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\title{Approximability of Lie groups}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{CSASC2016, Barcelona \\ September 22, 2016}
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\begin{frame}{Zilber's program}
Treatment of quantum mechanics (and foundations of physics in general) from the
point of view of model theory. The main goal is to (properly) handle infinities
in physics.
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\begin{frame}{Zilber's program}
Some highlights:
\begin{itemize}
\item Physics practice (finite computations) dictate that it should
be treated in the scope of a first-order, and not higher order, theory.
\item
Instead of analytic continuity consider continuity in terms of Zariski topology.
\item Physics should be based on mathematical structures which are categorical
(for example, $\mathbb C$ qualifies, while $\mathbb R$ does not).
Reasons: such structures 1) exhibit homogeneity 2) allow notion of dimension.
\item The physical principle ``quantum mechanics applied to 'large' structures
degenerates to classical mechanics'' translates to mathematical principle of
``finite approximation'': 'large' finite structures are treated as infinite
ones having first-order categorical theory.
\end{itemize}
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\begin{frame}{Some parallel developments}
(All around 2013--2014)
\begin{itemize}
\item Chalons \& Ressayre
Phenomenological approach to quantum mechanics:
mathematical objects exist up to an error in their transmission.
Technical tool: predicate-like calculus of binary relations.
\item Kapustin, Moldoveanu
System of axioms, basing on category theory, and involving some Lie-algebraic
structures, leading to Quantum Mechanics as the only possible theory.
(Sort of) continuity is assumed, appearance of $\mathbb C$ follows from axioms.
\item Kornyak
``Quantum discrete dynamical systems''.
Discretize everything; instead of $\mathbb R$ or $\mathbb C$, use cyclotomic
fields.
\end{itemize}
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\begin{frame}{Approximation according to Zilber}
\begin{block}{Definition}
A structure $A$ is approximated by a family of structures $\{B_i\}$
if there is a surjection $\prod_{\mathscr U} B_i$ (ultraproduct) $\to A$.
\end{block}
\bigskip
Compare with other (more traditional?) notions of approximation:
\emph{embedding} into direct product or ultraproduct.
\bigskip
\begin{block}{Question}
Whether (say) $SO(3)$ is approximated by finite groups?
\end{block}
\bigskip
\textbf{No}, if finite groups are simple (Pillay, 2015).
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\begin{frame}{From ultraproducts to direct products}
\begin{block}{Theorem (Bergman--Nahlus)}
For any cardinal $\kappa > 2$, and any algebraic system $A$ consisting of more
than one element, the following are equivalent:
\begin{enumerate}[(i)]
\item
For any surjective homomorphism $f: \prod_{i\in \mathbb I} B_i \to A$,
$\modulus{\mathbb I} < \kappa$, there is $i_0 \in \mathbb I$ such that $f$ factors through the canonical projection
$\prod_{i\in \mathbb I} B_i \to B_{i_0}$.
\item
For any surjective homomorphism $f: \prod_{i\in \mathbb I} B_i \to A$, there is
a $\kappa$-complete ultrafilter $\mathscr U$ on $\mathbb I$ such that $f$
factors through the canonical homomorphism $\prod_{i\in \mathbb I} B_i \to \prod_{\mathscr U} B_i$.
\end{enumerate}
\end{block}
\begin{block}{Corollary}
Zilber's question is equivalent to: whether the \emph{direct product} of
finite groups can be mapped surjectively onto $SO(3)$?
\end{block}
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\begin{frame}{References}
B. Zilber, \emph{Perfect infinities and finite approximation},
Infinity and Truth (ed. C. Chong et al.), World Scientific, 2014, 199--223.
\bigskip
P. Zusmanovich, \emph{On the utility of Robinson--Amitsur ultrafilters. I, II},
J. Algebra \textbf{388} (2013), 268--286; \textbf{466} (2016), 370--377;
arXiv: 0911.5414, 1508.07496
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{\Huge
That's all. Thank you.
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