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\title % (optional, use only with long paper titles)
{Definability in the group of infinitesimal rotations}

\author%[Author, Another] % (optional, use only with lots of authors)
{Martin Bays\inst{1},
Kobi Peterzil\inst{2}
}
% - Use the \inst{?} command only if the authors have different
%   affiliation.

\institute{\inst{1} WWU M\"{u}nster \\\
\inst{2} University of Haifa}

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\date{Logic Colloquium, Stockholm, 2017-08-17}
%\date[07-12-14] % (optional)
%{Friday}

!newcal U
!newfrak g
!newfrak m
!newcal A
!newcal B
!newop st
!newop mod
!newop Ad
!newop ad
!newop SO
!newop so
!newop su
!newop GL
!newop pr

\renewcommand{\O}{\mathcal{O}}
\renewcommand{\v}{\operatorname{v}}

\renewcommand{\o}{\circ}

\renewcommand{\O}{\mathcal{O}}
\newcommand{\Ru}{\mathcal{R}}
\newcommand{\Goo}{{G^{00}}}
\newcommand{\Soo}{{\SO_3^{00}}}

\begin{document}
Frame:
  \titlepage
  %  \tableofcontents
  %  % You might wish to add the option [pausesections]
  .

Frame:
  %2:30
  \frametitle{$\SO_3$}
  Theorem[Nesin-Pillay 1991]:
    ***) $X (= \SO_3(\R)^n$ is definable in the pure group $(\SO_3(\R);*)$ iff 
    it is definable in the field $(\R;+,\cdot)$.
    ** More generally, same for any simple centerless compact Lie group $G$
    definable in $(\R;+,\cdot)$.
    *e)*
    .
  Example:
    $\{ (A,B) (- \SO_3(\R) : \det(A-B)>0 \}$ is definable in $(\SO_3(\R);*)$.
    .
  Sketch of proof:
    ***) Define a copy of $\SO_3(\R)$ in $(G;*)$;
    ** Reconstruct the field from the projective plane of involutions of 
    $\SO_3(\R)$.
    ** See that this yields a bi-interpretation of $(G;*)$ with 
    $(\R;+,\cdot)$.
    *e)*
    .

Frame:
  %3:00
  \frametitle{$\Soo$}
  \[ (\Ru;+,\cdot) := (\R;+,\cdot)^\U \elemext (\R;+,\cdot) \]

  \[ 0 --> \m --> \O -->^{\st} \R --> 0 \]

  \[ 1 --> \Soo --> \SO_3(\Ru) -->^{\st} \SO_3(\R) --> 1 \]

  \[ \Soo = \SO_3(\Ru) \cap \begin{pmatrix}
    1+\m & \m   & \m   \\\
    \m   & 1+\m & \m   \\\
    \m   & \m   & 1+\m
    \end{pmatrix} \]

  $(\Soo;*)$ is interpretable in $(\Ru;+,\cdot,\O) |= RCVF$.
  Problem:
    Which $(\Ru;+,\cdot,\O)$-definable subsets of $(\Soo)^n$ are 
    $(\Soo;*)$-definable?
    .
  .

Frame:
  %6:30
  \frametitle{$(\Soo;*)$}
  ***) Lie algebra $\g(\Ru) = \so_3(\Ru) =
  \left\{ \begin{pmatrix}
     0 && -z &&  y \\\
     z &&  0 && -x \\\
    -y &&  x &&  0
  \end{pmatrix} \right\} ~= \Ru^3 = \{ (x,y,z) \}$.

  ** Infinitesimal Lie algebra:
    $\g_\m := \st^{-1}(0) ~= \m^3 <= \Ru^3$.

  ** Matrix exponentiation yields a homeomorphism
    $\exp_\m : \g_\m -~-> \Soo$.

  ** $\exp_\m(X)*\exp_\m(Y) = \exp_\m(X+Y) + \epsilon$ where $v(||\epsilon||) 
  >= v(||X||)+v(||Y||)$.

  ** If $X$ and $Y$ are collinear then
    $ \exp_\m(X)*\exp_\m(Y) = \exp_\m(X+Y)$.

  ** For $x (- \Soo$ and $h (- \SO_3(\Ru)$, group conjugation $x |-> x^h := 
  h*x*h^{-1}$ agrees with the matrix action of $\SO_3(\Ru)$ on $\m^3$:
  %(which is also the adjoint action):
    \[ \exp_\m(X)^h = \exp_\m(h X) .\] %\;\; ( = \exp_\m(\Ad(h) X) ) .\]
  *e)*
  % Now draw $\Soo$ as $\m^3$ and indicate conjugation action
  % (giving them time to absorb the above)
  % and recall the problem, maybe try to convince them it could be quite 
  % weak...
  .


Frame:
  %2:30
  \frametitle{Main theorem}
  Theorem:
    **(i) $X (= (\Soo)^n$ is $(\Soo;*)$-definable iff it is 
    $(\Ru;+,\cdot,\O)$-definable.

    **(ii)
      Moreover, the interpretation of $(\Soo;*)$ in $(\Ru;+,\cdot,\O)$ can be 
      completed to a bi-interpretation.
    *ee*
    .
    
  Example:
    $\{ (A,B) (- \Soo : v(\det(A-B))>\alpha \}$ is definable in $(\Soo;*)$.
    .

  Outline of proof:
  ***) Find an $(\Soo;*)$-definable ordered interval $J$;
  ** apply trichotomy to get a field $K$ in $J$;
  ** use adjoint representation to see the pair $\Soo <= \SO_3(\Ru)$ in $K$,
    yielding a bi-interpretation;
  ** the characterisation of definable sets follows.
  *e)*
  .

Frame:
  %6:00
  \frametitle{Finding an ordered interval}
  ***) Let $G := \SO_3(\Ru)$ and $\Goo := \Soo$.
  ** Let $b (- \Goo \\ \{e\}$.
  ** $C_b^G := \{ h (- G : h*b=b*h \} ~= \SO_2(\Ru)$;
  ** $C_b^{\Goo} := C_b^G \cap \Goo ~= \SO_2^{00} ~= \m$.
  ** $b^Gb^G = \xi(G^2)$ where $\xi(h,h') = b^h*b^{h'}$.
  ** $b^Gb^G = \exp_\m(B)$ where $B (= \m^3$ is the closed ball of radius 
  $||b^2||$.
  ** $b^Gb^G \cap C_b^{\Goo}$ is the interval $[b^{-2},b^2]$.
  ** By definable choice for the $(\Ru;+,\cdot)$-definable map $\xi$,
  $X := b^{\Goo}b^{\Goo} \cap C_b^{\Goo}$ contains some interval $[h,b^2]$.
  ** Translating, get $(\Goo;*)$-definable interval $[e,p) (= C_b^{\Goo}$, 
  hence $J := (p^{-1},p)$ as an ordered interval.
  ** Explicitly: $p := b^2h^{-1}$, then $[e,p] = h^{-1}X \cap b^2X^{-1}$.
  *e)*
  .

Frame:
  %2:30
  \frametitle{Trichotomy}
  ***) $b^{\Goo}$ spans $\Ru^3$, so for appropriate $h_1,h_2 (- \Goo$ and 
  after shrinking $J$,
  \[ \phi : J^3 --> \Goo; \phi(x_0,x_1,x_2) = x_0*x_1^{h_1}*x_2^{h_2} \]
  is a bijection with a neighbourhood of $e (- \Goo$.
  ** $(J;*,<)$ and $\phi$ are definable both in $(\Goo;*)$ and in 
  $(\Ru;+,\cdot)$.
  ** Pulling back the $\Goo$ group structure via $\phi$ puts ``non-linear'' 
  structure on $J$.
  ** By the Peterzil-Starchenko o-minimal trichotomy, a real closed field 
  $(K;+,\cdot)$ on an interval $K (= J$ is definable in this structure on $J$. 
  ** So $(K;+,\cdot)$ is definable both in $(\Goo;*)$ and in $(\Ru;+,\cdot)$.
  *e)*
  .

Frame:
  %3:00
  \frametitle{Bi-interpretation}
  ***) $(K;+,\cdot)$ is definable both in $(\Goo;*)$ and in $(\Ru;+,\cdot)$.
  ** Otero-Peterzil-Pillay: exists $(\Ru;+,\cdot)$-definable isomorphism 
  $\theta : (\Ru;+,\cdot) -~-> (K;+,\cdot)$.
  ** $\theta$ induces $\theta_G : G = \SO_3(\Ru) -~-> \SO_3(K)$.
  *e)*
  Claim:
    $\theta_G|_{\Goo} : \SO_3(\Ru)^{00} -~-> \SO_3(K)^{00}$ is 
    $(\Goo;*)$-definable.
    .
  Proof[Proof of main theorem]:
    ***) $\O$ is definable in $(\Ru;+,\cdot,\Goo)$,
    ** so $(\Ru;+,\cdot,\O)$ is interpreted on $K$ in $(\Goo;*)$ via $\theta$, 
    since $\SO_3(K)^{00}$ is $(\Goo;*)$-definable by the claim.
    ** $(\Goo;*)$ is interpreted in $(\Ru;+,\cdot,\O)$ tautologically.
    ** The composed interpretations are $\theta$ and $\theta_G|_{\Goo}$, which 
    are definable in $(\Ru;+,\cdot,\Goo)$ resp. $(\Goo;*)$.
    *e)*
    .
  .

Frame:
  %4:00
  \frametitle{Proof of claim}
  Claim:
    $\theta|_{\Goo} : \SO_3(\Ru)^{00} -~-> \SO_3(K)^{00}$ is 
    $(\Goo;*)$-definable.
    .
  Proof:
    ***) Differentiation in $K$ yields via $\phi$ an adjoint embedding
    \[ \Ad : \SO_3(\Ru) --> \GL_3(K) \]
    ** $\Ad$ is $(\Ru;+,\cdot)$-definable.
    ** $\Ad|_{\Goo}$ is $(\Goo;*)$-definable.
    ** $\eta := \Ad \o \theta_G^{-1} : \SO_3(K) --> \GL_3(K)$ is 
    $(K;+,\cdot)$-definable by purity, hence $(\Goo;*)$-definable.
    ** So $\theta_G|_{\Goo} = \eta^{-1} \o \Ad|_{\Goo}$ is $(\Goo;*)$-definable.
    *e)*
    .
  .

\end{document}
