{\mathcal{G}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\calI}{\mathcal{\altI}}
\newcommand{\calJ}{\mathcal{J}}
\newcommand{\calL}{\mathcal{L}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calO}{\mathcal{O}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calS}{\mathcal{S}}
\newcommand{\calT}{\mathcal{T}}
\newcommand{\calU}{\mathcal{U}}
\newcommand{\calV}{\mathcal{V}}
\newcommand{\calVo}{\mathcal{V}_{\mathrm{o}}}
\newcommand{\calVr}{\mathcal{V}_{\mathrm{r}}}
\newcommand{\calY}{\mathcal{Y}}

\newcommand{\fraka}{\mathfrak{a}}
\newcommand{\frake}{\mathfrak{e}}
\newcommand{\frakg}{\mathfrak{g}}
\newcommand{\frakh}{\mathfrak{h}}
\newcommand{\frakk}{\mathfrak{k}}
\newcommand{\fraki}{\mathfrak{i}}
\newcommand{\frakl}{\mathfrak{l}}
\newcommand{\frakn}{\mathfrak{n}}
\newcommand{\frakN}{\mathfrak{N}}
\newcommand{\frakm}{\mathfrak{m}}
\newcommand{\frakM}{\mathfrak{M}}
\newcommand{\frakb}{\mathfrak{b}}
\newcommand{\frakd}{\mathfrak{d}}
\newcommand{\frakp}{\mathfrak{p}}
\newcommand{\frakr}{\mathfrak{r}}
\newcommand{\fraku}{\mathfrak{u}}
\newcommand{\frakz}{\mathfrak{z}}
\newcommand{\frakgl}{\mathfrak{gl}}
\newcommand{\fraksl}{\mathfrak{sl}}
\newcommand{\frako}{\mathfrak{o}}
\newcommand{\frakso}{\mathfrak{so}}
\newcommand{\fraksu}{\mathfrak{su}}
\newcommand{\fraksp}{\mathfrak{sp}}
\newcommand{\frakse}{\mathfrak{se}}

\newcommand{\scrA}{\mathscr{A}}
\newcommand{\scrC}{\mathscr{C}}
\newcommand{\scrF}{\mathscr{F}}
\newcommand{\scrH}{\mathscr{H}}
\newcommand{\scrJ}{\mathscr{J}}
\newcommand{\scrK}{\mathscr{K}}
\newcommand{\scrO}{\mathscr{O}}
\newcommand{\scrR}{\mathscr{R}}
\newcommand{\scrT}{\mathscr{T}}

\newcommand{\sfg}{\mathsf{g}}
\newcommand{\sfh}{\mathsf{h}}
\newcommand{\sfv}{\mathsf{v}}
\newcommand{\sfw}{\mathsf{w}}
\newcommand{\sfK}{\mathsf{K}}
\newcommand{\sfJ}{\mathsf{J}}
\newcommand{\sfR}{\mathsf{R}}
\newcommand{\sfT}{\mathsf{T}}


\newcommand{\acts}{%
  \mathchoice{\raisebox{1pt}{$\displaystyle\curvearrowright$}}
             {\raisebox{1pt}{$\curvearrowright$}}
             {\raisebox{0.5pt}{$\scriptstyle\curvearrowright$}}
             {\raisebox{0.2pt}{$\scriptscriptstyle\curvearrowright$}}
}

\newcommand{\lacts}{%
  \mathchoice{\raisebox{1pt}{$\displaystyle\curvearrowleft$}}
             {\raisebox{1pt}{$\curvearrowleft$}}
             {\raisebox{0.5pt}{$\scriptstyle\curvearrowleft$}}
             {\raisebox{0.2pt}{$\scriptscriptstyle\curvearrowleft$}}
}

% \newcommand{\acts}{\curvearrowright}
% \newcommand{\lacts}{\curvearrowleft}
\newcommand{\dd}{\mathop{}\!\mathrm{d}}

\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}
\newcommand{\emb}{\mathop{\mathbin{\raisebox{.0ex}{\rotatebox[origin=c]{180}{$\ge$}}}}}
\newcommand{\embweak}{\mathop{\mathbin{\raisebox{.3ex}{\rotatebox[origin=c]{180}{$\gtrsim$}}}}}
\newcommand{\embsub}{\mathop{\mathbin{\raisebox{.2ex}{\rotatebox[origin=c]{180}{$\trianglerighteq$}}}}}
\newcommand{\embprop}{\mathop{\mathbin{\raisebox{.1ex}{\rotatebox[origin=c]{180}{$\sqsupseteq$}}}}}
% \newcommand{\embprop}{\mathrel{\mathpalette\bleqinn\relax}}
% \newcommand{\bleqinn}[2]{%
%   \ooalign{%
%     \raisebox{.0ex}{$#1\blacktriangleleft$}\cr
%     $#1\emb$\cr}}


\renewcommand{\i}{\mathrm{i}}
\renewcommand{\Re}{\operatorname{\mathsf{Re}}}
\renewcommand{\Im}{\operatorname{\mathsf{Im}}}
\newcommand{\Ad}{\mathrm{Ad}}
\newcommand{\ad}{\mathrm{ad}}
\newcommand{\Adg}{\mathrm{\mathbf{Ad}}}
\newcommand{\Fl}{\mathrm{Fl}}

\newcommand{\KP}{\mathbb{K}\mathbb{P}}
\newcommand{\RP}{\mathbb{R}\mathbb{P}}
\newcommand{\CP}{\mathbb{C}\mathbb{P}}

\newcommand{\const}{\mathrm{const}}
% \newcommand{\Lin}{\bm\mathrm{L}}

\newcommand{\mFB}{\mathsf{-FB}}
\newcommand{\FB}{\mathsf{FB}}
\newcommand{\PFB}{\mathsf{PFB}}
\newcommand{\VB}{\mathsf{VB}}
\newcommand{\Car}{\mathsf{Car}}
% \DeclareMathOperator{\ad}{ad}
% \DeclareMathOperator{\Ad}{Ad}
% \DeclareMathOperator{\Adg}{\mathbf{Ad}}
% \DeclareMathOperator{\Span}{span}
\newcommand{\diag}{\mathrm{diag}}
\renewcommand{\implies}{\Rightarrow}
\DeclareMathOperator{\graph}{graph}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\ver}{ver}
\DeclareMathOperator{\hor}{hor}
\DeclareMathOperator{\mor}{Mor}
\DeclareMathOperator{\rel}{rel\,}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\ob}{Ob}
\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\St}{St}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Inn}{Inn}
\DeclareMathOperator{\Mat}{Mat}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\aut}{\mathfrak{aut}}
\DeclareMathOperator{\cha}{\mathfrak{char}}
\DeclareMathOperator{\sym}{\mathfrak{sym}}
\DeclareMathOperator{\Gau}{Gau}
\DeclareMathOperator{\gau}{\mathfrak{gau}}
\DeclareMathOperator{\Fr}{Fr}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\rank}{rk}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\pr}{pr}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Lin}{Lin}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Diff}{Diff}
\DeclareMathOperator{\Pin}{Pin}
\DeclareMathOperator{\Spin}{Spin}
\DeclareMathOperator{\Hol}{Hol}
\DeclareMathOperator{\Ric}{\mathsf{Ric}}
\DeclareMathOperator{\Mon}{Mon}
\DeclareMathOperator{\Aff}{Aff}
\DeclareMathOperator{\SAff}{ASL}
\DeclareMathOperator{\Heis}{Heis}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\PGL}{\mathbb{P}GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\PSL}{\mathbb{P}SL}
\DeclareMathOperator{\Or}{O}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\CO}{CO}
\DeclareMathOperator{\SE}{SE}
\DeclareMathOperator{\U}{U}
\DeclareMathOperator{\SU}{SU}
\DeclareMathOperator{\PSU}{\mathbb{P}SU}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\Alt}{Alt}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\coim}{coim}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\ext}{ext}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\tor}{tor}
\DeclareMathOperator{\lcm}{lcm}
\let\div\relax
\DeclareMathOperator{\div}{div}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\curl}{curl}
\let\grad\relax
\DeclareMathOperator{\grad}{grad}
\DeclareMathOperator*{\res}{res}
\DeclareMathOperator{\Lief}{\bf{Lie}}
\DeclareMathOperator{\PDO}{PDO}
\DeclareMathOperator{\OP}{OP}
\DeclareMathOperator{\tra}{\mathsf{tra}}
\DeclareMathOperator{\dev}{\mathsf{dev}}
\DeclareMathOperator{\hol}{\mathsf{hol}}
\DeclareMathOperator{\vol}{\mathsf{vol}}
\DeclareMathOperator{\frakhol}{\mathfrak{hol}}



\newcommand{\Lie}{\mathcal{L}}
\newcommand{\colimit}{\varinjlim}
\newcommand{\limit}{\varprojlim}
\newcommand{\fX}{\mathfrak{X}}
\newcommand{\sub}{\vartriangleleft}
\newcommand{\sube}{\trianglelefteqslant}
\newcommand{\<}{\left\langle}
\renewcommand{\>}{\right\rangle}
\renewcommand{\mod}{\ \mathrm{mod}\ }
% \newcommand{\bull}{{\scriptscriptstyle\bullet}}

\usepackage{scalerel}
\makeatletter
  \newcommand\sbullet[1][.5]{\mathbin{\ThisStyle{\vcenter{\hbox{%
    \scalebox{#1}{$\SavedStyle\bullet$}}}}}%
  }
  \newcommand\smbullet[1][.8]{\mathbin{\ThisStyle{\vcenter{\hbox{%
    \scalebox{#1}{$\SavedStyle\bullet$}}}}}%
  }
\makeatother

\newcommand{\bigslant}[2]{{\raisebox{.2em}{$#1$}\left/\raisebox{-.2em}{$#2$}\right.}}

\usepackage{xparse}
\DeclareDocumentCommand{\restr}{m m o}  
{%
  \IfNoValueTF{#3}
  {
  \left.\kern-\nulldelimiterspace % automatically resize the bar with \right
  #1 % the function
  \vphantom{\big|} % pretend it's a little taller at normal size
  \right|_{#2}
  }{
  \left.\kern-\nulldelimiterspace % automatically resize the bar with \right
  #1 % the function
  \vphantom{\big|} % pretend it's a little taller at normal size
  \right|_{#2}^{#3} }
}

% This redefines \middle to include spacing similar to \mid
\let\originalmiddle=\middle
\def\middle#1{\mathrel{}\originalmiddle#1\mathrel{}}

% This definition needs graphicx
\newcommand{\PRLsep}{   %decorative line used to separate sections
    \noindent\makebox[\linewidth]{
      \resizebox{0.5\linewidth}{1pt}{$\blacklozenge$}
    }
}

% define \tuple and \set which allow line breaks at commas in inline math
% from https://tex.stackexchange.com/a/309558/44577
\newcommand{\splitatcommas}[1]{%
  \begingroup
  \ifnum\mathcode`,="8000
  \else
    \begingroup\lccode`~=`, \lowercase{\endgroup
      \edef~{\mathchar\the\mathcode`, \penalty0 \noexpand\hspace{0pt plus 1em}}%
    }\mathcode`,="8000
  \fi
  #1%
  \endgroup
}
\newcommand{\tuple}[1]{\splitatcommas{#1}}


% Define \widebar[widthpercent]{text}, custom sized \overline
% You may need to replace \renewcommand with \newcommand depending on other packages
\makeatletter 
    \newsavebox\myboxA
    \newsavebox\myboxB
    \newlength\mylenA
    \renewcommand*\widebar[2][0.9]{%
        \sbox{\myboxA}{$\m@th#2$}%
        \setbox\myboxB\null% Phantom box
        \ht\myboxB=\ht\myboxA%
        \dp\myboxB=\dp\myboxA%
        \wd\myboxB=#1\wd\myboxA% Scale phantom
        \sbox\myboxB{$\m@th\overline{\copy\myboxB}$}%  Overlined phantom
        \setlength\mylenA{\the\wd\myboxA}%   calc width diff
        \addtolength\mylenA{-\the\wd\myboxB}%
        \ifdim\wd\myboxB<\wd\myboxA%
        \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA}%
        \else
            \hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB}%
        \fi}
\makeatother


% WIDTH OF CAPTIONS
\usepackage{caption}
\usepackage{subcaption}
% \captionsetup{width=0.6\columnwidth}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%   BIBLIOGRAPHY  %%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\usepackage{csquotes}

\AtBeginDocument{

  %\AtEveryBibitem{\clearfield{month}} %remove day and month from dates
  %\AtEveryBibitem{\clearfield{day}}

  \renewbibmacro*{doi+eprint+url}{%
    \iftoggle{bbx:doi}
      {\iffieldundef{url}{\printfield{doi}}{}}
      {}%
    \newunit\newblock
    \iftoggle{bbx:eprint}
      {\usebibmacro{eprint}}
      {}%
    \newunit\newblock
    \iftoggle{bbx:url}
      {\usebibmacro{url+urldate}}
      {}
    }

  \newbibmacro{string+doiurlisbn}[1]{%
    \iffieldundef{url}{%
      \iffieldundef{doi}{%
      \iffieldundef{eprint}{%
        \iffieldundef{isbn}{%
          \iffieldundef{issn}{%
            #1%
          }{%
            \href{http://books.google.com/books?vid=ISSN\thefield{issn}}{#1}%
          }%
        }{%
          \href{http://books.google.com/books?vid=ISBN\thefield{isbn}}{#1}%
        }%
      }{
          \href{http://arxiv.org/abs/\thefield{eprint}}{#1}
      }%
      }{%
        \href{http://dx.doi.org/\thefield{doi}}{#1}%
      }%
    }{%
      \href{\thefield{url}}{#1}%
    }%
  }


  \renewbibmacro*{journal}{%
    \ifboolexpr{ test {\ifcitation} and not test {\iffieldundef{shortjournal}} }
      {\printfield[journaltitle]{shortjournal}}
      {\iffieldundef{journaltitle}
        {}
        {\printtext[journaltitle]{%
            \printfield[titlecase]{journaltitle}%
            \setunit{\subtitlepunct}%
            \printfield[titlecase]{journalsubtitle}}}}}

  \newbibmacro*{journal+sub}{%
    \ifboolexpr{ test {\ifcitation} and not test {\iffieldundef{shortjournal}} }
      {\printfield[journaltitle]{shortjournal}}
      {\iffieldundef{journaltitle}
        {}
        {\printtext[journaltitle]{%
          \printfield[noformat]{journaltitle}%
          \setunit{\addcolon\addspace}%
          \printfield[noformat]{journalsubtitle}}}}}

  \DeclareSourcemap{%
    \maps[datatype=bibtex]{
      % Journal abbreviations
      \map[overwrite]{
        \step[fieldsource=shortjournal]
        \step[fieldset=journaltitle,origfieldval]
      }
    }
  }

  % remove "in" word before journal name for articles
  %\renewbibmacro{in:}{}
  \renewbibmacro{in:}{\ifentrytype{article}{}{\printtext{\bibstring{in}\intitlepunct}}}

  % removes period at the very end of bibliographic record
  \renewcommand{\finentrypunct}{}

  % back reference text preceding the page number ("see p.")
  \DefineBibliographyStrings{english}{%
      backrefpage  = {\lowercase{s}ee p.}, % for single page number
      backrefpages = {\lowercase{s}ee pp.} % for multiple page numbers
  }

  % removes period after DOI and suppresses capitalization
  % of the word following DOI ("See p. xx" -> "see p. xx")
  %\renewcommand{\newunitpunct}{\addspace\midsentence}

  \DeclareFieldFormat{title}{\usebibmacro{string+doiurlisbn}{#1}}
  \DeclareFieldFormat[article, incollection, inproceedings]{title}{\usebibmacro{string+doiurlisbn}{\mkbibquote{#1}}}
  \DeclareFieldFormat{journaltitle}{\mkbibemph{#1},} % italic journal title with comma
  \DeclareFieldFormat[inbook,thesis]{title}{\mkbibemph{#1}\addperiod} % italic title with period
  \DeclareFieldFormat[article]{volume}{\textbf{#1}} % makes volume of journal bold and adds colon
  \DeclareFieldFormat{pages}{#1} % removes pagination (p./pp.) before page numbers

  \usepackage{setspace} % to change line spacing in bibliography with \setstretch
  \renewcommand*{\bibfont}{\small} % set font size to small

} % end of AtBeginDocument{...}

\usepackage[backend=biber, bibstyle=custom-numeric-comp2,citestyle=numeric, sorting=nyt, backref=false, sortcites=true, giveninits=true, language=english, maxbibnames=6, isbn=false, url=false, doi=false, eprint=false]{biblatex}
\renewcommand*{\mkbibnamefamily}[1]{\textsc{#1}}

\addbibresource{bib.bib}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%   GLOSSARY  %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\usepackage[acronym,toc,section=section,nogroupskip]{glossaries}
\usepackage[automake]{glossaries-extra}
\renewcommand{\glsnamefont}[1]{\textbf{#1}}
\makeglossaries
% abbreviations:
\newacronym{hlp}{HLP}{homotopy lifting property}
\newacronym{hep}{HEP}{homotopy extension property}
\newacronym{pou}{POU}{partition of unity}
\newacronym{fb}{FB}{fiber bundle}
\newacronym{vb}{VB}{vector bundle}
\newacronym{pfb}{PFB}{principal fiber bundle}
\newacronym{wlog}{WLOG}{without loss of generality}
\newacronym{tfae}{TFAE}{the following are equivalent}
\newacronym{inmt}{InMT}{inverse mapping theorem}
\newacronym{immt}{ImMT}{implicit mapping theorem}
\newacronym{mv}{MV}{Mayer-Vietoris}
\newacronym{ops}{OPS}{one-parameter subgroup}
\newacronym{eds}{EDS}{exterior differential system}
\newacronym{flt}{FLT}{fractional linear (M\"obius) transformation}
\newacronym{ma}{MA}{Monge-Amp\`ere}
\newacronym{cmc}{CMC}{constant mean curvature}
\newacronym{kdv}{KdV}{Korteweg-de~Vries equation}
\newacronym{sge}{SGE}{sine-Gordon equation}
\newacronym{bch}{BCH}{Baker-Campbell-Hausdorff}