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\usepackage{indentfirst}
\usepackage{amsmath}
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\usepackage{calc}
\usepackage{amssymb}
%\usepackage{mathptm}
\usepackage{amsthm}
%\usepackage{concrete}


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%2005 MACROS

\mathversion{bold}

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%FROM texpower slides


%%%%%%%%%%%%%%%%%%%

\def\ot{ {\tiny\otimes} }

\def\Rax{\mathbb R\langle a,x\rangle}


\def\Mng{(\mathbb R^{n \times n})^g}

\def\Sng{(\mathbb {SR}^{n \times n })^g }

%\def\Sngg{\mathbb S_n(\mathbb R^g \times \mathbb R^g)}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\vb}{\vskip .15cm}


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
%BILLs classical macros

\newcommand{\matX}{{\color{red}X}}
\newcommand{\matY}{{\color{red}Y}}
\newcommand{\RS}{\mathbb{S}}



\def\uu{{\color{red}{\upA \unku}}}


\def\uy{{\color{red}{\unky}}}
%\def\uy{{\color{red}{\upA \unky}}}
%\def\ux{{\color{red}{\upA \unkx}}}
\def\uz{{\color{red}{\upA \unkz}}}


\def\uU{{\color{red}{\upA \unkU}}}
\def\uY{{\color{red}{\upA \unkY}}}

\def\uX{{\color{red}{\upA \unkX}}}
\def\uZ{{\color{red}{\upA \unkZ}}}
\def\unku{{\color{red}{u}}}
\def\unkx{{\color{red}{x}}}
\def\unkz{{\color{red}{z}}}
\def\unkU{{\color{red}{U}}}

\newcommand{\unky}{{\textcolor{red}{y}}}

   \def\unkxj{{\color{red}{x_j}}}
\def\unkxone{{\color{red}{x_1}}}
\def\unkxtwo{{\color{red}{x_2}}}
\def\unkxthree{{\color{red}{x_3}}}
\def\unkxg{{\color{red}{x_g}}}
\def\unkxm{{\color{red}{x_g}}}

\def\XXj{{\color{red}{X_j}}}
\def\unkXone{{\color{red}{X_1}}}
\def\unkXtwo{{\color{red}{X_2}}}
\def\unkXthree{{\color{red}{X_3}}}

\def\unkXm{{\color{red}{X_m}}}
\def\unkT{{\color{red}{T}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\newcommand{\CR}{ \textcolor{red}}
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}
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\mathversion{bold}

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\renewcommand{\SS}{\scriptscriptstyle}
\newcommand{\rank}{\mathop{rank}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%2005
\def\gtupn{\mathbb{SR}^{n
\times n}}
\def\DOMp{\mathcal{D}_p}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\def\borCo{L}
\def\SID{{Symbolic Inequality Domain}}
\def\POD{\succeq}
\def\upA{\stackrel{\rightarrow} }
\def\cQ{{\cal Q}}

\def\cA{{\cal A}}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\cH{{\cal H}}
\def\sidG{{\cal G}}
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\def\cMGA{{\cal G}}
\def\cMGM{{\cal MG}}
\def\cMG{{\cal MG}}
\def\cG{{\cal G}}
\def\cL{{\cal L}}
\def\cS{{\cal S}}
\def\cU{\mathcal U}
\def\cV{\mathcal V}
\def\uM{{\upA M}}
\def\uZ{{\color{blue}{\upA \miZ}}}
\def\uH{{\color{blue}{\upA \updH}}}

\def\ux{{\color{red}{\upA \unkx}}}
\def\uX{{\color{red}{\upA \unkX}}}
\def\unkx{{\color{red}{x}}}
\newcommand{\unkX}{{\textcolor{red}{X}}}
   
   \iffalse
     \def\unkxj{{\color{red}{x_j}}}
\def\unkxone{{\color{red}{x_1}}}
\def\unkxtwo{{\color{red}{x_2}}}
\def\unkxthree{{\color{red}{x_3}}}
\def\unkxm{{\color{red}{x_m}}}

\def\unkXj{{\color{red}{X_j}}}
\def\unkXone{{\color{red}{X_1}}}
\def\unkXtwo{{\color{red}{X_2}}}
\def\unkXm{{\color{red}{X_m}}}
\def\unkT{{\color{red}{T}}}
\fi

\def\uMtil{{\upA{\tilde{M}}}}
\def\bLM{{\stackrel{\rightarrow}\borCo(\upA M)}}
\def\bL{{\stackrel{\rightarrow}\borCo}}
%\newcommand{\hess}[2]{{\mathcal{H}\Gamma(#1)[#2]}}
\newcommand{\hess}[2]{{\mathcal{H}p(#1)[#2]}}
\newcommand{\Q}{{\cal Q}}
\newcommand{\hra}{\hookrightarrow}
\newcommand{\LDL}{\ensuremath{\mathrm{LDL^T}}}
\newcommand{\LDU}{\ensuremath{\mathrm{LDU}}}
\newtheorem{thm}{Theorem}

\newcommand{\uA}{\textcolor{green}{ \upA A}}
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\newcommand{\updHH}{{\textcolor{blue}{H_1}}}
\newcommand{\updK}{{\textcolor{blue}{H_2}}}
\newcommand{\miZ}{{\textcolor{blue}{Z}}}
\newcommand{\Find}{{\textcolor{red}{Find}}}
\newcommand{\CMxIs}{{\textcolor{red}{CONVEX MATRIX INEQUALITIES}}}
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\newcommand{\efuY}{\textcolor{red}{\ESF{ \upA  Y}}}
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\newcommand{\efuZ}{\textcolor{blue}{\ESF{ \upA  Z}}}
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\newcommand{\efXj}{\textcolor{red}{\ESF{X}_j}}


\def\DEF{ \bf}
\def\cCp{ {\cal C}_p}

\def\demi{\frac{1}{2} }

\def\bbR{ {\bf R}}
\def\bR2{ {{\bf R}^2} }
\def\mRrr{ $\bbR^{r \times r}$ }
\def\mSRrr{ $ S \bbR^{r \times r}$ }

%\def\updh{\CB{h}}
\def\updH{\CB{H}}


\def\efupdH{{\color{MyGreen}{\upA {H} }}}
\def\uAll{{A_1, A_2 ,\unkX, \unkY}}
\def\uAllX{{A_1, A_2 ,\unkXone, \unkXtwo}}

\def\unkcL{\color{blue}{\cal L} }
\def\udelta{\color{orange}{\delta}}

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% Definition of new colors
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\def\smthm{ {\tiny\color{red} THM \color{black} }}


\newcommand{\updk}{{\textcolor{blue}{k}}}
\newcommand{\updh}{{\textcolor{blue}{h}}}

%-----------------------------------------


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\newcommand{\RR}{\mathbb{R}}
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\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}

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\newcommand{\nn}{\nonumber}
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\newcommand{\vareps}{\varepsilon}
\newcommand{\dt}{\delta}
\newcommand{\Dt}{\Delta}
\def\af{\alpha}
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\def\ss{\smallskip}


\def\tM{\tilde M}
\def\rs{\CR{s}}
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\def\iq{\indent \quad}



\def\tg{{\tilde g}}
\def\rx{\CR{x}}

\def\MC{M(\mathbb C)}
\def\matg{\MC^g}

\def\math{\MC^{\tg} }

\def\matnh{matnh??}
\def\matng{matng??}
\def\fn{f[n]}

%-----------------------------------

% FOR THESE SLIDES ONLY -OVERWRITES
% MIT

\def\ux{{\color{red}{ \unkx}}}
\def\uX{{\color{red}{ \unkX}}}
\def\unkx{{\color{red}{x}}}

%\newcommand{\unkX}{{\textcolor{red}{X}}}
     \iffalse
     \def\unkxj{{\color{red}{x_j}}}
\def\unkxone{{\color{red}{x_1}}}
\def\unkxtwo{{\color{red}{x_2}}}
\def\unkxthree{{\color{red}{x_3}}}
\def\unkxm{{\color{red}{x_m}}}
\fi
\def\unkXj{{\color{red}{X_j}}}
\def\unkXone{{\color{red}{X_1}}}
\def\unkXtwo{{\color{red}{X_2}}}

\def\unkXm{{\color{red}{X_g}}}

\def\unkT{{\color{red}{T}}}

\def\ft{\frametitle}



\begin{document}

\bfseries



%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pagecolor{MyGray}

%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%

\begin{frame}
    %\addtocounter{slide}{-1}
   

\centerline{\large
{ \color{blue} Linear Matrix Inequalities vs Convex Sets } }
\bigskip


\centerline{Harry Dym \quad \quad Weitzman Inst.  }

\centerline{Damon Hay \quad \quad Florida $\to$ Texas  }

\centerline{Igor Klep \quad \quad Everywhere in Solvenia }

\centerline{Scott McCullough \quad \quad University of Florida }

%\centerline{Jiawang Nie  \quad \quad UCSD, Cal Tech}

\centerline{Victor Vinnikov \quad \quad Ben Gurion U of the Negev}


\bigskip {\small
\centerline{I am Bill Helton \quad \quad  UCSD } }


\bigskip
\bigskip
{\small \CB{Advertisement:}}  \ \ \ \  Try noncommutative computation
 \centerline{ \color{red}{ NCAlgebra}\footnote{ \color{red}{
Helton, deOliveira (UCSD), Stankus (CalPoly SanLObispo ), Miller}}
\color{black} }
\centerline{ \color{red}{ NCSoSTools}\footnote{ \color{red}{
Igor Klep}}
\color{black} }
\end{frame}



%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Outline}

\tableofcontents
\end{frame}

\section{Semidefinite Programming and LMIs}

\begin{frame}
\frametitle{SDP and LMIs}


A \CB{linear pencil}  is a matrix valued function $L$ of the form
$$ L(\ux): = L_0 +  L_1  \unkxone +  \cdots  +  L_g \unkxg, $$
where
$L_0, L_1, L_2, \cdots, L_g$ are
symmetric matrices and
 $\ux:= \{ \unkxone, \cdots, \unkxm \}$ are
$m$ real parameters.

\ms 
A \CB{Linear Matrix Inequality (LMI)}  is one of the form:
$$ L(\ux) \succeq 0.$$
The set of solutions
$${\mathcal G}: =
\{ (\unkxone, \unkxtwo, \cdots, \unkxm): L_0 +  L_1  \unkxone +
\cdots  +  L_g \unkxm \quad \mbox{is PosSD}\}$$
is a convex set.
Solutions can be found numerically for problems of modest
size. This is called

\centerline{\CB{Semidefinite Programming \ \ \ SDP}}

Notation: \
a {\bf monic}  LMI  is one with $L_0 = I$.



\end{frame}

\begin{frame}[t]
\frametitle{SDP is Everywhere}

\CR{Main development in convex programming in the last 20 years.}

$\bullet$ Many problems from many areas can be converted to SDPs.

$\CB \bullet$ Max -Cut, etc, etc   is approximable by SDP
%\CR{(big now combinatorics)}

$\CR \bullet$ Linear systems theory,  classic problems
convert to Linear Matrix Inequalities LMIs.
Compromises opens many uses of SDP

$\CR \bullet$  Computational Real Algebraic Geometry RAG \\
 (RAG is critical in this talk)

  $\CG \bullet$ Lyapunov functions  for  Nonlin Sys.
 \qquad
  Statistics

 $\CG \bullet$  Types of Chemical Problems
\qquad
  \CR{Many others}
 \\

\bigskip
 \  See Nemirovski's Plenary ICM address 2006

 "Semidefinite Programming" gets about \CB{70,000 hits on google}\\
  %in June 2010
"Symmetric Matrix"
gets about \CB{300,000 hits on google}

\bigskip

In \CR{linear engineering systems} a special class of SDP is fundamental:
\ \
SDP with \CR{matrix (not scalar) variables}.

\end{frame}
%%%%%%


\input{sliLMIrepVVPartB.tex}
%%%%%%%%%%
%%%%%%



\begin{frame}
\ft{NC (free) LMI}

\CB{LMI in Matrix Variables}
\bs

\CR{OUTLINE}
\tableofcontents
\end{frame}
%%%%%%%%%%%%


%%%%%%%%
\section{Linear Systems give nc poly inequalities}
%%%%%%%%%%

%\input{ncDimlessPartB.tex}
%%%%%%%%


\begin{frame}
\frametitle{\vb Systems problems $\rightarrow $ Matrix Ineq}

\vspace*{.2cm}
\begin{center}
\unitlength=0.1mm
\begin{picture}(0,0)(200,90)
\put(100,100){\framebox(160,85){}} \put( 60,160){\vector( 1, 0){
40}} \put( 60,120){\vector( 1, 0){ 40}}
\put(175,145){\makebox(0,0){{  { Given}}}} \put(260,160){\vector(
1, 0){ 40}} \put(260,120){\vector( 1, 0){ 40}}
\put(135,35){\framebox(100,50){}} \put(60,60){\line(0,1){60}}
\put(300,60){\line(0,1){60}} \put(300,60){\vector(-1,0){65}}
\put(135,60){\vector(-1,0){75}} \put(180,60){\makebox(0,0){{
{\Find}}}}
\end{picture}
\end{center}
\vspace{.4cm}

Many such problems Eg. $H^\infty$ control


%%%%%%%%%%%%%%%
%\iffalse  OUT MTNS

\begin{tabbing}
Freq. domain \quad \= Nev-Pick ~~~ \= \kill
Freq. domain \> \textcolor{green}{Nev-Pick} \> \quad 1975-85 \\
State space \> \textcolor{green}{tricks} \> Riccatis 1985-95\\
Matrix ineqs \> \textcolor{green}{tricks} \> LMIs \quad  1990 -
\end{tabbing}


Solve numerical LMIs, { \small up to  $50 \times 50$
(unstructured) matrices},
via semidef prog., %2nd order cone prog.,
bundle methods, etc.  \ \
$$
%\begin{equation*}
\left.
\begin{array}{lcr}
F_1(\CR{X}) := A  \CR{X} + \CR{X} A^T + BB^T & \succeq & 0 \\
F_2(\CR{X}, \CR{Y} ) & \succeq & 0
\end{array} \right\} \qquad
\text{PosSD}
%\end{equation*}
$$

%%%%%%%%%%%%%%%%%%%%%%%

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{MATRIX INEQUALITIES }

    Polynomial or Rational function of matrices are PosSDef.

%\medskip

%IN MTNS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\fi
%
{\small{ Example:\  Get Riccati expressions like
$$
 A \unkX + \unkX A^T -  \unkX BB^T \unkX  + CC^T \succ 0
$$
OR Linear Matrix Inequalities (LMI) like
$$
\left(
\begin{array}{cc}
A \unkX + \unkX A^T + C^T C &  \unkX B\\
B ^T\unkX & I
\end{array}\right) \succ 0
$$
which is equivalent to the Riccati inequality. }}

\tiny{\CG{ncDimlessPartB}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\adg{{ \bf \color{red} a \color{black}}}
\def\bdg{{ \bf \color{red} b \color{black}}}
\def\cdg{{ \bf \color{red} c \color{black}}}
\def\bQ{{\color{blue} Q \color{black}}}
\def\Xdg{ {\bf \color{blue} X \color{black}}}
\def\Ydg{{ \bf \color{blue} Y \color{black}}}
\def\Edg{{ \bf \color{blue} E \color{black}}}
\def\EOneOne{{ \bf \color{blue} E_{11} \color{black}}}
\def\EOneTwo{{ \bf \color{blue} E_{12} \color{black}}}
\def\ETwoOne{{ \bf \color{blue} E_{21} \color{black}}}
\def\ETwoTwo{{ \bf \color{blue} E_{22} \color{black}}}
\def\Mdg{{ \bf \color{blue} M \color{black}}}
\def\Zdg{{ \bf \color{blue} Z\color{black}}}
\def\Qdg{{ \bf \color{blue} Q\color{black}}}
\def\ydg{{ \bf \color{blue} y \color{black}}}


\begin{frame}

\centerline{\Large \bf $H^\infty$ Control}

\bigskip

{\bf ALGEBRA PROBLEM: }\\Given the polynomials:

{\small

$H_{xx} =
\EOneOne \, A + A^T \, \EOneOne + C_1^T \, C_1 + \EOneTwo^T
\,
  \bdg \, C_2 + C_2^T \, \bdg^T \, \EOneTwo^T
   + \EOneOne \, B_1 \, \bdg^T \, \EOneTwo^T +
   \EOneOne \, B_1 \, B_1^T \, \EOneOne +
\EOneTwo \, \bdg \, \bdg^T \, \EOneTwo^T +
   \EOneTwo \, \bdg \, B_1^T \, \EOneOne
$

$H_{xz}=\ETwoOne \, A + \frac{a^T \, (\ETwoOne + \EOneTwo^T)}{2} +
  \cdg^T \, C_1 + \ETwoTwo \, \bdg \, C_2 +
   \cdg^T \, B_2^T \,
  \EOneOne^T +
\frac{\ETwoOne \, B_1 \, \bdg^T\, (\ETwoOne + \EOneTwo^T)}{2} +
   \ETwoOne \,
  B_1 \, B_1^T \, \EOneOne^T +
\frac{\ETwoTwo \, \bdg \, \bdg^T \, (\ETwoOne + \EOneTwo^T)}{2} +
   \ETwoTwo \, \bdg \, B_1^T \, \EOneOne^T $

$H_{zx}=A^T \, \ETwoOne^T + C_1^T \, \cdg + \frac{(\EOneTwo +
\ETwoOne^T) \, \adg}{2} +
  \EOneOne \,
B_2 \, \cdg +
   C_2^T \, \bdg^T \, \ETwoTwo^T +
  \EOneOne \, B_1 \, \bdg^T \, \ETwoTwo^T +
  \EOneOne \, B_1 \, B_1^T \, \ETwoOne^T +
   \frac{(\EOneTwo + \ETwoOne^T) \, \bdg \, \bdg^T \, \ETwoTwo^T}{2} +
   \frac{(\EOneTwo + \ETwoOne^T) \, \bdg \, B_1^T \, \ETwoOne^T}{2}$

$H_{zz}=\ETwoTwo \, \adg + \adg^T \, \ETwoTwo^T + \cdg^T \, \cdg +
\ETwoOne
\, B_2
\, \cdg +
   \cdg^T \, B_2^T \, \ETwoOne^T +
\ETwoOne \, B_1 \, \bdg^T \, \ETwoTwo^T + \ETwoOne \, B_1 \, B_1^T \,
\ETwoOne^T +
  \ETwoTwo \, \bdg \, \bdg^T \, \ETwoTwo^T + \ETwoTwo \, \bdg \, B_1^T \,
\ETwoOne^T
$
  }
\bigskip

{
{\bf (PROB) }
$A$, $B_1$, $B_2$, $C_1$, $C_2$ are knowns.

Solve the inequality
$\left(%
\begin{array}{cc}
  H_{xx} & H_{xz} \\
  H_{zx} &  H_{zz} \\
\end{array}%
\right) \preceq 0
$
for unknowns \\
\\
\centerline{
$\adg$, $\bdg$, $\cdg$ and  for $\EOneOne$, $\EOneTwo$, $\ETwoOne$ and
$\ETwoTwo$ }
}
\end{frame}

%%%%%%

\begin{frame}
%\frametitle{ENGINEERING MOTIVATION}

\bcen

{\huge{ \CB{Engineering problems  defined entirely by signal flow diagrams
and $L^2$ performance specs \\
\CR{are equivalent to} \\ Polynomial Matrix Inequalities}}
}
\ecen

\bigskip

How and why is a long story but the 
correspondence
\footnote{Ask after the talk if you would like more detail.}
 between
linear systems and  noncommutative algebra is on \CB{the next slides}:
  %On Separate slides to convince one of 
  %inherent messiness engineers face\\
 %(some algebraic algorithms would help)

\end{frame}
%%%%%%

\def\rX{ \CR{X} }
%%%%%%
\begin{frame}
\frametitle{Notation}

$\unkx= (\rx_1, \cdots, \rx_g)$ algebraic noncommuting variables
\bs

$p(\unkx)$ and nc polynomial

Eg. \ \  $p(\rx)= \rx_1\rx_2 + \rx_2 \rx_1$
\bs

$\rX= (\rX_1, \cdots \rX_g)$ a tuple of matrices.  

\bs
Substitute a matrix for each variable


Eg. \ \  $p(\rX)= \rX_1\rX_2 + \rX_2 \rX_1$



\end{frame}

%%%%%%
\begin{frame}
\frametitle{Linear Systems and Algebra Synopsis}
\vspace{-0.5in}

A Signal Flow Diagram with $L^2$ based performance, eg $H^\infty$

gives precisely a nc polynomial
  $$
  p(a,\ux):= \left(
                \begin{array}{ccc}
                  p_{11}(a,\ux) & \cdots & p_{1k}(a,\ux) \\
                  \vdots & \ddots & \vdots \\
                  p_{k1}(a,\ux) & \cdots & p_{kk}(a,\ux) \\
                \end{array}
              \right)
$$

The linear systems problem becomes exactly:\\
\ms
%{\Large{
\iq \CB{Given matrices} $A$. \\
\iq \CB{ Find matrices $\uX$ so that  $P(\color{black}{A},\uX)$ is PosSemiDef.}
%}}

\ms
WHY? Turn the crank using quadratic storage functions.

\ms
\CB{BAD} \  \CR{Typically $p$ is a mess}, until a hundred people work on it
and maybe \CR{convert  it to convex Matrix Inequalities}.
\end{frame}
%%%%%%

%%%%%%
\begin{frame}
QUESTIONS (Vague) :

\begin{center}

\CR{WHICH SUCH PROBLEMS "ARE" LMI PROBLEMS.} 

{Clearly, such a problem must be convex and
"semialgebraic".}

\CB{Which convex nc problems are  NC LMIS?}

\bs
\smallskip


 { \color{red} WHICH  PROBLEMS
\quad  ARE TREATABLE WITH LMI's? }
\\
\iq This requires some kind of \CB{ change of variables theory}.

\bs 
\ss

\CR{ARE LMIs INTERESTING MATHEMATICAL OBJECTS?}

\CB{Free Real Algebraic Geometry:  Polynomials positive on NC Convex sets}

\ms
\bs

\CB{
These are topics of this talk}

\CB{
We shall suppress the two classes of variables $a, \rx$.}

\end{center}

\end{frame}
%%%%%%


\begin{frame}
     \frametitle{ Outline}
     \tableofcontents
     \end{frame}
%%%%%%%%
\iffalse
\begin{frame}
     \frametitle{ \CR{SUMMARY of  OLD RESULTS??DELETE}   }

\iq \qquad \qquad Hay, H,  McCullough, Putinar,  Vinnikov

\begin{enumerate}
\item DIMENSIONLESS SYS PROBS ARE NONCOMMUTATIVE


\item \smthm \color{red} 2 \ \ \color{black}
\ EVERY  ``POSITIVE" NC POLYNOMIAL
IS A SUM OF SQUARES.
   \color{green}$\leftarrow$ COMPUTABLE. \color{black}

\item \smthm \color{red} 3 \ \ \color{black}
 EVERY ``CONVEX" NC POLYNOMIAL HAS DEGREE $ \leq 2$.
  {  \color{green} Equiv. to an LMI. \color{black} }

\item \smthm  \color{red} 4 \ \ \color{black}
THMs 2 AND 3 LOCALIZE, eg NC POSITIVSTELLENSATZ




\item  {  \color{red}  {\tiny{ALGORTHIM}} \color{black} }CHECKS  CONVEXITY OF NC RATIONAL FUNCTION.
  \color{green} TRY NCAlgebra \color{black}

\item
 \CR{ALL CONVEX DIMENSIONLESS SYSTEMS PROBLEMS ARE
TREATABLE BY LMIs.- comming soon}
%>AUTOMATICALLY GENERATE THE  LMI?


\end{enumerate}
\end{frame}
\fi
%%%%%%%%%%%%%
\section{LMIs and Convex Sets}

\input{sliConvSetHBB10.tex}
%%%%%%%%%%%%%%%%%%%%

\section{NC Real Algebraic Geometry: \ Positivstellensatz}

\begin{frame}
\frametitle{NC Real Algebraic Geometry}
\centerline{\CB{\Large{NC convexity yields a Perfect Positvstellensatz}}}
\bs

\CR{OUTLINE}  Joint work with Klep and McCullough
\tableofcontents
\end{frame}


\begin{frame}
\ft{NC convex Positivstellensatz}
\framesubtitle{It is \CB{perfect}}

\iffalse
A nc symmetric polynomial  $p$ is \emph{positive semidefinite}
on an nc LMI domain $\cD_L$ if and only if it
has a weighted sum of squares representation with optimal degree bounds.
\fi

\bs

\CR{THM} \quad
\CR{SUPPOSE} $L$ is a monic linear pencil; \
$\cD_L$ has interior.\\
\bs

 \indent  \  $L(X) \succeq 0$ implies $p(X) \succeq 0$ \
\CR{IFF}
$$
p(x)
= \rs(x)^* \rs(x)  + \sum_j^{\rm finite} \rf_j(x)^* L(x) \rf_j(x),
$$
\indent where
 $\rs, \rf_j$ are vectors of noncommutative polynomials 
  each
  of 
  
  \centerline{degree no greater than
 $\CR{ \frac{\deg(p) }{2} ]_+ }$. }
 \end{frame}

\begin{frame}
\ft{Compare to commutative PosSS}
 \framesubtitle{Commutativity has blemishes}
 
This result contrasts sharply with the commutative setting:

\bs

\CR{THM} \CB{(COMMUTATIVE)} 

\indent \quad
\CR{IF} \  $L(X) \succeq 0$ implies $p(X) \succ 0$, \
\CR{THEN}
$$
p(x)
= \rs(x)^* \rs(x)  + \sum_j^{\rm finite} \rf_j(x)^* L(x) \rf_j(x),
$$
\indent \quad
where
 $\rs, \rf_j$ are vectors of  polynomials 
  each
  of \\ 
\indent \quad(\CR{high}) degree  
  with bound depending on  
  $\CR{\frac{1}{\min_{\{x: L(x) \succeq 0\}}
    p(x)}}$.
  
  \bs
  
  Thus we call the nc convex PosSS a ``\CB{Perfect PosSS}"
  
\end{frame}


\def\bet{\begin{theorem}}
\def\eet{\end{theorem}}

\def\bel{\begin{lemma}}
\def\eel{\end{lemma}}

\newcommand{\ax}{\langle\x\rangle}
\def\x{x}


\begin{frame}
\ft{Duality: PosSS $\leftarrow \rightarrow$ Moment Problems}
 
 \bcen
 Matrices of Moments have Hankel structure
 \ecen
 
 \bs
 
 {\Large{ 
  The main ingredient of the proof is an analysis of
 rank preserving extensions of truncated
noncommutative
 Hankel matrices.}}
 \bs  \bs \bs  \bs

 
 \bcen
 \CB{\huge{Voil\`a}}
\ecen

\end{frame}


\begin{frame}
\ft{nc Hankel Matrices}
%\bcen
%A moment matrix construction: eg. $S =$ TV Screen
%\ecen


%Tautology:\\
%\small
%The convex set $S =\{x\in\re^2:  1-x_1^4-x_2^4\geq 0\}$ equals

{\scriptsize
\[
%\left\{(x_1,x_2):\,
\baray{c}
%1-x_1^4-x_2^4 \geq 0  \ \ \ \ and \\  \qquad  \\
\underbrace{
\bbm
1      & x_1     & x_2    & x_1^2 & x_1x_2 & x_2^2 \\
x_1    & x_1^2   & x_1x_2 & x_1^3 & x_1^2x_2 & x_1x_2^2 \\
x_2    & x_1x_2  & x_2^2  & x_1^2x_2 & x_1x_2^2 & x_2^3 \\
x_1^2 & x_1^3  & x_1^2x_1 & x_1^4 & x_1^3x_2 & x_1^2x_2^2 \\
x_1x_2 & x_1^2x_2  & x_1x_2^2 & x_1^3x_2 & x_1^2x_2^2 & x_1x_2^3 \\
x_2^2 & x_1x_2^2  & x_2^3 & x_1^2x_2^2 & x_1x_2^3 & x_2^4 \\
\ebm}_{ = \,\, \mbox{\small \quad
$ first \ column \ \times \  (first \ column)^T $ } }
%\succeq 0
\earay
%\right\}.
\]
}
Hankel: \
Replace each monomial $x^\alpha$ by a number $y_\alpha= \ell(x^\alpha)$,
we get:
%\end{frame}
%\begin{frame}
%\bcen
%{Relaxation $x^\alpha \to y_\alpha$}
%\ecen

%\bit
%\item
%The convex set $S =\{x:  1-x_1^4-x_2^4\geq 0\}$ is contained in
{\scriptsize
\[
Hank:=
%\left\{(x_1,x_2):\,
\baray{c}
%1-y_{40}-y_{04} \geq 0 \\ \qquad  \\
\bbm
1      & { y_{10}}     & { y_{01}}   & y_{20} & y_{11} & y_{02} \\
{ y_{10 } }  & y_{20}  & y_{11} & y_{30} & y_{21} & y_{12} \\
{ y_{01 } }  & y_{11}  & y_{02} & y_{21} & y_{12} & y_{03} \\
y_{20} & y_{30}  & y_{21} & y_{40} & y_{31} & y_{22} \\
y_{11} & y_{21}  & y_{12} & y_{31} & y_{22} & y_{13} \\
y_{02} & y_{12}  & y_{03} & y_{22} & y_{13} & y_{04} \\
\ebm 
%\succeq 0
\earay
%\right\},
\]
}
%since for every $x\in S$, we can choose $y_{ij} = x_1^ix_2^j$.

%\vskip -1cm

%\item This is linear. Is a lifted LMI rep
%if $ S = \tilde S $.\\
%\CR{ Does $ S = \tilde S $ ?}
%\eit
\bs
{\small{True confession- the Hankels above are commutative.
\\ nc Hankels need lots of laptop battery on the plane}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{frame}
\ft{The nc world is flat}

\CR{THM} \ \
Every positive definite noncommutative \CB{Hankel} matrix has a 
``$1$-step flat" extension.

\ms
\CB{meaning}
\ms

Given $\CB{y_{ij} }$ up to a certain degree, find the $\CR{y_{ij}}$ making
{\scriptsize
\[
\CG{Hank}:=
%\left\{(x_1,x_2):\,
\CR{
\baray{c}
%1-y_{40}-y_{04} \geq 0 \\ \qquad  \\
\bbm
\CB 1      & \CB { y_{10}}     & \CB { y_{01}}   & \CB{y_{20}} &\CB{ y_{11}} 
&\CB{ y_{02}}  \\
\CB { y_{10 } }  & \CB {y_{20} } &\CB{  y_{11} }& y_{30} & y_{21} & y_{12} \\
\CB { y_{01 } }  & \CB {y_{11} } &\CB{  y_{02}} & y_{21} & y_{12} & y_{03} \\
\CB{y_{20}} & y_{30}  & y_{21} & y_{40} & y_{31} & y_{22} \\
\CB{y_{11}} & y_{21}  & y_{12} & y_{31} & y_{22} & y_{13} \\
\CB{y_{02}} & y_{12}  & y_{03} & y_{22} & y_{13} & y_{04} \\
\ebm 
%\succeq 0
\earay}
%\right\},
\]
}
\bcen
\CG{Hank} is PosSemiDef, \ 

 and 

rank \CB{Hankel} $=$ rank \CG{Hank}.
\ecen
%%%%%%%%%%%
\iffalse
\bet[Existence of Rank Preserving Extensions-REPEATS??]
\label{thm:invHank}
Any given  positive definite  Hankel operator $A$ on $\cH_k$, 
has an extension to a Hankel operator $A_m$ 
on $\cH_{k+ m}$ whose rank is the same as the rank of $A$.
There are many  rank preserving Hankel extensions $A_1$,
 however all  $A_m$ with $m>1$ are uniquely determined by $A_1$. 
\eet
\fi
\end{frame}

\begin{frame}
\ft{Proof of free convex PosSS}
Patch flat extension for nc Hankel theorem
together  with the usual  Hahn Banach separating linear
functional plus GNS (Putinar) proof.

\bs \bs
Long story.
\end{frame}

\def\tg{{\tilde g}}

\section{Change of Variables to achieve Free Convexity}

\begin{frame}
\centerline{\CB{\Large{Changing variables to achieve NC convexity}}}

\bs
\bs

\tableofcontents

\end{frame}


\def\arxs{\langle\rx,\rx^*\rangle}

\def\arx{\langle\rx\rangle}

\begin{frame}[t]
\frametitle{Change of variables}

\begin{block}{Main issues}
\begin{itemize}
\item
Which sets can be transformed into nc convex sets?
\bcen  \CR{Equivalently}
\ecen 
\ms

\item
\CB{Which sets can be transformed into nc LMI domains?}
\end{itemize}

\vspace{1cm}

\includegraphics[scale=.5]{pic.png}

\end{block}

\vfill
Based on joint work with Igor Klep and Scott McCullough.
\end{frame}

\subsection{nc maps}

\iffalse
\begin{frame}
\frametitle{Change of variables}
\framesubtitle{nc maps}

DELETE???

 An \CR{nc map} 
  $f: \cU\to \math$ is a sequence
 of functions $\fn:\cU(n) \to\matnh$ which
\ben[\rm (1)]
\item
\CB{respects direct sums}: 
for each $X\in \cU(n)$ and  $Y\in \cU(m),$
 \[
   f(X\oplus Y)=f(X) \oplus f(Y);
 \]
\item
\CB{respects similarity}: for each 
  $X,Y\in \cU(n )$ and invertible $n\times n$
  matrix $S$
  such that $S^{-1}XS=Y$,
\[
 S^{-1}  f(X) S= f(Y).
\]
\een
\end{frame}
\fi
%%%%%%%%%%

\def\ben{\begin{enumerate}}
\def\een{\end{enumerate}}

\begin{frame}
\framesubtitle{Change of variables}
\frametitle{NC (free) analytic maps}

\ben[\rm(1)]
\item
Analytic nc polynomials have no $\rx^*$.
\pause
\item
$f(\rx_1)=\rx_1^*$ is \CB{not} an nc analytic map.
\pause
\item
Convergent \CB{power series}
$$
f= \sum_{w\in\arx} c_w w, \quad c_w\in\C
$$
are \CB{nc analytic  maps}. Here $w$ are words in $x_j$.
\een
Convergence works manageably, but
we do not discuss convergence in this talk.
\CR{Think polynomials!}
See
Dan Voiculescu \ \ \ Dima KV and Victor V,
or see our  HKMc  paper for a light version
{\small{
$$
f=
\left(
\begin{array}{c}
  f_1    \\
  \vdots   \\
  f_\tg 
\end{array}
\right)
$$
}}
\end{frame}
%%%%%%%%%%%%
\iffalse
\begin{frame}
\frametitle{Change of variables}
\framesubtitle{nc maps. Properties}

\begin{block}{Proposition}
Suppose $\cU$ is an nc domain.
\ben[\rm (1)]
\item
  A \CB{continuous} nc map $f:\cU\to \math$ \CB{is analytic}.
\item If $X\in\cU(n)$, and $H\in\matng$ has sufficiently small norm,
then
 \[
  f\begin{pmatrix} X & H \\ 0 & X\end{pmatrix}
   = \begin{pmatrix} f(X) & f^\prime(X)[H] \\ 0 & f(X)\end{pmatrix}.
 \]
\een
\end{block}
\pause

\begin{corollary}
A continuous nc map admits a convergent power series expansion.
\end{corollary}

\end{frame}
\fi
%%%%%%

\subsection{Proper nc maps}

\begin{frame}
\frametitle{Change of variables}
\framesubtitle{Proper nc maps}

$$\matg:= \cup_n \ ({\mathbb C}^{n\times n})^g$$ 

\ms 
Let $\cU\subset\matg$ and $\cV\subset\math$ be 
given nc basic semialgebraic  domains.


\ms

\begin{itemize}

\ss

\item
 An
  nc map $f:\cU\to\cV$ is \CR{proper} 
  if each $\fn:\cU(n)\to \cV(n)$ is proper.
That is, 
  if $K\subset \cV(n)$ is compact, then
  $f^{-1}(K)$ is compact.

\ss

\pause
\item
  In other words: for all $n$,
  if $(z_j)$ is a sequence in $\cU(n)$
  and $z_j\to\partial\cU(n)$, then
  $f(z_j)\to\partial\cV(n)$.

\ss

\pause
\item
 In the case $g=h$ and both $f$ and $f^{-1}$ are (proper)
nc maps,
 we say $f$ is 
a \CR{bianalytic} nc map.
\end{itemize}
\end{frame}



\def\Lam{\Lambda}
\def\cB{\mathcal B}
\def\NN{\mathbb N}
\subsection{Pencil balls and maps}

\begin{frame}
\frametitle{Change of variables}
\framesubtitle{Pencil balls}



\begin{itemize}
\item
Let $\mathbb B^g:=\{X\in \matg :\  \|X\|<1\}$ be the \CR{operator ball}.
\item
To a linear pencil 
$$\Lam(\ux):= L_1 \ux_1 +\cdot + L_g \ux_g,  $$ 
$L_j$ not necc symmetric, we associate its \CR{pencil ball}
$$
\cB_\Lam:=\{X\mid \|\Lam(X)\|<1\}= \cD_{\scriptsize\begin{pmatrix}I&\Lam\\\Lam^*&I\end{pmatrix}}.
$$
\end{itemize}\pause

\CR{THM}
    \begin{quote}
     \CR{SUPPOSE}
      $\Lam$ is a 
   \CR{minimal dimensional defining pencil} for $\cB_\Lam$. \
% 
   %   $\Lam$ is a linear pencil and \\
  %   $\tilde \Lam$ is a \CR{minimal dimensional 
  %defining pencil for $\cB_\Lam$}.\\
\CR{IF} $f:\cB_\Lam \to \mathbb B^\tg$ is a proper analytic nc map
(with matrix coefficients)
     with $f(0)=0$,\
     \CR{THEN}   there is a contraction-valued analytic
     ${J}:\cB_\Lam \to \mathbb B$ such that
 \begin{equation*}
    f(x) =U \begin{pmatrix}  \Lam(x) & 0\\ 0&{J}(x)  \end{pmatrix} V^*
 \end{equation*}
 for some 
 unitaries $U,V$.
\end{quote}
\end{frame}

\def\cF{\mathcal F}


\begin{frame}
\frametitle{Change of variables}
\framesubtitle{Pencil balls. An important special case}
\begin{itemize}
\item
Let $$\Lam=\sum_{i,j}E_{ij}\, \rx_{ij},$$ 
where $E_{ij}$ are the matrix units.\pause
\item
Then 
$$
\cB_\Lam=\{ X = (X_{ij})_{i,j} \mid \|X\|<1\}.
$$
\item
The nc automorphism group of $\cB_\Lam$ is transitive and contains
maps of the form
$$
 \cF_v (u):=v-(I-vv^* )^{1/2}u(I-v^* u)^{-1}(I-v^* v)^{1/2},
$$
where $v$ is a scalar matrix of norm $<1$.
\end{itemize}
Generalizes Muhly -Solel fin dim path algebras result.
\end{frame}



\subsection{NC Proper maps are bianalytic}
\begin{frame}
%\frametitle{Change of variables}
\frametitle{Proper nc maps tend to be bianalytic}

\begin{theorem}
Let $\cU$ be an nc domain in $g$ variables,\\
let $\cV$ be an nc domain in $\tg$ variables, \\
and  suppose $f:\cU\to \cV$ is an nc analytic map.
\begin{enumerate}[\rm (1)]
\item
If $f$ is proper, then it is one-to-one,
  and\\ $f^{-1}:f(\cU)\to \cU$ is an nc map.
\item
If $g=\tg$ and $f:\cU\to\cV$ is
 proper and continuous,  then\\ $f$ is bianalytic.
\end{enumerate}
\end{theorem}
\end{frame}


\begin{frame}
\frametitle{Change of variables $g=\tilde g$ is rigid}
\framesubtitle{\only<1>{Circular domains:}\only<2>
{Proper nc self-maps tend to be linear}}
\begin{itemize}
\item
A subset $S$ of a complex vector space is
  \CR{circular} if $\exp(it) s\in S$  whenever
  $s\in S$ and $t\in\R$.
\item
  An nc domain $\cU$ is \CR{circular}
  if each $\cU(n)$ is circular.
\end{itemize}\pause
\bigskip


\begin{theorem}
Let $\cU,\cV$  be nc domains in $g$ variables containing $0$,
and  suppose $f:\cU\to \cV$ is a proper analytic nc map
with $f(0)=0$.\\
If $\cU$ and $\cV$ are \CB{circular}, then $f$ is \CB{linear}.
\end{theorem}

Generalizes:

Gelu Popescu map result for Reinhardt Domains

Muhly- Solel path algebras.

\end{frame}


\section{Summary}

\begin{frame}
\frametitle{Summary}

Highlights to remember:
\ben[\rm (1)]
\item
A free convex bounded basic open semialgebraic  set is 
the set of solutions to some LMI.
\item
Much of classical real algebraic geometry goes thru to free cases.
For free convex situations there is a "perfect" PosSS.
\item
A proper free analytic map is one-to-one.(Partly in progress.)\\
A proper free analytic map between domains of the same dimension is bianalytic.
If the domains are circular and $f(0)=0$, it is linear.
\item
A proper free analytic map $f$ on pencil balls with $f(0)=0$ is
the direct sum of   a linear pencil ball isometry plus junk.
\item
\item
\een


\end{frame}





\end{document}



%%%%%%%%%%%% 