\usepackage{amsthm, amsfonts, amssymb, amsmath, amscd,  amsrefs, 
nameref,mathtools, thmtools,comment,
mathrsfs}
\usepackage{graphics, slashed, multicol, latexsym, float, todonotes, enumitem, tensor,lipsum, graphicx, tikz, tikz-cd, tkz-berge, tkz-graph, color, cancel}

%The following must be loaded in this order
\usepackage[hidelinks,hypertexnames=false]{hyperref}
\usepackage[nameinlink]{cleveref}
%suppress equation numbers unless labeled 
\usepackage{autonum}

% some heavy packages I don't always use
%\usepackage{gensymb,polynom,relsize,dsfont,marvosym,mdframed}
\usepackage{stmaryrd,framed,mdframed}
\usepackage{moreenum}

%\usepackage{eqnnumwarn}
\hypersetup{colorlinks = true,
            linkcolor = black,
            urlcolor  = blue,
            citecolor = black,
}

%used to silence warnings
%\usepackage{silence}

%suppress hbox overfull warnings
\hfuzz=20pt 
%suppress all hbox issues
%\hbadness=10000
%suppress all "vbox" issues
%\vbadness

%suppress pagenumbers:
%\usepackage[margin=1in]{geometry}
%\pagenumbering{gobble}

%suppress numbering subsections within sections:
% \usepackage{chngcntr}
% \counterwithout{subsection}{section}
%determine what to show in toc:
%-1 part     1 section     3 subsubsection  5 subparagraph
%0 chapter  2 subsection  4 paragraph
%\setcounter{tocdepth}{4}

%\usepackage[compat=1.1.0 ]{tikz-feynman}
%\usepackage{simpler-wick}
\usetikzlibrary{backgrounds}
%\usetikzlibrary{knots}
%this is for the crazier arrows in tikcd (squiggly etc)
%\usetikzlibrary{decorations.pathmorphing}

%this is for smiley and frownie
\usepackage{wasysym}

\graphicspath{{visuals/}}

%ENVIRONMENTS
\usepackage{array}
\newcolumntype{L}{>{$}l<{$}}
\newcolumntype{C}{>{$}c<{$}}
\newcolumntype{R}{>{$}r<{$}}

\newcommand*{\rom}[1]{\expandafter\@slowromancap\romannumeral #1@}
\newcommand{\mathtabular}[1]{\[ \renewcommand{\arraystretch}{1.3} #1 \]}
\newcommand{\cyc}[1]{\setlength\arraycolsep{0.1em}\begin{pmatrix}#1\end{pmatrix}}
\newenvironment{cd}
    {\begin{tikzcd}[column sep=small,row sep=small]}
    {\end{tikzcd}}
\newenvironment{tinycd}
    {\begin{tikzcd}[column sep=tiny,row sep=small]}
    {\end{tikzcd}}
\tikzset{
    rot/.style={rotate=-90}
}
\newcommand{\numbers}{\arabic*}
\newcommand{\abc}{\alph*}
\newcommand{\ABC}{\Alph*}
\newcommand{\iii}{\roman*}
\newcommand{\III}{\Roman*}
\newcommand{\albe}{\greek*}
\newenvironment{enum}
    {\begin{enumerate}}
    {\end{enumerate}}
\newenvironment{qenum}
    {\begin{enumerate}[label=\alph*.]}
    {\end{enumerate}}
\newenvironment{blank_enum}
    {\begin{enumerate}[label={}]}
    {\end{enumerate}}
\newenvironment{Proof}
    {\begin{proof}}
    {\renewcommand{\qedsymbol}{$\blacksquare$}
    \end{proof}
    \renewcommand{\qedsymbol}{$\square$}}
\newenvironment{Matrix}
    {\renewcommand{\arraystretch}{1.3}\begin{pmatrix}}
    {\end{pmatrix}}
\newenvironment{environment}{}{}

\newenvironment{largebox}{\medskip\begin{mdframed}}{\end{mdframed}\medskip}
%workaround which does not require the package 'mdframed' only 'framed'
%\newenvironment{largebox}[1][0.9]{\medskip\begin{minipage}[l]{#1\textwidth}\begin{framed}}{\end{framed}\end{minipage}\medskip}
%\newenvironment{Largebox}{\medskip\begin{oframed}}{\end{oframed}\medskip}

\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother


%environments get:
%	definition
%	crefname
%	numberwithin setting(in cls files)
%	Snippet
\newtheorem{thm}{Theorem}
\Crefname{thm}{Theorem}{Theorems}
\newtheorem{lem}{Lemma}
\Crefname{lem}{Lemma}{Lemmata}
\newtheorem{prop}{Proposition}
\Crefname{prop}{Proposition}{Propositions}
\newtheorem{cor}{Corollary}
\Crefname{cor}{Corollary}{Corollaries}
\newtheorem{con}{Conjecture}
\Crefname{con}{Conjecture}{Conjectures}
\newtheorem{fact}{Fact}
\Crefname{fact}{Fact}{Facts}
\newtheorem{obs}{Observation}
\Crefname{obs}{Observation}{Observations}
%%%
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\Crefname{defn}{Definition}{Definitions}
\newtheorem{exr}{Exercise}
\Crefname{exr}{Exercise}{Exercises}
\newtheorem*{sol}{Solution}
% \Crefname{sol}{Solution}{Solutions}
\newtheorem{exm}{Example}
\Crefname{exm}{Example}{Examples}
\newtheorem{cexm}{Counterexample}
\Crefname{cexm}{Counterexample}{Counterexamples}
\newtheorem{wrn}{Warning}
\Crefname{wrn}{Warning}{Warnings}
\newtheorem{qn}{Question}
\Crefname{qn}{Question}{Questions}
\newtheorem{clm}{Claim}
\Crefname{clm}{Claim}{Claims}
%%%
\theoremstyle{remark}
\newtheorem{rmk}{Remark}
\Crefname{rmk}{Remark}{Remarks}
%%%
\Crefname{axiom}{Axiom}{Axioms}
%%%
%\numberwithin{clm}{thm}
\crefformat{footnote}{#2\footnotemark[#1]#3}

%make arxiv links actually link to the website
\renewcommand{\eprint}[1]{\href{https://arxiv.org/abs/#1}{arXiv:#1}}

%LETTERS

\newcommand{\CP}{\mathbb{CP}}
\newcommand{\RP}{\mathbb{RP}}

\renewcommand{\AA}{\mathbb{A}}
\newcommand{\BB}{\mathbb{B}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\DD}{\mathbb{D}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\GG}{\mathbb{G}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\II}{\mathbb{I}}
\newcommand{\JJ}{\mathbb{J}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\LL}{\mathbb{L}}
\newcommand{\MM}{\mathbb{M}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\renewcommand{\SS}{\mathbb{S}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\UU}{\mathbb{U}}
\newcommand{\VV}{\mathbb{V}}
\newcommand{\WW}{\mathbb{W}}
\newcommand{\XX}{\mathbb{X}}
\newcommand{\YY}{\mathbb{Y}}
\newcommand{\ZZ}{\mathbb{Z}}

\newcommand{\bA}{\mathbf{A}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bC}{\mathbf{C}}
\newcommand{\bD}{\mathbf{D}}
\newcommand{\bE}{\mathbf{E}}
\newcommand{\bF}{\mathbf{F}}
\newcommand{\bG}{\mathbf{G}}
\newcommand{\bH}{\mathbf{H}}
\newcommand{\bI}{\mathbf{I}}
\newcommand{\bJ}{\mathbf{J}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bL}{\mathbf{L}}
\newcommand{\bM}{\mathbf{M}}
\newcommand{\bN}{\mathbf{N}}
\newcommand{\bO}{\mathbf{O}}
\newcommand{\bP}{\mathbf{P}}
\newcommand{\bQ}{\mathbf{Q}}
\newcommand{\bR}{\mathbf{R}}
\newcommand{\bS}{\mathbf{S}}
\newcommand{\bT}{\mathbf{T}}
\newcommand{\bU}{\mathbf{U}}
\newcommand{\bV}{\mathbf{V}}
\newcommand{\bW}{\mathbf{W}}
\newcommand{\bX}{\mathbf{X}}
\newcommand{\bY}{\mathbf{Y}}
\newcommand{\bZ}{\mathbf{Z}}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\cl}{\ell}

\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fb}{\mathfrak{b}}
\newcommand{\fc}{\mathfrak{c}}
\newcommand{\fd}{\mathfrak{d}}
\newcommand{\fe}{\mathfrak{e}}
\newcommand{\ff}{\mathfrak{f}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\fh}{\mathfrak{h}}
% \renewcommand{\fi}{\mathfrak{i}}
\newcommand{\fj}{\mathfrak{j}}
\newcommand{\fk}{\mathfrak{k}}
\newcommand{\fl}{\mathfrak{l}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fn}{\mathfrak{n}}
\newcommand{\fo}{\mathfrak{o}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}}
\newcommand{\fr}{\mathfrak{r}}
\newcommand{\fs}{\mathfrak{s}}
\newcommand{\ft}{\mathfrak{t}}
\newcommand{\fu}{\mathfrak{u}}
\newcommand{\fv}{\mathfrak{v}}
\newcommand{\fw}{\mathfrak{w}}
\newcommand{\fx}{\mathfrak{x}}
\newcommand{\fy}{\mathfrak{y}}
\newcommand{\fz}{\mathfrak{z}}

\newcommand{\ba}{\mathbf{a}}
% \newcommand{\bb}{\mathbf{b}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bd}{\mathbf{d}}
\newcommand{\be}{\mathbf{e}}
% \newcommand{\bf}{\mathbf{f}}
\newcommand{\bg}{\mathbf{g}}
\newcommand{\bh}{\mathbf{h}}
\newcommand{\bi}{\mathbf{i}}
\newcommand{\bj}{\mathbf{j}}
\newcommand{\bk}{\mathbf{k}}
\newcommand{\bl}{\mathbf{l}}
\newcommand{\bm}{\mathbf{m}}
\newcommand{\bn}{\mathbf{n}}
\newcommand{\bo}{\mathbf{o}}
\newcommand{\bp}{\mathbf{p}}
\newcommand{\bq}{\mathbf{q}}
\newcommand{\br}{\mathbf{r}}
\newcommand{\bs}{\mathbf{s}}
\newcommand{\bt}{\mathbf{t}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bw}{\mathbf{w}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bz}{\mathbf{z}}

\renewcommand{\b}{\beta}
\newcommand{\dd}{\delta}
\newcommand{\Del}{\Delta}
\newcommand{\e}{\epsilon}
\newcommand{\g}{\gamma}
\newcommand{\Gam}{\Gamma}
\renewcommand{\k}{\kappa}
\newcommand{\h}{\hbar}
\newcommand{\io}{\iota}
\newcommand{\Lam}{\Lambda}
\newcommand{\lam}{\lambda}
\renewcommand{\l}{\lambda}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\p}{\partial}
\newcommand{\z}{\zeta}
\newcommand{\gam}{\gamma}


% TEXT 
\newcommand{\Fraisse}{Fra\"{i}ss\'{e}\ }
\newcommand{\aff}{\text{aff}}
\newcommand{\DLO}{\text{DLO}}
\newcommand{\alg}{\text{alg}}
\newcommand{\topo}{\text{top}}
\newcommand{\hw}{\text{hw}}
\newcommand{\findim}{\text{fd}}
\newcommand{\splitt}{\text{split}}
\newcommand{\red}{\text{red}}
\newcommand{\eqs}{ \text{eqs}}
\newcommand{\std}{ \text{std}}
\newcommand{\even}{ \text{even}}
\newcommand{\odd}{ \text{odd}}
\newcommand{\solv}{\text{solv}}
\newcommand{\orb}{\text{orb}}
\newcommand{\divis}{\text{div}}
\newcommand{\nil}{\text{nil}}
\newcommand{\ucsf}{\text{ucsf}}
\newcommand{\spin}{\text{spin}}
\newcommand{\finite}{\text{finite}}
\newcommand{\PB}{\text{PB}}
\newcommand{\tot}{\text{tot}}
\newcommand{\pl}{\text{pl}}
\newcommand{\eff}{\text{eff}}
\newcommand{\ym}{\text{ym}}
\newcommand{\tym}{\text{tym}}
\newcommand{\orbits}{\text{orbits}}
\newcommand{\cycl}{\text{cycl}}
\newcommand{\BH}{\text{BH}}
\newcommand{\orbit}{\text{orbit}}
\newcommand{\suchthat}{\;\text{s.t.}\;}
%\newcommand{\st}{\, \vert \,}
\newcommand{\st}{\, \middle\vert \,}
\newcommand{\loc}{\text{loc}}
\newcommand{\inn}{\text{in}}
\newcommand{\out}{\text{out}}
\newcommand{\RG}{\text{RG}}
\newcommand{\op}{\text{op}}
\newcommand{\ini}{\textbf{in}}
\newcommand{\fin}{\textbf{fin}}
\newcommand{\EFT}{\text{EFT}}
\newcommand{\QED}{\text{QED}}
\newcommand{\SM}{\text{SM}}
\newcommand{\FS}{\text{FS}}
\newcommand{\brst}{\text{brst}}
\newcommand{\phys}{\text{physical}}
\newcommand{\closed}{\text{closed}}
\newcommand{\tcl}{\text{cl}}
\newcommand{\exact}{\text{exact}}
\newcommand{\ass}{\text{ass}}
\newcommand{\aux}{\text{aux}}
\newcommand{\GW}{\text{GW}}
\newcommand{\redvir}{\text{red vir}}
\newcommand{\AdS}{\text{AdS}}
\newcommand{\pt}{\text{pt}}
\newcommand{\graded}{\text{grd}}
\newcommand{\preq}{/_{\text{pre}} \, }

%ARROWS
\newcommand{\rarrow}{\rightarrow}
\newcommand{\fromto}{\rightarrow}
\newcommand{\lfromto}{\xrightarrow}
%\newcommand{\to}{\rightarrow}
\newcommand{\lto}{\xrightarrow}
\newcommand{\inj}{\hookrightarrow}
\newcommand{\linj}{\xhookrightarrow}
\newcommand{\surj}{\twoheadrightarrow}
\newcommand{\lsurj}{\xtwoheadrightarrow}
\newcommand{\epi}{\twoheadrightarrow}
\newcommand{\mono}{\hookrightarrow}
\newcommand{\embed}{\hookrightarrow}
\newcommand{\larrow}{\leftarrow}
\newcommand{\tofrom}{\leftarrow}
\newcommand{\ltofrom}{\xleftarrow}
\newcommand{\from}{\leftarrow}
\newcommand{\lfrom}{\xleftarrow}
\newcommand{\invinj}{\hookleftarrow}
\newcommand{\linvinj}{\xhookleftarrow}
\newcommand{\invsurj}{\twoheadleftarrow}
\newcommand{\linvsurj}{\xtwoheadleftarrow}
\newcommand{\bij}{\leftrightarrow}
\newcommand{\lbij}{\xleftrightarrow}
\newcommand{\backto}{ \ \rotatebox[origin=c]{90}{$\circlearrowleft$}\ }
\newcommand{\acts}{ \ \rotatebox[origin=c]{-90}{$\circlearrowright$}\ }
\newcommand{\racts}{\backto}
\newcommand{\acted}{\backto}
\newcommand{\tvi}{\looparrowright}%this is for self transverse immersion (princeton summer 18, Pardon)
\newcommand{\paut}{\dashrightarrow}%partial automorphism (ucb model theory (229) spring 19)
\newcommand{\rinf}{\rightarrow \infty}
\newcommand{\rhup}{\rightharpoonup}
\newcommand{\rhdown}{\rightharpoondown}
\newcommand{\down}{\downarrow}
\newcommand{\up}{\uparrow}
\newcommand{\bleadsto}{\leftrightsquigarrow}

% SYMBOLS
\newcommand{\frowny}{\frownie}
\newcommand{\lineheight}{\baselineskip}
\newcommand{\emp}{\emptyset}
\renewcommand{\phi}{\varphi}
\newcommand{\bphi}{\bar\varphi}
\newcommand{\bpsi}{\bar\psi}
\newcommand{\trans}{\pitchfork}
\newcommand{\minus}{\setminus}
\newcommand{\corr}{\;\, \widehat{ = } \;\,}
\newcommand{\dalem}{\square}
\newcommand{\nab}{\nabla}
\newcommand{\determines}{\vdash}
\newcommand{\satisfies}{\models}
\newcommand{\proves}{\vdash}
\newcommand{\bland}{\bigwedge}
\newcommand{\bands}{\bigwedge}
\newcommand{\ands}{\wedge}
\newcommand{\ors}{\vee}
\newcommand{\bors}{\bigvee}
\newcommand{\blor}{\bigvee}
\newcommand{\comp}{\circ}
\newcommand{\converse}{\Longleftarrow}
\newcommand{\conv}{*}
\newcommand{\btimes}{\bigtimes}
\newcommand{\bdsum}{\bigoplus\limits}
\newcommand{\dsum}{\oplus}
\newcommand{\btp}{\bigotimes}
\newcommand{\tp}{\otimes}
\newcommand{\boxtp}{\boxtimes}
\newcommand{\dun}{\amalg}
\newcommand{\bdun}{\coprod}
\newcommand{\un}{\cup}
\newcommand{\bun}{\bigcup}
\newcommand{\inter}{\cap}
\newcommand{\binter}{\bigcap}
\newcommand{\al}{\alpha}
\newcommand{\compl}{\mathrm{c}}
\newcommand{\clos}[1]{\overline{#1}}
\newcommand{\tclos}[1]{\overline{#1}}
\newcommand{\tclosure}[1]{\overline{#1}}
\newcommand{\closure}[1]{\operatorname{cl}\left(#1\right)}
%\newcommand{\interior}[1]{#1^{\circ}}
\newcommand{\interior}[1]{\operatorname{int}\left(#1\right)}
\newcommand{\dirlim}{\varinjlim}
\newcommand{\invlim}{\varprojlim}
\newcommand{\ceq}{\coloneq}
\newcommand{\ceqq}{\coloneqq}
\newcommand{\eqqc}{\eqqcolon}
\newcommand{\simeqq}{\cong}
\newcommand{\orth}{\perp}
\newcommand{\msin}{\perp}
\newcommand{\act}{\cdot}
\newcommand{\something}{\bullet}
\newcommand{\point}{\bullet}
\newcommand{\somethingg}{*}
\newcommand{\cp}{\amalg}
\newcommand{\trianglerightneq}{\mathrel{\ooalign{\raisebox{-0.5ex}{\reflectbox{\rotatebox{90}{$\nshortmid$}}}\cr$\triangleright$\cr}\mkern-3mu}}
\newcommand{\triangleleftneq}{\mathrel{\reflectbox{$\trianglerightneq$}}}
\newcommand{\order}{\trianglelefteq}
\newcommand{\orderl}{\trianglelefteq}
\newcommand{\sorderl}{\triangleleftneq}
\newcommand{\sorderg}{\trianglerightneq}
\newcommand{\orderr}{\preccurlyeq}
\newcommand{\orderrl}{\preccurlyeq}
\newcommand{\orderrr}{\underline{\lessdot}}
\newcommand{\orderrrl}{\underline{\lessdot}}
\newcommand{\nsup}{\triangleright}
\newcommand{\nsub}{\triangleleft}
\newcommand{\nsubeq}{\ \underline{\triangleleft}\ }
\newcommand{\charsub}{\triangleleft \! \Mapsfromchar}
\newcommand{\charsup}{\Mapstochar \! \triangleright}
\newcommand{\subeq}{\subseteq}
\newcommand{\sub}{\subset}
\newcommand{\subneq}{\subsetneq}
\newcommand{\sups}{\supset}
\newcommand{\supeq}{\supseteq}
\newcommand{\supneq}{\supsetneq}
\newcommand{\cupp}{\smile}
\newcommand{\symd}{\triangle}
 \newcommand{\given}{\vert}
\renewcommand{\div}{\vert}
\newcommand{\ndiv}{\hspace{-0.2ex}\not\hspace{0.6ex}\mid}
\newcommand{\dx}{\,dx\,}
\newcommand{\dy}{\,dy\,}
\newcommand{\dz}{\,dz\,}
%define new symbol for forking independent (model theory)
\newcommand{\forkindep}[1][]{%
  \mathrel{
    \mathop{
      \vcenter{
        \hbox{\oalign{\noalign{\kern-.3ex}\hfil$\vert$\hfil\cr
              \noalign{\kern-.7ex}
              $\smile$\cr\noalign{\kern-.3ex}}}}}\displaylimits_{#1}}}
%\def\forkindep{\mathrel{\raise0.2ex\hbox{\ooalign{\hidewidth$\vert$\hidewidth\cr\raise-0.9ex\hbox{$\smile$}}}}}
\newcommand{\indfrom}{\forkindep}

%COMMAND WITH ARGUMENTS
\renewcommand{\d}[1]{\, d #1 \,}
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\lr}[1]{\left\langle #1 \right\rangle}
\newcommand{\braket}[3]{\left\langle #1 \middle\vert #2 \middle\vert #3 \right\rangle}
\newcommand{\brkt}[2]{\left\langle #1 \vert #2  \right\rangle}
\newcommand{\bra}[1]{\left\langle #1  \right\vert}
\newcommand{\ket}[1]{\left\vert #1  \right\rangle}
\newcommand{\fsl}[1]{\slashed{#1}}
%\newcommand{\fsl}[1]{\ensuremath{\mathrlap{\!\not{\phantom{#1}}}#1}}
\newcommand{\libr}[1]{\left[ #1 \right]}
\newcommand{\cop}[1]{\left[ #1 \right) }
\newcommand{\opc}[1]{\left( #1 \right] }
\newcommand{\restr}[2]{\left.#1 \right\vert_{#2}}
\newcommand{\conj}[1]{#1^*}
\newcommand{\leftdual}[1]{ \, ^\vee #1 }
\newcommand{\rightdual}[1]{ #1^\vee }
\newcommand{\rdual}{\rightdual}
\newcommand{\dual}[1]{ #1^* }
\newcommand{\hodge}[1]{\star #1}
\newcommand{\ddual}[1]{ #1^{**} }
\newcommand{\tpose}[1]{#1^{T}}
\newcommand{\tens}[2]{\tensor{#1}{#2}}
\newcommand{\ttilde}[1]{\widetilde{#1}}
\newcommand{\uc}[1]{\widetilde{#1}}
\newcommand{\barr}[1]{\overline{#1}}
\newcommand{\hatt}[1]{\widehat{#1}}
\newcommand{\shff}[1]{#1^{\textrm{Sh}}}
\newcommand{\squo}[2]{#1/\!\!/#2}
\newcommand{\multunder}[1]{\substack{#1}}
\newcommand{\mcal}[1]{\mathcal{#1}}
\newcommand{\mfrak}[1]{\mathfrak{#1}}
\newcommand{\bb}[1]{\mathbb{#1}}
\newcommand{\rone}{\text{\rom{1}}}
\newcommand{\rtwo}{\text{\rom{2}}}
\newcommand{\rthree}{\text{\rom{3}}}
\newcommand{\rfour}{\text{\rom{4}}}
\newcommand{\one}{\mathbf{1}}
%\newcommand{\one}{\mathds{1}}
%\newcommand{\two}{\mathds{2}}
%\newcommand{\mb}[1]{\mathbf{#1}}
\newcommand{\mb}{\boldsymbol}
\newcommand{\mscr}[1]{\mathscr{#1}}
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{\node[shape=circle,draw,inner sep=2pt](char){#1};}}
\DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
\DeclareFontShape{U}{mathx}{m}{n}{
      <5> <6> <7> <8> <9> <10>
      <10.95> <12> <14.4> <17.28> <20.74> <24.88>
      mathx10
      }{}
\DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
\DeclareFontSubstitution{U}{mathx}{m}{n}
\DeclareMathAccent{\widecheck}{0}{mathx}{"71}
\DeclareMathAccent{\wideparen}{0}{mathx}{"75}
%\newcommand{\checkk}[1]{\widecheck{#1}}
\newcommand{\checkk}[1]{\check #1}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\under}[1]{\underline{#1}}
\newcommand{\overl}[1]{\overline{#1}}
\newcommand{\oline}[1]{\overline{#1}}
\newcommand{\modp}[1]{ \, \left(\mathrm{mod} \, #1\right) \, }
\renewcommand{\mod}[1]{\, \mathrm{mod} \, #1 \,}
\newcommand{\cmod}[2]{ \, \equiv \, #1 \, \left(\mathrm{mod} \, #2\right)\,}
\newcommand{\ubr}[2]{\underbrace{#1}_{#2}}
\newcommand{\obr}[2]{\overbrace{#1}^{#2}}
\newcommand{\units}[1]{#1^\times}
\newcommand{\ps}[1]{\left\llbracket #1 \right\rrbracket}
\newcommand{\ls}[1]{\left(\!\left( #1 \right)\!\right)}
\newcommand{\tv}[1]{\left\llbracket #1 \right\rrbracket}
\newcommand{\ext}{\wedge}
\newcommand{\bigext}{\bigwedge}
\newcommand{\biset}[3]{{}_{#2} {#1}_{#3}}
\newcommand{\dblcs}[3]{#2 \backslash #1  / #3}
\newcommand{\rightcs}[2]{#2 \backslash #1}
\newcommand{\leftorbits}[2]{\,_{#2}#1}
\newcommand{\gcl}[2]{\, ^{#2}#1}
\newcommand{\gcr}[2]{#1^{#2}}
\newcommand{\anti}[1]{{#1}^*}
\newcommand{\gnum}[1]{\text{gh}\#\left(#1\right)}
\newcommand{\agnum}[1]{\text{anti-gh}\#\left(#1\right)}
\newcommand{\consh}[1]{\underline{\underline{#1}}}
\newcommand{\dunder}[1]{\underline{\underline{#1}}}
\newcommand{\lmod}[2]{^{#2}#1}
\newcommand{\dmod}[2]{^{#2}#1}
\newcommand{\amod}[2]{#1_{#2}}
\newcommand{\damod}[3]{^{#2}{#1}_{#3}}
\newcommand{\rad}[1]{\sqrt{#1}}
%\newcommand{\rad}{\sqrt{\vphantom{x}}}

%CATEGORIES
\newcommand{\cat}{\mathbf}
\newcommand{\Rep}{\cat{Rep}}
\newcommand{\Sh}{\cat{Sh}}
\newcommand{\PreSh}{\cat{PreSh}}
\newcommand{\Grp}{\cat{Grp}}
\newcommand{\TopGrp}{\cat{TopGrp}}
\newcommand{\Mfd}{\cat{Mfd}}
\newcommand{\SymplMfd}{\cat{Sympl Mfd}}
\newcommand{\Mon}{\cat{Mon}}
\newcommand{\Cat}{\cat{Cat}}
\newcommand{\Ab}{\cat{Ab}}
\newcommand{\Fr}{\cat{Fr}}
\newcommand{\Set}{\cat{Set}} 
\newcommand{\Pos}{\cat{Pos}}
\newcommand{\Pre}{\cat{Pre}}
\newcommand{\Rg}{\cat{Rg}}
\newcommand{\Rig}{\cat{Rig}}
\newcommand{\Rng}{\cat{Rng}}
\newcommand{\Ring}{\cat{Ring}}
\newcommand{\Crg}{\cat{Crg}}
\newcommand{\Crig}{\cat{Crig}}
\newcommand{\Crng}{\cat{Crng}}
\newcommand{\Cring}{\cat{Cring}}
\newcommand{\Semiring}{\cat{Semiring}}
\newcommand{\Root}{\cat{Root}}
\newcommand{\Semi}{\cat{Semi}}
\newcommand{\Open}{\cat{Open}}
\newcommand{\Simp}{\cat{Simp}}
\newcommand{\SN}{\cat{SN}}
\newcommand{\Top}{\cat{Top}}
\newcommand{\TVS}{\cat{TVS}}
\newcommand{\ExtGr}{\cat{ExtGr}}
\newcommand{\LocSys}{\cat{LocSys}}
\newcommand{\Uni}{\cat{Uni}}
\newcommand{\Vect}{\cat{Vect}}
\newcommand{\FVect}{\cat{FDVect}}
\newcommand{\GVect}{\cat{GrdVect}}
\newcommand{\GFVect}{\cat{GrdFDVect}}
\newcommand{\DAMod}[2]{^{#1}\cat{Mod}_{#2}}
\newcommand{\Rmod}{R\text{-}\cat{Mod}}
\newcommand{\dMod}[1]{^{#1}\cat{Mod}}
\newcommand{\aMod}[1]{\cat{Mod}_{#1}}
\newcommand{\lMod}[1]{#1\text{-}\cat{Mod}}
\newcommand{\rMod}[1]{\cat{Mod}\text{-}#1}
\newcommand{\Mod}[2]{#1\text{-}\cat{Mod}\text{-}#2}
\newcommand{\Module}{\cat{Mod}}
\newcommand{\Alg}{\cat{Alg}}
\newcommand{\lAlg}[1]{#1\text{-}\cat{Alg}}
\newcommand{\rAlg}[1]{\cat{Alg}\text{-}#1}
\newcommand{\Gset}{G\text{-} \cat{set}}
\newcommand{\Biset}[2]{\cat{#1}\text{-}\Set\text{-}\cat{#2}}
\newcommand{\lbis}[1]{#1\text{-}\Set}
\newcommand{\rbis}[1]{\Set\text{-}#1}
\newcommand{\kcoalg}[1]{ k \text{-}\cat{Coalg} }
\newcommand{\Chain}[1]{\cat{Ch\left( #1 \right)}}
\newcommand{\Ch}{\cat{Ch}}
\newcommand{\Coalg}{\cat{Coalg}}
\newcommand{\Bialg}{\cat{Bialg}}
\newcommand{\Hopf}{\cat{Hopf}}
\newcommand{\AlgVar}{\cat{Alg Var}}
\newcommand{\Lie}{\cat{Lie}}
\newcommand{\LieAlg}{\cat{Lie}\text{-}\cat{Alg}}
\newcommand{\LieGp}{\cat{Lie}\text{-}\cat{Gp}}
\newcommand{\Lrsp}{\cat{LRSp}}
\newcommand{\LRSp}{\cat{LRSp}}
\newcommand{\Affsch}{\cat{AffSch}}
\newcommand{\AffSch}{\cat{AffSch}}
\newcommand{\Sch}{\cat{Sch}}
\newcommand{\Schemes}{\cat{Sch}}
\newcommand{\Coh}{\cat{Coh}}
\newcommand{\Qco}{\cat{QCoh}}
\newcommand{\QCoh}{\cat{QCoh}}
\newcommand{\Qcoh}{\cat{QCoh}}
\newcommand{\Functors}{\cat{Functors}}
\newcommand{\Jac}{\cat{Jac}}

%OPERATORS
\DeclareMathOperator{\const}{const}
\DeclareMathOperator{\Lan}{Lan}
\DeclareMathOperator{\Ran}{Ran}
\DeclareMathOperator{\items}{Item}
\DeclareMathOperator{\titems}{t-Item}
\DeclareMathOperator{\crit}{crit}
\DeclareMathOperator{\hlim}{hlim}
\DeclareMathOperator{\hcolim}{hcolim}
\let\Bar\undefined
\DeclareMathOperator{\Bar}{Bar}
\DeclareMathOperator{\wind}{wind}
\DeclareMathOperator{\algcl}{acl}
\DeclareMathOperator{\acl}{acl}
\DeclareMathOperator{\dcl}{dcl}
\DeclareMathOperator{\Forb}{Forb}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\FT}{FT}
\DeclareMathOperator{\sh}{sh}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\Skel}{Skel}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Sob}{Sob}
\DeclareMathOperator{\Derivations}{Der}
\DeclareMathOperator{\Ad}{Ad}
\DeclareMathOperator{\codim}{Codim}
\DeclareMathOperator{\Spin}{Spin}
\DeclareMathOperator{\component}{comp}
\DeclareMathOperator{\Supp}{Supp}
\DeclareMathOperator{\centerr}{center}
\DeclareMathOperator{\Pin}{Pin}
\DeclareMathOperator{\Tw}{Tw}
\DeclareMathOperator{\Fuk}{Fuk}
\DeclareMathOperator{\ev}{ev}
\DeclareMathOperator{\Slope}{Sl}
\DeclareMathOperator{\grad}{grad}
\DeclareMathOperator{\nd}{nd}
\DeclareMathOperator{\czind}{cz}
\DeclareMathOperator{\PD}{PD}
\DeclareMathOperator{\SW}{SW}
%Symplectic homology
\DeclareMathOperator{\SH}{SH}
\DeclareMathOperator{\HFK}{HFK}
\DeclareMathOperator{\DFK}{DFK}
\DeclareMathOperator{\AFK}{AFK}
\DeclareMathOperator{\DAFK}{DAFK}
\DeclareMathOperator{\CFK}{CFK}
%Floer 
\let\HF\undefined
\DeclareMathOperator{\HF}{HF}
\DeclareMathOperator{\CF}{CF}
%Floer bimodules
\DeclareMathOperator{\CFD}{CFD}
\DeclareMathOperator{\CFDA}{CFDA}
%Wrapped
\DeclareMathOperator{\CW}{CW}
\DeclareMathOperator{\HW}{HW}
%Monopole
\DeclareMathOperator{\HM}{HM}
\DeclareMathOperator{\CM}{CM}
%Khovanov 
\DeclareMathOperator{\CKh}{CKh}
%cyclidrical contact 
\DeclareMathOperator{\CH}{CH}
\DeclareMathOperator{\cyC}{CC}
%
\DeclareMathOperator{\inv}{inv}
\DeclareMathOperator{\Foc}{Foc}
\DeclareMathOperator{\Fus}{Fus}
\DeclareMathOperator{\Fun}{Fun}
\DeclareMathOperator{\Cts}{Cts}
\DeclareMathOperator{\Tot}{Tot}
\DeclareMathOperator{\cone}{cone}
\DeclareMathOperator{\Fit}{Fit}
\DeclareMathOperator{\HB}{HB}
\DeclareMathOperator{\St}{St}
\DeclareMathOperator{\Bl}{Bl}
\DeclareMathOperator{\Map}{Map}
\DeclareMathOperator{\Maps}{Maps}
\DeclareMathOperator{\Hol}{Hol}
\DeclareMathOperator{\Sect}{Sect}
\DeclareMathOperator{\Solv}{Solv}
\DeclareMathOperator{\coinv}{coinv}
\DeclareMathOperator{\rep}{Rep}
\DeclareMathOperator{\height}{ht}
\DeclareMathOperator{\Diff}{Diff}
\DeclareMathOperator{\Diffeo}{Diffeo}
\DeclareMathOperator{\vect}{Vect}
\DeclareMathOperator{\Split}{Split}
\DeclareMathOperator{\Tors}{Tors}
\DeclareMathOperator{\val}{val}
\DeclareMathOperator{\Dehn}{Dehn}
\DeclareMathOperator{\Fl}{Fl}
\DeclareMathOperator{\LGr}{LGr}
\DeclareMathOperator{\Transversals}{Transversals}
\DeclareMathOperator{\Forget}{Forget}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\Syl}{Syl}
\DeclareMathOperator{\Symp}{Symp}
\DeclareMathOperator{\Ham}{Ham}
\DeclareMathOperator{\res}{res}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Fix}{Fix}
\DeclareMathOperator{\Nat}{Nat}
\DeclareMathOperator{\Mor}{Mor}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Out}{Out}
\DeclareMathOperator{\Inn}{Inn}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\Coind}{Coind}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Ass}{Ass}
\DeclareMathOperator{\Inf}{Inf}
\DeclareMathOperator{\Def}{Def}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\Iso}{Iso}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\mSpec}{m-Spec}
\DeclareMathOperator{\Obj}{Obj}
\DeclareMathOperator{\Perm}{Perm}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Alt}{Alt}
\DeclareMathOperator{\Irr}{Irr}
\DeclareMathOperator{\Sgr}{Sgr}
\DeclareMathOperator{\Free}{Free}
\DeclareMathOperator{\type}{tp}
\DeclareMathOperator{\cliff}{Cliff}
\DeclareMathOperator{\Cl}{Cl}
\DeclareMathOperator{\Op}{Op}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\discr}{discr}
\DeclareMathOperator{\crossing}{cr}
\DeclareMathOperator{\weight}{weight}
\DeclareMathOperator{\circles}{circ}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\coim}{coim}
\DeclareMathOperator{\domain}{dom}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\cont}{c}
\DeclareMathOperator{\EL}{EL}
\DeclareMathOperator{\Ball}{B}
\DeclareMathOperator{\colim}{colim}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\Dy}{Dy}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\acc}{acc}
\DeclareMathOperator{\specf}{sf}
\DeclareMathOperator{\Vol}{Vol}
\DeclareMathOperator{\Area}{area}
\DeclareMathOperator{\area}{area}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\theory}{Th}
\DeclareMathOperator{\Ima}{Im}
\let\Im\undefined
\DeclareMathOperator{\Im}{Im}
\DeclareMathOperator{\Rea}{Re}
\DeclareMathOperator*{\Residue}{Res}
\DeclareMathOperator{\ch}{ch}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\meas}{m}
\DeclareMathOperator{\ess}{ess}
\DeclareMathOperator{\chara}{char}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\PU}{PU}
\DeclareMathOperator{\PGL}{PGL}
\newcommand{\gl}{\mfrak{gl}}
\DeclareMathOperator{\PSL}{PSL}
\let\O\undefined
\DeclareMathOperator{\O}{O}
\renewcommand{\o}{\mfrak{o}}
\DeclareMathOperator{\SL}{SL}
\renewcommand{\sl}{\mfrak{sl}}
\let\SO\undefined
\DeclareMathOperator{\SO}{SO}
\newcommand{\so}{\mfrak{so}}
\DeclareMathOperator{\Sp}{Sp}
\renewcommand{\sp}{\mfrak{sp}}
\DeclareMathOperator{\SU}{SU}
\newcommand{\su}{\mfrak{su}}
\DeclareMathOperator{\U}{U}
\let\uu\u
\renewcommand{\u}{\mfrak{u}}
\newcommand{\lienil}{\mfrak{nil}}
\newcommand{\lierad}{\mfrak{rad}}

