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% file:         synopsis.tex
% purpose:      synopsis of ``selected problems'' Ostrava course
% created:      pasha mar 15 2016
% modified:     pasha sep 4 2016
% modification: refs
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\begin{document}

\title{Selected Problems in Mathematics \\ 
Synopsis of course at OU, Summer semester 2016}
\author{Pasha Zusmanovich}
\address{}
\email{pasha.zusmanovich@gmail.com}
\date{Last minor revision September 4, 2016}

\maketitle


{\small Each lecture lasted 1.5 hours}.


\section{Iterative correlation matrices} % lecture 1

\begin{problem}
Does the negative entropy of a matrix $A$, defined as 
\[
- E(A) = \sum_{\lambda \in Spec(A)} \lambda \> ln(\lambda)
\]
increase with iterative correlations?

If not (most probably), how we can modify it, or invent another function of 
matrix which will increase or decrease with iterative correlations?
\end{problem}


\begin{problem}
Can we use some form of contraction mapping theorem to
prove the convergence of iterative correlations?
\end{problem}


\begin{problem}
Do iterative correlations of the \(4 \times 4\) matrix
\[
\left(\begin{matrix}
\phantom{-}1\phantom{.73333333333} & -0.73333333333                      & -0.2\phantom{3333333333} & 0.2\phantom{3333333333} \\
          -0.73333333333           &  \phantom{-}1\phantom{.73333333333} & -0.2\phantom{3333333333} & 0.2\phantom{3333333333} \\
-0.2\phantom{3333333333}           & -0.2\phantom{3333333333}            & \phantom{-}1\phantom{.73333333333} & 0.33333333333 \\
\phantom{-}0.2\phantom{3333333333} & \phantom{-}0.2\phantom{3333333333}  & \phantom{-}0.33333333333 & 1\phantom{.33333333333}
\end{matrix}\right)
\]
converge? If not, what makes this matrix special? Are there other examples? 
\end{problem}



\vspace{-25pt}
\begin{thebibliography}{9}

\bibitem{chen} C.-H. Chen, \emph{Generalized association plots: information visualization via iterative generated correlation matrices}, 
Statistica Sinica \textbf{12} (2002), N1, 7--29.

\bibitem{concor} J.B. Kruskal, \emph{A theorem about CONCOR},
Technical Report MH 2C-571, Bell Laboratories, 1978. 

\end{thebibliography}







\medskip
\section{Gravitational approach to clustering and alternative laws of gravitation
} % lecture 2

\vspace{-20pt}
\begin{thebibliography}{9}

\bibitem{AHN}
E.G. Adelberger, B.R. Heckel, and A.E. Nelson,
\emph{Tests of the gravitational inverse-square law}, 
Ann. Rev. Nucl. Part. \textbf{53} (2003), 77--121; arXiv:hep-ph/0307284.

\bibitem{D} A.K. Dewdney, \emph{The Planiverse}, Springer, 2000.

\bibitem{M} M. Milgrom, 
\emph{General virial theorem for modified-gravity MOND},
Phys. Rev. D \textbf{89} (2014), 024016; arXiv:1311.2579.

\end{thebibliography}






\medskip
\section{Composition of binary maps into ternary maps} % lecture 3

\begin{problem}
Give a closed form formula, or recurrence formula for the numbers of ternary
maps on an \(n\)-element set represented in the form \((x*y)*z\) and/or 
\(x*(y*z)\) for some binary map \(*\). Do the patterns one observes for small 
\(n\) remain? In particular, is it true that there is a relatively small number
of "overlaps", i.e. these maps are mostly different for different \(*\)'s?
\end{problem}

\begin{problem}
Compute further few terms of the number of commutative (= symmetric) ternary
maps represented in the form \((x*y)*z\), and the number of such maps 
represented in such form for \emph{commutative} \(*\). Are these two numbers 
coincide?
\end{problem}

The same questions may be posed when the maps are counted up to permutation.

%\vspace{-20pt}
\begin{thebibliography}{9}

\bibitem{banach} P. Zusmanovich,
\emph{On the last question of Stefan Banach}, Expos. Math., to appear; 
arXiv:1408.2982. 

\end{thebibliography}






\medskip
\section{Algebraic systems} % lecture 4

General algebraic systems, groups, semigroups, rings, fields, vector spaces, 
algebras. Simple algebras. Isomorphism.

\vspace{-20pt}
\begin{thebibliography}{9}

\bibitem{schafer} R.D. Schafer, 
\emph{An Introduction to Nonassociative Algebras},
Academic Press, 1966; reprinted in a slightly corrected form by Dover, 1995.

\end{thebibliography}





\medskip
\section{Low-dimensional algebras. Lie algebras} % lecture 5

Lie algebras. Any associative algebra $A$ can be turned into a Lie algebra
with respect to commutator $[a,b] = ab - ba$.
Commutant $[L,L]$ of a Lie algebra $L$.

\begin{claim}
Any $1$-dimensional algebra over a field $K$ is either trivial, or isomorphic
to $K$.
\end{claim}

\begin{theorem}
Any $2$-dimensional Lie algebra is either abelian, or isomorphic
to $\langle x, y \vertbar [x,y] = x \rangle$.
\end{theorem}

\vspace{-20pt}
\begin{thebibliography}{9}

\bibitem{jacobson} N. Jacobson, \emph{Lie Algebras}, Interscience Publ., 1962; reprinted by Dover, 1979.

\end{thebibliography}






\medskip
\section{Low-dimensional Lie algebras} % lecture 6

Center of a Lie algebra. Direct sum of Lie algebras.

Examples of $3$-dimensional Lie algebras:
$sl_2(K)$ as commutant or quotient of $gl_2(K)$, simplicity of $sl_2(K)$.
Heisenberg Lie algebra.

Sketch of classification of $3$-dimensional Lie algebras.

\begin{claim}
Over an algebraically closed field, every Lie algebra has a $2$-dimensional
subalgebra.
\end{claim}




\medskip
\section{Lie algebras with given properties of subalgebras. Commutators} 
% lecture 7

Quaternions, $so_3(\mathbb R)$. 

Extension of the base field, forms.

\begin{claim}\label{cl-so3}
All proper subalgebras of $so_3(\mathbb R)$ are $1$-dimensional.
\end{claim}

\begin{problem}
Is it true that any finite-dimensional Lie algebra over any field, all whose
proper subalgebras are $1$-dimensional, has dimension $\le 3$?
\end{problem}

Minimal nonabelian Lie algebras constructed from the vector space $V$ and a
linear map $f:V \to V$: they are of the form $V \oplus Kx$, where $V$ is an 
abelian subalgebra, and $[v,x] = f(v)$ for $v\in V$.

Generators and relations. Tarski's monsters.

\begin{problem}
Do there exist infinite-dimensional Lie algebras, all whose proper subalgebras
are $1$-dimensional?
\end{problem}

\begin{claim}
If in a Lie algebra $L$ every element of commutant is a commutator, and
$[L,L] \cap Z(L) \ne 0$, then $L$ contains a $3$-dimensional Heisenberg Lie 
subalgebra.
\end{claim}

\begin{claim}
In $sl_2(K)$ any element is a commutator.
\end{claim}

\begin{proof}[Sketch of the proof]
As $B^{-1}[X,Y]B = [B^{-1}XB, B^{-1}YB]$ (what amounts to saying that 
$X \mapsto B^{-1}XB$ is an automorphism of $sl_2(K)$), it is enough to prove
the statement for matrices in Jordan normal form. We have only two 
possibilities:
$$
\left(\begin{matrix}
\lambda & 0 \\
0       & -\lambda
\end{matrix}\right)
$$
and
$$
\left(\begin{matrix}
0 & 1 \\
0 & 0
\end{matrix}\right) .
$$
\end{proof}






\medskip
\section{Lie algebras -- various things} % lecture 8

Witt algebra as algebra of differential operators. $sl_2(K)$ as its subalgebra
spanned by $\langle \frac{d}{dx}, x \frac{d}{dx}, x^2 \frac{d}{dx} \rangle$.

Proof of Claim \ref{cl-so3}:

\begin{lemma}
Any $3$-dimensional perfect algebra does not have $2$-dimensional abelian 
subalgebras.
\end{lemma}

\begin{lemma}
Any Lie algebra of the form $D^{(-)}$, where $D$ is associative division algebra
over a field of characteristic zero, does not have $2$-dimensional nonabelian 
subalgebras.
\end{lemma}

\begin{proof}
Suppose we have for some $x,y\in D$, $xy - yx= x$. Then $[x,x^{-1}y] = 1$,
what can be handled, for example, by passing to the extension of the base field
and using trace argument.
\end{proof}


Structure constants of a Lie algebra. Problem to decide whether two Lie algebras
are isomorphic reduces to solving quadratic systems.

\begin{problem}
Develop a program attempting to decide whether two given Lie algebras are
isomorphic, by combining GAP capabilities with Gr\"obner bases calculations.
\end{problem}





\medskip
\section{Lie algebras -- various things. II} % lecture 9

In a free $3$-step nilpotent Lie algebra of rank $4$, generated by $x,y,z,t$,
the element $[x,y] + [z,t]$ is not represented as a commutator.

\begin{problem}
Describe a class of Lie algebras for which every element of commutant is a
commutator.
\end{problem}


Lie $p$-algebras. Example of $p$-structures: abelian algebra, Lie algebra
of the form $A^{(-)}$.

Lie derivation algebra of an algebra.

\begin{claim}
For an arbitrary algebra $A$ over a field of characteristic $p$, and 
$D \in \Der(A)$, $D^p \in \Der(A)$.
\end{claim}





\medskip
\section{Derivations, Lie \texorpdfstring{$p$}{p}-algebras} % lecture 10

$\Der K[x] \simeq $ Witt.

\begin{problem}
Does any finite-dimensional Lie algebra can be represented as a derivation
algebra of some (nonassociative) algebra?
\end{problem}

Inner derivations, outer derivations. Outer derivations as the first cohomology
group.

\begin{problem}[very vague]
Compute cohomology of various Lie algebras. 
\end{problem}

\begin{problem}
Improve existing computer algorithms for computation of cohomology of Lie 
algebras.
\end{problem}

$p$-structures on the $2$-dimensional nonabelian Lie algebra.

\begin{problem}[Jacobson]
Is it true that any (possibly infinite-dimensional) Lie algebra satisfying
the condition $x^{[p]^{n(x)}} = x$ is abelian?
\end{problem}





\medskip
\section{Lie \texorpdfstring{$p$}{p}-algebras, gradings} % lecture 11

\begin{theorem}
If a Lie $p$-algebra $L$ satisfies $x^{[p]} = x$ for any $x \in L$, then $L$
is abelian.
\end{theorem}

\begin{proof}[Sketch of the proof]
We have $\lambda^p = \lambda$ for any element $\lambda$ of the base field, hence
the base field coincides with $GF(p)$. As any $\ad x$ is annihilated by 
polynomial $f(t) = t^p - t$ with roots in the base field, then we have root
space decomposition with respect to $\ad x$. If we have non-zero roots, then
we have $2$-dimensional nonabelian subalgebra which also satisfies the same
property $x^{[p]} = x$, what is impossible by computation from the previous
lecture. Hence $\ad x$ is nilpotent, what implies $\ad x = 0$.
\end{proof}

Relationship between derivations and automorphisms: if $D$ is a derivation,
then $exp (D)$, suitably defined, is an automorphism.

Gradings. Appearance of gradings as root space decomposition with respect 
to adjoint maps, derivations, automorphisms.

Example of non-semigroup grading from \cite{elduque-2}.

\begin{problem}
Is any grading on a finite-dimensional simple Lie algebra over a field of
characteristic zero, a semigroup grading?
\end{problem}

\vspace{-25pt}
\begin{thebibliography}{9}

\bibitem{elduque-2} A. Elduque, \emph{More non-semigroup Lie gradings}, Lin. Algebra Appl. \textbf{431} (2009), N9, 1603--1606; arXiv:0809.4547.

\end{thebibliography}





\medskip
\section{Number of semisimple Lie algebras. Operads} % lecture 12

Semisimple Lie algebras in zero and positive characteristic. Example of a 
semisimple Lie algebra in characteristic $p$ which is not the direct sum of
simples ($\Sl(2) \otimes K[t]/(t^p) + \frac{d}{dt}$).

\begin{problem}
Investigate the function $f(n)$, where $f(n)$ is the number of non-isomorphic
finite-dimensional semisimple Lie algebras over an algebraically closed field
of characteristic zero, of dimension $n$.
\end{problem}


Operads. Koszul duality of binary quadratic operads. Tensor product of algebras
of dual operads as Lie algebras.




\medskip
\section{Operads} % lecture 13

Examples of Koszul dual operads: (associative, associative), 
(Lie, associative commutative). 

\begin{problem}
Find ``interesting'' Lie algebras represented as the tensor product of algebras
over Koszul dual operads.
\end{problem}

Poincar\'e series of an operad. Examples:
\begin{gather*}
g_{Ass}(t) = \sum_{n=1}^\infty (-1)^n \frac{n!}{n!} t^n = - \frac{t}{1+t} \\
g_{Comm}(t) = \sum_{n=1}^\infty (-1)^n \frac{1}{n!} t^n = e^{-t} - 1 \\
g_{Lie}(t)  = \sum_{n=1}^\infty (-1)^n \frac{(n-1)!}{n!} = - \ln (1+t) .
\end{gather*}

Ginzburg--Kapranov criterion for Koszulity of operads.

\begin{problem}
Is it true that for $n>8$ the inverse power series of the polynomial 
$t - t^n + t^{2n-1}$ has non-negative coefficients?
\end{problem}

\vspace{-20pt}
\begin{thebibliography}{9}

\bibitem{positivity} V. Dotsenko, M. Markl, and E. Remm, 
\emph{Non-koszulness of operads and positivity of Poincar\'e series},
arXiv:1604.08580.

\bibitem{alternative} A. Dzhumadil'daev and P. Zusmanovich,
\emph{The alternative operad is not Koszul}, 
Experiment. Math. \textbf{20} (2011), N2, 138--144; 
Corrigendum: \textbf{21} (2012), N4, 418; 
\href{http://arxiv.org/abs/0906.1272}{arXiv:0906.1272}.

\end{thebibliography}





\end{document}
%end of synopsis.tex
