text:
\subsection*{Introduction}

In this paper I propose several problems in the theory of skew braces. I hope
that these problems will help to strengthen the interest in the theory of skew
braces and set-theoretic solutions to the Yang--Baxter equation. I have not
attempted to review the general theory. I do not discuss problems related to
homology of solutions~\cite{MR3558231}, semibraces~\cite{MR3649817} or
trusses~\cite{trusses}.  I concentrate only on skew braces and the Yang--Baxter
equation.

In~\cite[\S9]{MR1183474} Drinfeld wrote that maybe it would be interesting to
study set-theoretical solutions to the equation
\begin{equation}
	\label{eq:YB}
	R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12},
\end{equation}
where $X$ is a set, $R\colon X\times X\to X\times X$ and $R_{ij}$ is the map acting
like $R$ on the two factors $i$ and $j$ and leaving the third factor alone.
Writing $r=\tau\circ R$, where $\tau$ is the map $(x,y)\mapsto (y,x)$,
Equation~\eqref{eq:YB} becomes
\begin{equation}
	\label{eq:braid}
	r_{12}r_{23}r_{12}=r_{23}r_{12}r_{23},
\end{equation}
where $r_{12}=r\times\mathrm{id}$ and $r_{23}=\mathrm{id}\times r$.

Let us say that a pair $(X,r)$ is a \emph{set-theoretic solution} to the Yang--Baxter
equation (YBE) if $X$ is a non-empty set and $r\colon X\times X\to X\times X$ is a
bijective map satisfying~\eqref{eq:braid}.
If $(X,r)$ is a solution of the YBE we write
\[
	r(x,y)=(\sigma_x(y),\tau_y(x)).
\]
We say that the solution $(X,r)$ is \emph{involutive} if $r^2=\mathrm{id}_{X\times X}$
and \emph{non-degenerate} if $\sigma_x\in\mathbb{S}_X$ and $\tau_x\in\mathbb{S}_X$ for all
$x\in X$, where $\mathbb{S}_X$ denotes the group of bijective maps $X\to X$. By
convention, our \emph{solutions} will always be non-degenerate solutions.

One tool to study set-theoretic solutions to the YBE is the theory of skew left braces.
Braces were introduced by Rump in~\cite{MR2278047}.
For a recent survey on braces we refer to~\cite{cedo}.
Skew braces appeared later
in~\cite{MR3647970}.  A \emph{skew left brace} is a triple $(A,+,\circ)$, where
$(A,+)$ and $(A,\circ)$ are groups
(not necessarily abelian) such that
\begin{equation}
	\label{eq:left}
	a\circ (b+c)=a\circ b-a+a\circ c
\end{equation}
for all $a,b,c\in A$.
To define skew right braces one needs to
replace~\eqref{eq:left} by
\begin{equation}
	\label{eq:right}
	(a+b)\circ c=a\circ c-c+b\circ c.
\end{equation}
The inverse of an element $a\in A$ with respect to the circle (or
multiplicative) group of $A$ will be denoted by $a'$. Examples of skew left braces can
be found in~\cite{MR3320237,MR3177933,MR3647970,MR3291816}.

A skew \emph{two-sided} brace is a skew left brace that is also a skew right
brace with respect to the same pair of operations.

If $\mathcal{X}$ is a class of groups, a skew left brace will be called of type
$\mathcal{X}$ if its additive group belongs to $\mathcal{X}$.  Skew left braces
of abelian type are those braces introduced by Rump in~\cite{MR2278047}.  By
convention, a \emph{left brace} will always be a skew left brace of abelian
type, i.e. with abelian additive group.

Braces and skew left braces have several interesting connections, see for
example~\cite{MR3291816,MR3763907}. The connection between skew left braces and
non-degenerate solutions of the YBE is given in the following theorem,
see~\cite[Theorem 3.1]{MR3647970}:

\begin{thm*}
	If $A$ is a skew left
	brace, then the map
	\[
		r_A\colon A\times A\to A\times A,\quad
		r_A(a,b)=(-a+a\circ b,(-a+a\circ b)'\circ a\circ b),
	\]
	is a non-degenerate
	set-theoretic solution of the YBE. Moreover, $r_A$ is involutive if and
	only if $A$ is of abelian type.
\end{thm*}

Solutions produced from skew left braces are universal. In a
different language, one finds the following result in
\cite{MR1722951,MR1769723,MR1809284}:

\begin{thm*}
	If $(X,r)$ be a solution of the YBE, then there
	exists a unique skew left brace structure over the group
	\[
		G=G(X,r)=\langle X:x\circ y=u\circ v\text{ whenever $r(x,y)=(u,v)$}\rangle
	\]
	such that the diagram
	\[
		\begin{tikzcd}
			X\times X\arrow[r, "r"] \arrow["\iota\times\iota"', d]
			& X\times X \arrow[d, "\iota\times\iota" ] \\
			G\times G \arrow[r, "r_{G}"]
			& G\times G
		\end{tikzcd}
	\]
	commutes, where $\iota\colon X\to G(X,r)$ is the canonical map.
\end{thm*}

One proves that the pair $(G(X,r),\iota)$ satisfies a universal property,
see~\cite[Proposition 3.9]{MR3647970} and~\cite[Theorem 3.5]{MR3763907}.  The
group $G(X,r)$ is infinite since for example the degree map $X\to\mathbb{Z}$,
$x\mapsto 1$, extends to a group homomorphism $G(X,r)\to\mathbb{Z}$.  Several
properties of $G(X,r)$
are discussed in~\cite{MR2301033}.
If the set $X$ of the solution $(X,r)$ is finite, then one can realize this
solution using a finite skew left brace, see~\cite{Bachiller3,MR3177933}.

\subsection*{Problems}

It was observed by Rump~\cite{MR2132760} that non-degenerate involutive
solutions are in bijective correspondence with non-degenerate cycle sets. A
\emph{cycle set} is a pair $(X,\cdot)$, where $X\times X\to X$, $(x,y)\mapsto
x\cdot y$, is a map such that each $\varphi_x\colon X\to X$, $y\mapsto x\cdot
y$, is bijective, and
\[
(x\cdot y)\cdot (x\cdot z)=(y\cdot
x)\cdot (y\cdot z)
\]
for all $x,y,z\in X$.

A cycle set $(X,\cdot)$ is said to be \emph{non-degenerate} if the map
$x\mapsto x\cdot x$ is bijective. Finite cycle sets are non-degenerate,
see~\cite[Theorem~2]{MR2132760}. The correspondence between non-degenerate
cycle sets and involutive solutions is given by
\[
	r(x,y)=((y*x)\cdot y,y*x)
\]
where $x*y=z$ if and only if $x\cdot z=y$.
Homomorphisms of cycle sets are defined in the usual manner.

\begin{problem}
	Construct the free cycle set.
\end{problem}

It would be interesting to have a nice description of the free cycle set. This
nice description could be used for example to compute homology.

In~\cite{MR1722951}, Etingof, Schedler and Soloviev constructed all
non-degenerate involutive solutions of cardinality $\leq8$, see
Table~\ref{tab:numbers}. To construct finite involutive solutions one needs to
construct finite cycle sets.

\begin{table}[h]
        \centering
        \caption{Involutive non-degenerate solutions of cardinality $\leq8$.}
        \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
            \hline
            cardinality & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\
			\hline
            solutions & 1 & 2 & 5 & 23 & 88 & 595 & 3456& 34528\\
            square-free & 1 & 1 & 2 & 5 & 17 & 68 & 336& 2041\\
            indecomposable & 1 & 1 & 1 & 5 & 1 & 10 & 1 & 98\\
            \hline
        \end{tabular}
        \label{tab:numbers}
\end{table}

\begin{problem}
	\label{pro:size9}
	Construct all cycle sets of small cardinality.
\end{problem}

One can try for example with the construction of cycle sets of cardinality nine. Of
course Problem~\ref{pro:size9} refers to the construction of all isomorphism
classes of cycle sets.  It would also be interesting to construct all cycle
sets (or involutive solutions) of small cardinality under particular assumptions. For
example, a solution $(X,r)$ of the YBE is said to be \emph{square free} if
$r(x,x)=(x,x)$ for all $x\in X$, this means that one needs cycle sets
$(X,\cdot)$ such that $x\cdot x=x$ for all $x\in X$.

\begin{problem}
	\label{pro:squarefree}
	Construct all square-free cycle sets
	of small cardinality.
\end{problem}

An involutive solution $(X,r)$ is said to be \emph{indecomposable} if the group
$\mathcal{G}(X,r)$ defined as the subgroup of $\mathbb{S}_X$ generated by
$\{\sigma_x:x\in X\}$ acts transitively on $X$.  Similarly one defines
indecomposable cycle sets.  The classification of indecomposable solutions with
a prime number of elements appears in~\cite{MR1783929,MR1848966}.

\begin{problem}
	\label{pro:indecomposable}
	Construct all indecomposable
	cycle sets of small cardinality.
\end{problem}

Maybe Problems~\ref{pro:size9}--\ref{pro:indecomposable} and similar problems
could be studied using constraint satisfaction methods.

The number of finite
involutive solutions increases rapidly with the number of elements of the
underlying set.  Therefore it makes sense to ask for an estimation:

\begin{problem}
	Estimate the number of cycle sets of cardinality $n$ for $n\to\infty$.
\end{problem}

Much less is known for non-involutive solutions. Before going into the general
problem of constructing non-involutive solutions, one could start with
injective solutions. A solution $(X,r)$ is said to be \emph{injective} if the canonical
map $X\to G(X,r)$ is injective.  As it was proved in~\cite{MR1722951},
involutive solutions are always injective:
\[
	\{\text{involutive solutions}\}\subsetneq
	\{\text{injective solutions}\}\subsetneq
	\{\text{solutions}\}.
\]

\begin{problem}
	Construct all injective non-involutive solutions of
	low cardinality.
\end{problem}

\begin{problem}
	\label{pro:all}
	Construct all solutions of small cardinality.
\end{problem}

Problem~\ref{pro:all} is related to the construction of finite biquandles of
small cardinality, see for example~\cite{MR2819176,MR3666513}.
A different approach to construct all solutions could be based on skew left
braces, see~\cite{Bachiller3};  this method
requires the classification of finite skew left braces.

Simple involutive solutions were defined in~\cite[\S2]{MR3437282}. A surjective
map $p\colon X\to Y$ of involutive solutions is said to be a \emph{covering} if
all the fibers $p^{-1}(y)$ have the same cardinality. A covering $X\to Y$ is
said to be \emph{trivial} if either $|Y|=1$ or $|Y|=|X|$.  An involutive
solution $(X,r)$ is said to be \emph{simple} if $|X|>1$ and any covering $X\to
Y$ is trivial.

\begin{problem}
	Classify finite simple involutive solutions.
\end{problem}

One could simply ask for examples of small involutive simple solutions:

\begin{problem}
	Construct finite simple involutive solutions of small cardinality.
\end{problem}

It would be interesting to understand the simplicity of an involutive solution in the
language of left braces:

\begin{problem}
	Is it possible to read off the simplicity of an involutive solution $(X,r)$ in
	terms of the left brace $G(X,r)$?
\end{problem}

It would be also interesting to develop the theory of non-involutive simple
solutions. Some ideas could be obtained if one reads off the simplicity of a
solution in terms of a skew left brace where it is realized.

\begin{problem}
	\label{pro:torsion}
	Let $(X,r)$ be a solution to the YBE.  When does the multiplicative group of
	$G(X,r)$ have torsion?
\end{problem}

Problem~\ref{pro:torsion} was posed by Bachiller in~\cite{Bachiller3}.
Gateva--Ivanova and Van den Bergh proved in~\cite{MR1637256} that in the case
of involutive solutions the multiplicative group of $G(X,r)$ is always
torsion-free. It is easy to construct examples of non-involutive solutions
$(X,r)$ such that the multiplicative group of $G(X,r)$ has torsion.

\begin{problem}
	Construct the free (skew) left brace.
\end{problem}

Based on the work of Bachiller~\cite{MR3465351} and Catino and
Rizzo~\cite{MR2486886} on regular subgroups, an algorithm to construct finite
skew left braces was introduced in~\cite{MR3647970}.  With some exceptions,
this algorithm was used to construct small (skew) left braces.  For $n\in\mathbb{N}$
let $s(n)$ be the number of non-isomorphic skew left braces of size $n$ and let
$b(n)$ be the number of non-isomorphic left braces of size $n$.  Table
\ref{tab:braces} shows some values of $s(n)$ and $b(n)$ and several open cases.
Maybe some of these open cases are
fairly straightforward computational projects.

\begin{table}[h]
	\caption{The number of non-isomorphic left braces and skew left braces.}
	\begin{tabular}{|c|cccccccccccc|}
		\hline
		$n$ & 1 & 2 & 3 & 4 & 5  & 6 & 7 & 8 & 9 & 10 & 11 & 12\tabularnewline
		$b(n)$ & 1  & 1  & 1  & 4  & 1  & 2  & 1  & 27  & 4  & 2  & 1  & 10\tabularnewline
		$s(n)$ & 1 & 1 & 1 & 4 & 1 & 6 & 1 & 47 & 4 & 6 & 1 & 38 \tabularnewline
		\hline
		$n$ & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24\tabularnewline
		$b(n)$ & 1  & 2  & 1  & 357  & 1  & 8  & 1  & 11  & 2  & 2  & 1  & 96\tabularnewline
		$s(n)$ & 1 & 6 & 1 & 1605 & 1 & 49 & 1 & 43 & 8 & 6 & 1 & 855 \tabularnewline
		\hline
		$n$ & 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 & 33 & 34 & 35 & 36\tabularnewline
		$b(n)$ & 4  & 2  & 37  & 9  & 1  & 4 & 1  & ?  & 1  & 2  & 1  & 46\tabularnewline
		$s(n)$ & 4 & 6 & 101 & 29 & 1 & 36 & 1 & ? &  1 & 6 & 1 & 400\tabularnewline
		\hline
		$n$ & 37 & 38 & 39 & 40 & 41 & 42 &  43 & 44 & 45 & 46 & 47 & 48\tabularnewline
		$b(n)$  & 1  & 2  & 2  & 106  & 1  & 6  & 1  & 9  & 4 & 2  & 1  & 1708\tabularnewline
		$s(n)$ & 1 & 6 & 8 & 944 & 1 & 78 & 1 & 29 & 4 & 6 & 1 & 66209 \tabularnewline
		\hline
		$n$    & 49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60\tabularnewline
		$b(n)$ &  4 & 8 & 1 & 11 & 1 & 80 & 2 & 91 & 2 & 2 & 1 & 28\tabularnewline
		$s(n)$ & 4 & 51 & 1 & 43 & 1 & ? &  12 & 815 & 2 & 6 & 1 & 418\tabularnewline

		\hline
		$n$ & 61 & 62 & 63 & 64 & 65 & 66 & 67 & 68 & 69 & 70 & 71 & 72\tabularnewline
		$b(n)$ & 1 & 2 & 11 & ? & 1 & 4 & 1 & 11 & 1 & 4 & 1 & 489\tabularnewline
		$s(n)$ & 1 & 6 & 11 & ? &  1 & 36 & 1 & 43 & 1 & 36 & 1 & 17790 \tabularnewline

		\hline
		$n$ & 73 & 74 & 75 & 76 & 77 & 78 & 79 & 80 & 81 & 82 & 83 & 84\tabularnewline
		$b(n)$ & 1 & 2 & 5 & 9 & 1 & 6 & 1 & 1985 & ? & 2 & 1 & 34\tabularnewline
		$s(n)$ & 1 & 6 & 14 & 29 & 1 & 78 & 1 & ? &  ? &  6 & 1 & 606\tabularnewline

		\hline
		$n$ & 85 & 86 & 87 & 88 & 89 & 90 & 91 & 92 & 93 & 94 & 95 & 96\tabularnewline
		$b(n)$ & 1 & 2 & 1 & 90 & 1 & 16 & 1 & 9 & 2 & 2 & 1 & ?\tabularnewline
		$s(n)$ & 1 & 6 & 1 & 800 & 1 & 294 & 1 & 29 & 8 & 6 & 1 & ? \tabularnewline

		\hline
		$n$ & 97 & 98 & 99 & 100 & 101 & 102 & 103 & 104 & 105 & 106 & 107 & 108\tabularnewline
		$b(n)$ & 1 & 8 & 4 & 51 & 1 & 4 & 1 & 106 & 2 & 2 & 1 & 494 \tabularnewline
		$s(n)$ & 1 & 53 &  4 & 711 & 1 & 36 & 1 & 944 &  8 &  6 &  1 & ? \tabularnewline

		\hline
		$n$ & 109 & 110 & 111 & 112 & 113 & 114 & 115 & 116 & 117 & 118 & 119 & 120\tabularnewline
		$b(n)$ & 1 & 6 & 2 & 1671 & 1 & 6 & 1 & 11 & 11 & 2 & 1 & 395\tabularnewline
		$s(n)$ & 1 & 94 &  8 &  ? &  1 & 78 &  1 & 43 &  47 &  6 &  1 & ? \tabularnewline
		\hline

		$n$ & 121 & 122 & 123 & 124 & 125 & 126 & 127 & 128 & 129 & 130 & 131 & 132\tabularnewline
		$b(n)$ & 4 & 2 & 1 &  9  & 49 & 36 & 1 & ? & 2 &  4 & 1 & 24\tabularnewline
		$s(n)$ & 4 & 6 & 1 & 29 & 213 & 990 & 1 & ? & 8 & 36 & 1 & 324 \tabularnewline
		\hline

		$n$  & 133 & 134 & 135 & 136 & 137 & 138 & 139 & 140 & 141 & 142 & 143 & 144\tabularnewline
		$b(n)$ & 1 & 2 & 37 & 108 & 1 & 4 & 1 & 27 & 1 & 2 & 1 & ?\tabularnewline
		$s(n)$ & 1 & 6 & 101 & ? & 1 & 36 & 1 & 395 & 1 & 6 & 1 & ? \tabularnewline
		\hline

		$n$ & 145 & 146 & 147 & 148 & 149 & 150 & 151 & 152 & 153 & 154 & 155 & 156\tabularnewline
		$b(n)$ & 1 & 2 & 9 & 11 & 1 & 19 & 1 & 90 & 4 & 4 & 2 & 40\tabularnewline
		$s(n)$ & 1 & 6 & ? & 43 & 1 & ? & 1 & ? & 4 & 36 & 12 & 782 \tabularnewline
		\hline

		$n$ & 157 & 158 & 159 & 160 & 161 & 162 & 163 & 164 & 165 & 166 & 167 & 168\tabularnewline
		$b(n)$ & 1 & 2 & 1 & ? & 1 & ? & 1 & 11 & 2 & 2 & 1 & 443\tabularnewline
		$s(n)$ & 1 & 6 & 1 & ? & 1 & ? & 1 & 43 & 12 & 6 & 1 & ? \tabularnewline
		\hline
	\end{tabular}
	\label{tab:braces}
\end{table}

\begin{problem}
	\label{pro:32}
	Construct all left braces of size $32$.
\end{problem}

Some results related to Problem~\ref{pro:32} are shown in
Table~\ref{tab:braces32}.  According to these results, one needs to construct
left braces of size $32$ with additive group isomorphic to $C_2^{5}=C_2\times
C_2\times C_2\times C_2\times C_2$, where $C_2$ denotes the cyclic
(multiplicative) group with two elements.  Apparently, this particular case
cannot be done in reasonable time with the techniques of~\cite{MR3647970}.  The
number of (skew) left braces of size $64$, $96$ or $128$ seems to be extremely
large and our computational methods are not strong enough to construct them
all.

\begin{table}[h]
	\centering
	\caption{Some calculations of braces of size $32$.}
	\begin{tabular}{|c|c|c|}
		\hline
		additive group & number & time needed\tabularnewline
		\hline
		$C_{32}$ & 9 & 0.1\tabularnewline
		$C_8\times C_4$ & 1334 & 4 hours\tabularnewline
		$C_{16}\times C_2$ &  120 & 30 minutes\tabularnewline
		$C_4^2\times C_2$ & 13512 & 13 days\tabularnewline
		$C_8\times C_2^2$ & 1547 & 10 hours\tabularnewline
		\hline
	\end{tabular}
	\label{tab:braces32}
\end{table}

\begin{problem}
	\label{pro:p^n}
	Let $p$ be a prime number.  Classify
	skew left braces of size $p^n$.
\end{problem}

Problem~\ref{pro:p^n} is important because is the key step in the
classification of skew left braces of nilpotent type.

Left braces of size $p^2$ and $p^3$, where $p$ is a prime number, were
classified by Bachiller in~\cite{MR3320237}. He proved that $b(p^2)=4$ and
$b(p^3)=6p+19$.  Since groups of order $p^2$ are abelian, it follows that
$s(p^2)=4$.  Skew left braces of size $p^3$ were classified by Nejabati Zenouz
in~\cite{NZ}. He proved that $s(p^3)=6p^2+8p+23$ whenever $p>3$. From
Table~\ref{tab:braces} one gets $s(8)=47$ and $s(27)=101$.

\begin{problem}
	\label{pro:pq}
	Let $p$ and $q$ be different prime numbers.
	Construct all skew left braces of size $pq$.
\end{problem}

It should be fairly easy to solve Problem~\ref{pro:pq} using the techniques
of~\cite{MR2030805}.

\begin{problem}
	\label{pro:p^2q}
	Let $p$ and $q$ be different prime numbers. Consruct all skew left braces
	of size $p^2q$.
\end{problem}

Left braces of size $p^2q$ for prime numbers $p$ and $q$ such that $q>p+1$ were
classified by Dietzel in~\cite{Dietzel}. He proved that
\[
	b(4q)=\begin{cases}
		9 & \text{if $4\nmid q-1$},\\
		11 & \text{if $4\mid q-1$},
	\end{cases}
\]
if $q>3$, and that
\[
	b(p^2q)=\begin{cases}
		4 & \text{if $p\nmid q-1$},\\
		p+8 & \text{if $p\mid q-1$ and $p^2\nmid q-1$},\\
		2p+8 & \text{if $p^2\mid q-1$},
	\end{cases}
\]
if $q>p+1>3$.

\begin{problem}
	\label{pro:estimate_braces}
	Estimate the number of (skew) left braces of size $n$ for $n\to\infty$.
\end{problem}

The following problem appears in~\cite{cedo}:

\begin{problem}
	\label{pro:aut}
	Compute automorphism groups of skew left braces of size $p^n$.
\end{problem}

Automorphism groups of skew left braces of size $p^3$
were computed in~\cite{NZpaper}.

In~\cite[2.28(I)]{MR2095675} Gateva--Ivanova made the following conjecture: For
each finite involutive square-free solution $(X,r)$ there exist $x,y\in X$ such
that $x\ne y$ and $\sigma_x=\sigma_y$.  It was proved by Ced\'o, Jespers and
Okni\'nski in~\cite{MR2652212} that the conjecture is true if the group
$\mathcal{G}(X,r)$ generated by $\{\sigma_x:x\in X\}$ is abelian.  Later in~\cite{MR2885602} and
in~\cite{MR3177933} it was shown using other methods that the result is also valid
if $\mathcal{G}(X,r)$ is infinite abelian. In full generality, the conjecture is
now known to be false. The first counterexample appeared in~\cite{MR3437282};
other counterexamples were later constructed in~\cite{MR3719300,MR3780533}.  It
would be interesting to find essentially new counterexamples.

\begin{problem}
	\label{pro:GI}
	Construct counterexamples to Gateva--Ivanova conjecture.
\end{problem}

One could ask, for example, for counterexamples of size nine.  Computer
calculations show that there is only one counterexample to Gateva--Ivanova
conjecture among the 38698 involutive solutions of size $\leq8$.

It could be enlightening to attack Problem~\ref{pro:GI} using the theory of
braces.  Let $A$ be a brace and $X$ be a subset of $A$ such that the
restriction $r_X=r_A|_{X\times X}$ of $r_A$ to $X\times X$ is a solution to the
YBE. We say that $(A,X)$ is a \emph{Gateva--Ivanova pair} if the solution
$(X,r_X)$ is a counterexample to Gateva--Ivanova conjecture.

\begin{problem}
	\label{pro:pairs}
	Find Gateva--Ivanova pairs.
\end{problem}

One could start studying Problem~\ref{pro:pairs} by inspecting the database of
small braces. I should remark that counterexamples to Gateva--Ivanova
conjecture might provide new examples of
Artin-Schelter regular algebras with global dimension
$>3$ with interesting properties to study. Gateva--Ivanova conjecture motivated
a deeper study of the structure of braces and related objects, see for
example~\cite{MR3177933, GI15,MR2278047}.

Gateva--Ivanova conjecture is of course related with the retractability of
involutive solutions, introduced in~\cite{MR1722951}.  For an involutive
solution $(X,r)$, consider the equivalence relation on $X$ given by $x\sim y$ if
and only if $\sigma_x=\sigma_y$.
Denoting by $Y$ the set of equivalence classes of $X$, the
map $r$ induces a function $\overline{r}\colon Y\times Y\to Y\times Y$. The
pair $(Y,\overline{r})$ is a solution to the YBE which will be denoted by
$\operatorname{Ret}(X, r)$.
One defines inductively
\[
\operatorname{Ret}^{m+1}(X, r) = \operatorname{Ret}(\operatorname{Ret}^m (X, r))
\]
for all $m\geq1$.  The solution $(X, r)$ is said to be a \emph{multipermutation
solution} if there is a positive integer $m$ such that $\operatorname{Ret}^m(X,
r)$ has only one element. The solution $(X,r)$ is said to be
\emph{irretractable} if $\operatorname{Ret}(X, r) = (X, r)$.

A group $G$ admits a \emph{left ordering} if it admits a total
ordering $<$ such that if $x<y$ then $zx<zy$ for all $x,y,z\in G$.
In~\cite{MR3815290} it is proved that a finite involutive set-theoretic
solution $(X, r)$ of the YBE is a multipermutation solution if and only if its
structure group $G(X,r)$ admits a left ordering. One of the implications was
proved in ~\cite{MR3572046,MR2189580}.

\begin{problem}
	\label{pro:left}
	Let $(X,r)$ be an injective non-involutive solution of the YBE. When does the
	multiplicative group of $G(X,r)$ admit a left ordering?
\end{problem}

Problem~\ref{pro:left} appears in~\cite{LV3}.

A group $G$ satisfies the \emph{unique product property} if for every finite
non-empty subsets $A$ and $B$ of $G$ there is an element $x$ which can be
written uniquely as $x = ab$ with $a\in A$ and $b\in B$.

\begin{problem}
	\label{pro:UPP}
	Let $(X, r)$ be an involutive irretractable solution of the YBE.
	Can $G(X,r)$ satisfy the unique product property?
\end{problem}

In~\cite[Example 8.2.14]{MR2301033} Jespers and Okni\'nksi proved that the
structure group $G(X,r)$ of a certain involutive irretractable solution $(X,r)$
of cardinality four does not satisfy the unique product property.  They showed that
this structure group contains a subgroup isomorphic to the celebrated Promislow
group~\cite{MR940281}.

A group $G$ is said to be \emph{diffuse} if for every finite non-empty subset
$A$ of $G$ there exists $a\in A$ such that for all $g\in G\setminus\{1\}$,
either $ga\not\in A$ or $g^{-1}a\not\in A$.  Diffuse groups satisfy the unique
product property. However, the precise relation between diffuseness and unique
products is not clear at the moment.  In~\cite{LV3} it is proved that the
structure group $G(X,r)$ of an involutive solution $(X,r)$ is diffuse if and
only if $(X,r)$ is a multipermutation solution.  Therefore a positive answer to
Problem~\ref{pro:UPP} would give an example of a non-diffuse group with the
unique product property.

\begin{problem}
	\label{pro:MP}
	Study non-involutive multipermutation solutions.
\end{problem}

Some work related to Problem~\ref{pro:MP} can be found in~\cite{LV3,MR3763907}.

As it was observed by Rump, two-sided braces are equivalent to radical rings.
The multiplication of this radical ring is the operation
\[
a*b=-a+a\circ b-b.
\]
It makes sense to consider this operation for all skew left braces, although in
general it will not be associative.  The following problem appears
in~\cite{MR3818285}.

\begin{problem}
	\label{pro:star_associative}
	Let $A$ be a left brace such that the operation $*$ is associative. Is $A$
	a two-sided brace?
\end{problem}

It is proved in~\cite[Proposition 2.2]{MR3818285} that the answer is positive
if the additive group of the left brace has no elements of order two. For skew
braces, Problem~\ref{pro:star_associative} can be answered in the negative,
see~\cite[\S1]{KSV}.

A \emph{left ideal} of a skew left brace $A$ is a subgroup $L$ of the additive
group of $A$ such that $A*L\subseteq L$. An \emph{ideal} of $A$ is a left ideal
$I$ such that $a+I=I+a$ and $a\circ I=I\circ a$ for all $a\in A$. A skew left
brace $A$ is said to be \emph{simple} if it has no ideals different from
$\{0\}$ and~$A$.

\begin{problem}
	\label{pro:simple}
	Classify finite simple skew left braces.
\end{problem}

Problem~\ref{pro:simple} is intensively studied for finite left braces, see for
example~\cite{MR3763276,BCJO18,MR3812099}. However, not much is known about
finite simple skew left braces that are not of abelian type.

\begin{problem}
	\label{pro:representations}
	Study representation theory of skew left braces.
\end{problem}

Rump defined modules over right braces in~\cite{MR2320986}. A different
interpretation of Problem~\ref{pro:representations} could be the following: To
study consequences of having skew left brace homomorphisms between two given
skew left braces.

Left braces such that either the additive or the multiplicative group is
isomorphic to $\mathbb{Z}$ were classified in~\cite{CSV}. It is not clear how to extend
some of these results to skew left braces.  The operations $g^k + g^l =
g^{k+(-1)^kl}$ and $g^k\circ g^l = g^{k+l}$ turn the set $\{g^k:k\in\mathbb{Z}\}$ into
a skew left brace with multiplicative group isomorphic to $\mathbb{Z}$ and additive
group isomorphic to the infinite dihedral group.  It seems that there are no
other non-trivial skew left braces with multiplicative group isomorphic to
$\mathbb{Z}$.

\begin{problem}
	\label{pro:Z}
	Classify isomorphism classes of skew left braces with multiplicative group
	isomorphic to $\mathbb{Z}$.
\end{problem}

Problem~\ref{pro:Z} could be studied using techniques from factorizable groups,
see for example~\cite{MR1211633}. Similarly one could also ask for the
classification of skew left braces where one of its groups is isomorphic to the
infinite dihedral group.

Finite skew left braces with cyclic additive groups were classified by Rump
in~\cite{MR2298848,Rump}.
In the same vein, it would be interesting to know the classification of skew
left braces with multiplicative group isomorphic to the quaternion group.  This
problem is not even solved for left braces.  Such left braces are called
\emph{quaternion left braces}. For $m\in\mathbb{N}$ let $q(4m)$ be the number of
isomorphism classes of quaternion left braces of size $4m$.

\begin{problem}
	\label{pro:quaternion}
	Prove that
	\[
		q(4m)=\begin{cases}
			2 & \text{if $m$ is odd,}\\
			7 & \text{if $m\equiv0\bmod8$,}\\
			9 & \text{if $m\equiv4\bmod8$,}\\
			6 & \text{if $m\equiv2\bmod8$ or $m\equiv6\bmod8$}.\\
		\end{cases}
	\]
\end{problem}

The conjectural formula for $q(4m)$ of Problem~\ref{pro:quaternion} was verified by
computer for all $m\leq128$.

\begin{problem}
		Which finite abelian groups appear as the additive group of a
		quaternion left brace?
\end{problem}

For $m\in\{2,\dots,128\}$ the additive group of a quaternion left brace of order $4m$
is isomorphic to one of the following groups:
\begin{equation*}
        C_{4m},\,
        C_{2m}\times C_2,\,
        C_m\times C_2\times C_2,\,
        C_m\times C_4,\,
        C_{m/2}\times C_2\times C_2\times C_2.
\end{equation*}
By inspection, $C_{m}\times C_2^2$ appears if
$m\equiv2,4,6\bmod8$ and $C_m\times C_4$ and $C_{m/2}\times C_2^3$
appear if $m\equiv4\bmod8$.

\begin{problem}
    \label{con:Q}
	Is it true that there are seven classes of isomorphism of quaternion left braces of
	size $2^k$ for $k>4$?
\end{problem}

It is interesting to study properties of groups appearing as the multiplicative
groups of skew left braces. Let us start with the case of left braces. A
finite group $G$ is said to be an \emph{involutive Yang--Baxter} group (or
simply a IYB group) if it is isomorphic to the multiplicative group of a finite
left brace.  In~\cite{MR1722951}, Etingof, Schedler and Soloviev proved that IYB
groups are always solvable.  A natural problem arises: Is every finite solvable
group the multiplicative group of a left brace?  The answer is negative, as it was
shown by Bachiller~\cite{MR3465351} following the ideas of
Rump~\cite{MR3291816}.  However, it would be interesting to find other
counterexamples.

\begin{problem}
	Which is the minimal cardinality of a solvable group that is not a IYB group?
\end{problem}

Some results related to IYB-groups can be found in~\cite{
MR3500774,
MR2584610,MR3177933,MR3447734,MR3116652}.  Related problems are the following:

\begin{problem}
	\label{pro:Cedo1}
	Is every nilpotent group of nilpotecy class two the multiplicative group
	of a left brace?
\end{problem}

\begin{problem}
	\label{pro:Cedo2}
	Is every nilpotent group of nilpotecy class two the multiplicative group
	of a two-sided brace?
\end{problem}

\begin{problem}
	\label{pro:Cedo3}
	Which finite nilpotent groups are multiplicative groups of a
	two-sided (or left) brace?
\end{problem}

Problems~\ref{pro:Cedo1}--\ref{pro:Cedo3} appeared in the survey~\cite{cedo}.
Problem~\ref{pro:Cedo3} is interesting even in the particular case of groups of nilpotency class
$\leq3$.

I learned the following related problem from Rump:

\begin{problem}
	\label{pro:IYBSylow}
	Is there an example of a non-IYB finite group where all the Sylow subgroups
	are IYB groups?
\end{problem}

The solution to the YBE one obtains from a skew left brace is always a
biquandle. Biquandles are non-associative structures useful in combinatorial
knot theory.  We refer to~\cite{MR3379534,MR2896084,MR3444659} for nice
introductions to the subject.

\begin{problem}
	Study knot invariants produced from skew left braces.
\end{problem}

As biquandles produce knot invariants and these invariants can be strengthened
by using quandle and biquandle homology, it is then natural to ask if the
homology of braces of~\cite{MR3530867} can be used in combinatorial knot
theory:

\begin{problem}
	Is it possible to strengthen invariants produced from skew left braces by
	using brace homology?
\end{problem}

Left and right nilpotent left (and right) braces were defined by Rump in~\cite{MR2278047}.
Strongly nilpotent left braces were defined by Smoktunowicz in~\cite{MR3814340}.
These definitions extend to skew left braces, see~\cite{CSV}.  A skew brace
$A$ is said to be \emph{left nilpotent} if there exists $n\geq1$ such that
$A^n=0$, where $A^1=A$ and $A^{n+1}$ is defined as the subgroup $A*A^n$ of $(A,+)$
generated by $\{a*x:a\in A,x\in A^n\}$. Similarly $A$ is said to
be \emph{right nilpotent} if there exists $n\geq1$ such that $A^{(n)}=0$, where
$A^{(1)}=A$ and $A^{(n+1)}$ is defined as the subgroup $A^{(n)}*A$ of $(A,+)$ generated by
$\{x*a:x\in A^{(n)},a\in A\}$.

The following problem of Smoktunowicz appears in~\cite{MR3765444} in an equivalent formulation:

\begin{problem}
	\label{pro:Agata}
	Let $G$ be a finite group which is the multiplicative group of some left
	brace. Is $G$ is the multipicative group of a right nilpotent left brace?
\end{problem}

Problem~\ref{pro:Agata} also makes sense for skew left braces of nilpotent type.

\begin{problem}
	\label{pro:CSV1}
	Are there simple two-sided skew braces of nilpotent type?
\end{problem}

A skew brace $A$ is said to be \emph{prime}
if for all non-zero ideals $I$ and $J$, the subgroup $I*J$ of $(A,+)$ generated by
$\{u*v:u\in I,v\in J\}$ is non-zero.

\begin{problem}
	\label{pro:CSV2}
	Is every finite prime skew left brace of nilpotent type a simple skew left brace?
\end{problem}

\begin{problem}
	\label{pro:CSV3}
	Is every finite prime left brace a simple left brace?
\end{problem}

In ~\cite[\S5]{abundance} Ced\'o, Jespers and Okni\'nski found a
prime non-simple finite left brace. This example solves
Problems~\ref{pro:CSV2} and~\ref{pro:CSV3}.

\begin{problem}
	\label{pro:CSV4}
	Are there prime two-sided skew braces of nilpotent type?
\end{problem}

A skew left brace is said to be \emph{strongly nilpotent} if it is right and
left nilpotent.  A skew left brace $A$ is said to be \emph{strongly nil} if for
every element $a\in A$ there is a positive integer $n=n(a)$ such that any
$*$-product of $n$ copies of $a$ is zero.

\begin{problem}
	\label{pro:CSV5}
	Is every finite strongly nil skew left brace a strongly nilpotent skew left brace?
\end{problem}

For a skew left brace $A$ let $\rho_1(a)=a$ and $\rho_{k+1}(a)=\rho_k(a)*a$ for
$n\geq1$.  The skew left brace $A$ is said to be \emph{right nil} if there
exists a positive integer $n$ such that $\rho_n(a)=0$ for all $a\in A$.

\begin{problem}
	\label{pro:CSV6}
	Is every finite right nil skew left brace a right nilpotent skew left brace?
\end{problem}

Radical rings are the source of several other problems for skew left braces.
For example, it might make sense
to discuss an analog of the K\"othe conjecture in the context of skew left braces.
For rings one of the formulations of the conjecture is the following: the sum of two nil left ideals in a
ring is a left nil ideal. The conjecture was formulated arround 1930 and it is
still open, see~\cite{MR3439122,MR1879880}.

\begin{problem}
	\label{pro:CSV7}
	Is there an analog of the K\"othe conjecture for skew left braces?
\end{problem}

Problems~\ref{pro:CSV1},~\ref{pro:CSV2},~\ref{pro:CSV3},~\ref{pro:CSV4},~\ref{pro:CSV5}
and~\ref{pro:CSV6} appear in~\cite{CSV} and~\cite{KSV}. K\"othe conjecture has
been shown to be true for various classes of rings such as polynomial identity
rings, right Noetherian rings and radical rings.

Bachiller observed the connection between skew braces and Hopf--Galois
extensions. This connection is precisely described in the appendix of~\cite{MR3763907}.
In the theory of Hopf--Galois extensions the following problems are natural,
see~\cite{MR3425626}:

\begin{problem}
	\label{pro:SS}
	Is there a skew left brace with solvable additive group but non-solvable
	multiplicative group?
\end{problem}

\begin{problem}
	\label{pro:NS}
	Is there a skew left brace with non-solvable additive group but nilpotent
	multiplicative group?
\end{problem}

Problems~\ref{pro:SS} and~\ref{pro:NS} also appeared in~\cite{MR3763907}
and~\cite[Problem 19.90]{kourovka}. In general, it is interesting to know
relations between the multiplicative and the additive group of a skew left
brace. Several results in this direction can be found in the theory of
Hopf--Galois extensions, see for example~\cite{MR2011974,MR3030514,MR3425626}.
The following problem is mentioned in~\cite{LV3} and~\cite[Problem
19.49]{kourovka}:

\begin{problem}
	\label{pro:orderable}
	Let $A$ be a skew left brace with left-orderable multiplicative group. Is the
	additive group of $A$ left-orderable?
\end{problem}

Timur Nasybullov constructed an example of a left brace that answers
Problem~\ref{pro:orderable} negatively (private communication). Another example that answers Problem~\ref{pro:orderable} appears in~\cite{CSV}.

In~\cite{CSV} it is proved that skew left braces generated (as a skew left
brace) by a single element yield indecomposable solutions of the YBE. It is
then natural to ask if all finite indecomposable solutions are of this type:

\begin{problem}
	\label{pro:AA}
	Is it true that for each indecomposable involutive solution $(X,r)$
	there exists a one-generator left brace $A$ generated by $x$ such that $X$ is
	of the form $X=\{ax+x:a\in A\}$?

\end{problem}

Problem~\ref{pro:AA} appears in~\cite{MR3771874} for left braces.

Rump proved that braces are equivalent to linear cycle sets.  A \emph{linear
cycle set} is a triple $(A,+,\cdot)$, where $(A,+)$ is an abelian group
and $(A,\cdot)$ is a cycle set
such that
$a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ and
$(a+b)\cdot c=(a\cdot b)\cdot (a\cdot c)$
hold for all $a,b,c\in A$.

A theory of dynamical extensions of cycle sets was used in~\cite{MR3437282} to
produce a counterexample to Gateva--Ivanova conjecture.

\begin{problem}
	\label{pro:dynamical}
	Develop the theory of dynamical extensions of linear cycle sets.
\end{problem}

To study non-involutive solutions one replaces cycle sets by skew cycle sets.
According to~\cite[\S6.1]{MR3763907}, a skew cycle set is defined as a linear
cycle set but where the commutativity of the group $(A,+)$ is not assumed.
Problem~\ref{pro:dynamical} can be stated for skew braces and skew
cycle sets.

Almost all of the questions for skew left braces make sense in the particular
and highly interesting case of $k$-linear left braces, which are left braces
where the additive group is a $k$-vector space compatible with the
multiplicative group.  Those braces were introduced by Catino and Rizzo
in~\cite{MR2486886} as \emph{circle algebras}.

There are also several questions on one-generator skew left braces, as those
skew left braces are maybe easier to study than skew left braces. In
particular, one could also ask most of the previous questions on skew left braces
for one-generator skew left braces.

\subsection*{Acknowledgements}

This work is supported by PICT-201-0147 and MATH-AmSud 17MATH-01. The author
thanks F. Ced\'o, T. Gateva--Ivanova, E. Jespers, V. Lebed, K. Nejabati Zenouz,
J. Okni\'nski, A.  Smoktunowicz, W.  Rump, P. Zadunaisky for discussions,
comments and problems.

\nocite{MR3478858}

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


text:
JCC Krakow events during the 23rd Jewish Culture Festival
3 JULY WEDNESDAY
12:00-24:00 JCC OpenAir: Garden cafe in the unique space between the Tempel Synagogue and the JCC
13:00 Famous Rabbis from Krakow: A series of lectures organized by the Centre for the Study on the History and Culture of Krakow's Jews, Jagiellonian University in Krakow and JCC Krakow. Prof. Michał Galas, Rabbi Ozjasz Thon (1870-1936). (in English)
14:00 Sing Along with the JCC choir! Come and join the JCC choir and sing Mordechai Gebirtig's songs. Led by Michał Szostało. JCC WEST, Izaaka 5
14:00 How Many Jews are in Krakow? Unanswerable questions and questionable answers with JCC Executive Director Jonathan Ornstein (in English)
14:30 Two Generations, One Topic: What is a Good Jewish Home? Representatives of two Jewish generations will debate what is a good Jewish home and what are the differences between a prewar Jewish home and a contemporary one. (in Polish)
15:00 Young Scholars on the History and Culture of Polish Jews: A series of lectures organized by the Polish Association for Jewish Studies, the Institute of Jewish Studies of Jagiellonian University, and JCC Krakow. Agnieszka Mitas-Syrek (UWr), Jewish Autobiographies - Between Self-creation and Fact: The Diary in Jewish Culture. (in Polish)
16:30 Ruth Ellen Gruber and Annamaria Orla-Bukowska, Jewish. Jewish? "Jewish". Jewish! (in English)
What does the word "Jewish" signify in today's Central Eastern Europe, in Poland, and especially in Kraków? The ongoing transformations in this region have taken its Jews and their non-Jewish compatriots on an interesting ride. Major changes have become manifest in the decade since Ruth Ellen Gruber's classic work, Virtually Jewish: Reinventing Jewish Culture in Europe was published. In a dialogue between two who have been fellow travellers since the journey's start, Ruth Ellen Gruber and Annamaria Orla-Bukowska discuss where things stand now and where this might take us.
17:00 PRELIMD IN THE JCC Patryk Pufelski, Locusts! – The Plague You Don't Know Much About. (in Polish)
17:30 Dr. Ewa Węgrzyn, The Immigration of Polish Jews to Israel 1956-1959. The subject of the lecture is Jewish immigration from Poland to Israel in 1956-59, which is often called "The Aliyah of Gomulka". The aim of the presentation is to describe the reasons of immigration, the course of the journey and the process of adaptation of the new Jewish immigrants from Poland into Israeli society. (in English)
19:00 Premiere screenings of documentary films: Magda (9min.) directed by Slawomir Grünberg & Katka Reszke and Castaways (Wyrzutki) (17min.) directed by Slawomir Grünberg & Tomasz Wiśniewski. Q&A session with the filmmakers. (in Polish with English subtitles)
Festiwal Kultury Żydowskiej

redpajama_set_name: RedPajamaCommonCrawl


text:
Two Marines Who Met in a Burger Joint Celebrate Their 75th Wedding Anniversary
Ray and Dorothy Hill on their wedding day, Feb. 12, 1944. The couple, both 95-year-old Marine veterans, are celebrating their 75th wedding anniversary on Feb. 23 in Sun City Center. (Photo provided by the Hill family)
Tampa Bay Times, St. Petersburg, Fla. | By Howard Altman
SUN CITY CENTER -- He was a half-drunk Marine, shipped back stateside from the hellish battlefields of the South Pacific with a case of malaria.
She was a sassy private, one of the first women to join the Marine Corps.
It was 1943 in the Mojave Desert, on a military base where love would soon blossom for Staff Sgt. Ray Hill and PFC Dorothy Russell.
On Feb. 23, they celebrated their 75th wedding anniversary in a private ceremony in Sun City Center.
They have a great back story.
Ray and Dorothy Hill, 95-year-old Marine veterans who served during World War II, were honored at Tropicana Field in 2016.
In 1940, with war drums beating over both horizons, Ray Hill was a 16-year-old living in Indiana. He and his older cousin wanted to join the military so they went to a Navy recruiter, but were told to come back when they were older.
They went next door to the Marine recruiter. In short order, Ray was a leatherneck.
After Pearl Harbor, he was shipped off to the South Pacific and sent to Guadalcanal, where the U.S. launched a surprise attack in August of 1942. It was a bloody campaign, with the Japanese losing about 20,000 troops and the Americans about 2,000.
"The thing that made Guadalcanal so bad was the living conditions," Ray told the Times in an interview. "We ran out of supplies. We got so hungry we would even eat Japanese fish heads and rice. That's not very good eating. But nothing else bothered me."
He said he saw "blood and guts but it never got to me. Some of the guys couldn't take it, but I guess it was the way I was made. I was only 17 and I had to lie about my age in order to get into the Marine Corps. I was just having fun."
But after catching malaria, he was sent to Mojave Marine Air Base in California in June of 1943. There he served as a guard and spent his off-hours at a local joint where he downed burgers and beer. Sometimes a little too much beer, which was his condition one fateful November evening.
Enter Dorothy Russell.
Russell was the second youngest of 12 children raised by an adventurous widower in a town near Spokane, Wash.
In 1943, Dorothy was 19 and working in an office. She pined for more.
"The war was raging and none of my brothers were serving. They were all older and married," she said. "I was very patriotic and wanted to serve my country."
One by one, the military services opened programs for women. But Dorothy had her heart set on joining the Marines. When the Marines opened a program for women in February 1943, Russell knew where she wanted to be. But she needed her father's permission because of the age requirements.
He granted it, with words of advice.
"'Make me proud of you,'" he told her.
On July 24, 1943, Dorothy walked into the Marine recruiting station in Spokane and raised her right hand, thus becoming the first woman from her town, and one of the first in the nation, to become a Marine.
Already a sharpshooter, having spent her weekends shooting squirrels in the woods, the new private wanted to see some action. Much to her dismay, the only action she saw was the keys of a typewriter hitting paper.
Eventually shipped out to the Marine air base in Mojave, things picked up for Dorothy when she wandered into the burger place with a few other women Marines. As they sat around wondering what they were going to do for Thanksgiving, Dorothy walked over to a table full of guys sitting nearby.
"I sat down with them and Ray was at the table," she recalled with a laugh. "He was slumped down in a chair, not having anything to say."
When she decided to leave, Ray and another Marine stood up and offered to walk her home.
"When Ray stood up, he looked pretty good," Dorothy said. "He had dark hair and brown eyes. The other one was light hair and blue eyes. I was a sucker for dark hair."
Ray, too, laughs at the memory.
"I was half-loaded on beer," he said. "She sashays up and wants to talk to the boys. She got tired of talking to the girls. She came over, introduced herself, and I kind of thought, 'She's a pretty cute chick. I'll look this babe over.' "
After a short separation while Ray went for training in Baltimore, the two realized they were meant for each other and decided to marry, with the understanding that Ray would leave the service because of his malaria. They were married Feb. 12, 1944, and Dorothy soon became pregnant with the couple's daughter, Sandra. As a result, Dorothy was honorably discharged.
But then a doctor cured Ray, and back off to war he went, missing his first child's birth. Stationed in Okinawa, Ray convinced a fighter pilot to take him over the A-bombed city of Hiroshima.
"I saw the devastation and it was a lesson in what Hell would look like," he said.
Ray left the service in November 1945 and the couple had a second child, Gary, in 1946. For several years, the couple moved around, often living in a trailer until Ray finally settled down in Ventura, Calif., working for Bank of America. They moved to Florida 16 years ago and on February 23 will celebrate their diamond wedding anniversary.
When asked the secret to a 75-year marriage, Dorothy spelled it out in four letters.
"L-o-v-e," she said.
This article is written by Howard Altman from Tampa Bay Times, St. Petersburg, Fla. and was legally licensed via the Tribune Content Agency through the NewsCred publisher network. Please direct all licensing questions to legal@newscred.com.
Military Headlines World War II Women Veterans
© Copyright 2019 Tampa Bay Times, St. Petersburg, Fla.. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.


Hanging out by the teach trails is brunette amateur Briana Lee posing in a hot bluish suit. Her hip are exceedingly lovely and I like it whenever that babe pulls forward her grey suit within order to flash marvelous white clothes. I love the look of her supple and silky skin. Caressing her all over would be a lot of our fun. That babe is a flirt and proves this can by pulling down the upfront of her grey costume for halloween to expose bouncy large meatballs. Briana Lee's gazoo a muslim hypnotizing. Watching her pull-over down these white clothes and bend over gives put me in a apuro.


text:
1961 Photo sent by Sudhir Kashyap [13th Nov 2021]
CLASS PHOTO
Chairs left to right: Sukhinder Singh, S. Kashyap, H.S. Atwal, Gurmit Singh, Mr. F.M. Brown, G.S. Anand, Preharan Singh, S. Khanna, S.M. Nanda.
1st row standing from left to right: A.K. Anand, Himmat Singh, K.S. Dugal, Rupinder Singh, S.S. Mann, J.S. Rarewala, Gurdyal Singh, G.S. Gill, R. Sawhney, R. Rovshen.
Standing last row from left to right: V.K. Chhabra, Deshpal Singh, A. Motwane, J.K. Wason, G.S. Gill, S.K. Narula, A. Mulchandani.
The photograph was sent to OCA Website by: Sudhir Khanna
CRICKET TEAM 1961:
Standing left to right: A Anand, G Prakash, Rupinder Singh, Preharan Singh, Sukhinder Singh, S.M. Nanda.
Sitting left to right: R.S. Sodhi, G.S. Anand, H.S. Mamik [Capt], S. Kashyap, J.S. Gill.
The 1961 Cricket Team Photo contributed by H.S. Mamik and Sudhir Kashyap.
Curzon 1961 picture provided by Brig Arun Kochhar
Photo added September 30th 2014:
RIVAZ HOUSE [Photo provided by Himmat Singh]. Text by Vijay Khurana:
Himmat Singh sent three photographs of Rivaz House for the years 1953, 1961, and 1962.
Two observations. With Mr Farhat Paul as Rivaz House Master, all the boys had a handkerchief in their top left coat pockets (see the photographs). He loved these little touches and the expression, "You Fat Cow!!" He bellowed these words and stretched them over a good ten seconds for the impact that he wanted to deliver depending on his mood. His other talents were exceptional but he left in me an enduring interest in history with details of Babur that the Baburnama never recorded !! He embellished every class with fascinating details that made all the Mughal emperors come alive !! He had several other attributes but his large garnet ring, his penmanship, and his well endowed behind I do remember. There was more, much more, to this flamboyant man but I leave that to your imagination.
I see faces and remember with affection Mohan (M.M.) Sachdev, Ashok Dina Nath, Huppy Grewal, the Gideon brothers, Preharan Singh, Motwani, my dear friend Girish Ghai, the body builder who frequented my house in Bombay on so many occasions. His parents (address 69, Warden Road) were always warm and hospitable. There are others who revive memories aplenty.
LEFROY HOUSE 1961
Photos provided by Vijay Khurana who writes "Two attachments accompany this mail, one is a high resolution photograph and the other a scan of a photo copy of the original with the names listed below. No guess work required. The size of the original is much too large to fit on the bed of the scanner I possess and hence this additional effort.
You will notice some of the really grew between 1954 and 1961!!"


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Peter Kerly gets set for his fifth London Marathon
63-year-old Peter Kerly from Kent is busy fundraising for Orbis ahead of the London Marathon next month - and this year, for the first time ever, he will be running alongside his son.
Peter works as a Head of Business Development near his home in Ulcombe, Kent, and over the past few years has run the London Marathon five times. This year, we are very pleased that he is running in aid of Orbis – and his impressive fundraising efforts include a Big Marathon Quiz in his local village hall on 13th April. Good luck to those quiz participants with seasoned marathon veteran Peter in charge!
London Marathon runner Peter Kerly and his medal collection
This year, Peter will also be running the marathon alongside his son for the first time. His preparations for the marathon include following a rigorous Virgin Intermediate training programme, and his top training tip is to listen to one's body and take extra time to recover if need be.
Peter is aiming to complete the marathon in four and a half hours and hopes to raise more than £2000, a sum that will be doubled as part of our See My Future appeal - an appeal that will see the UK government double all public donations made before 23rd June.
Orbis is delivering clinical support to citizens that do not have the luxury of an NHS - and more importantly [they are] sharing their knowledge and experiences with like-minded clinicians locally to ensure the good work continues.
Peter Kerly
London Marathon 2019 runner
The London Marathon takes place on Sunday 28th April, and Peter is one of five runners fundraising for Orbis. We will be cheering them all on from the sidelines!
We are grateful to have supporters like Peter who are so committed to helping us tackle avoidable blindness. The London Marathon is no small task and this year, as part of our See My Future appeal, the money raised by our amazing runners will have twice the impact – proving the people we treat with the possibility of a much brighter future.
Vani Krishnaswamy
Senior Fundraising Officer, Orbis UK
To help Peter smash past his fundraising target and help restore sight to even more people, you can support him here.


School-based telehealth provides access to high-quality healthcare in the school setting. A nurse practitioner or a pediatrician examines the student with the assistance of the school nurse or a telepresenter.
A telepresenter is an individual who is trained on the technology to facilitate the exam under the guidance of a provider. The school nurse or another trained individual can serve as the telepresenter during a telemedicine visit.
Students who are enrolled in the program at the below participating schools. Click on the county for a list of participating schools.
How do I enroll my student in the program?
Please complete the enrollment packet and return it to your school nurse.
Contact your school nurse to schedule an appointment.
The program treats common childhood illnesses such as sore throat, rashes and chronic diseases such as asthma. If the student's condition is not able to be diagnosed via telemedicine, the provider will work with the family to connect them to an in-person provider.
Please note that services vary by school to include acute care, chronic disease management, mental health, and group health education.
Does insurance cover the cost of the school-based telehealth program?
Insurance carriers are billed, and copays and deductibles apply. SC Medicaid covers the visits. Other insurance company coverage varies. It is important for patients to consult their insurance plan prior to a visit to ensure coverage for school-based telehealth services.
I represent a provider group or school district that is interested in starting school-based telehealth in my community in SC. What is the first step?
Please complete the Program Request Form.
How do I find out about on-going initiatives for school-based health in South Carolina?
Learn more about the South Carolina School-Based Health Collaborative.


Join Us for the Harlem Wizards Game!
Due to the overwhelming success of the last 3 years, this event will no doubt be a sellout. Please purchase your tickets early for this event. Also, in honor of Veterans Day, all veterans in attendance will be recognized before the game.
Grandfriends' Day Visits Scheduled for October!
This is an occasion for a special adult in a student's life--grandparent, aunt, uncle, parent, family friend, etc.--to visit and share in a special activity. This is a day that our staff and students look forward to each year!
In order to maintain the flow of the line at drop-off and pick-up, please do not get out of your vehicle. Please allow your child to exit/enter the car independently. This will make the process more efficient and help everyone with their busy schedules. Thank you for your help!
Our teachers will be wearing jeans on Mondays. In order to enjoy this added comfort, they will donate a dollar each Monday to support a variety of charities. So, please know that our teachers are supporting worthy causes while also enjoying the opportunity to dress down!
We hope that you will plan to attend. Below is a flyer for the event as well as a quick synopsis of the movie.
SCREENAGERS Synopsis: Through poignant, and unexpectedly funny stories, along with surprising insights from authors, psychologists, and brain scientists, SCREENAGERS reveals how tech time impacts kids' development and offers solutions on how adults can empower kids to best navigate the digital world and find balance.
Launched at the end of the last school year, the new School Messenger (Safe Arrival) Absence Reporting System makes it easy for parents/guardians to report their child's absence, reduces the time it takes to verify student attendance and will allow our staff to respond to unexplained student absences more efficiently. Please click here for directions on how to use your phone, app, or computer to report an absence.
Important Announcements from the BSES HSA!
Parents' Night out at Board & Brush: The HSA is very excited to announce that BSES is doing our first ever Parents' Night out at Glen Mills Board & Brush on Friday, November 30th from 6:30-9:30pm. We have a limited number of seats, so book your spot before it fills up! All signs in the gallery are $65, and the HSA will receive a portion of the profits from this event!
Yankee Candle and Otis Spunkmeyer Fundraiser: Yankee Candle and Otis Spunkmeyer fundraising packets came home last week! In one of our most popular fundraisers, you can help our school while purchasing the candles, cookies, wrapping paper, popcorn, gifts, and more that you would have bought even it was it wasn't a fundraiser! Use the online options to share the sale with friends and family and to take advantage of direct shipping to your home. For traditional ordering, send the order sheets and payment into school by Tuesday, October 9th and pick up your orders on Wednesday, November 14th. Please contact Dawn Wyzen with any questions: dawnw109@verizon.net.
Orders are expected to arrive in mid October and will be delivered to the classroom you select in the ordering process.
This Week's Fact About Greece: The highest mountain in Greece is Mount Olympus. it has over 50 peaks with the highest reaching 2917 metres (9570 feet).


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The Southern Critique of Centralization
By Donald LivingstonJanuary 28, 2019Blog
The Southern political tradition, in practice and theory, is one of its most valuable contributions to America and the world. The one constant theme of that tradition from 1776–through Jefferson, Madison, John Taylor, St George Tucker, Abel Upshur, John C. Calhoun, the Nashville Agrarians, Richard Weaver, M. E. Bradford, down to the scholars of the Abbeville Institute–is a systematic critique of centralization. Nothing comparable to it exists elsewhere in America or in Europe.
A criticism of centralization presupposes that decentralization is a good thing. But why is that? The answer is complex and requires viewing what was happened in 1776 from a trans Atlantic perspective. The Declaration of Independence is merely the American version of a conflict that had been going on in Europe since at least the 17th century between the emerging centralized modern state and a revived interest in the classical republican tradition which goes back to the ancient Greeks.
There are four principles to this republican tradition: First, republican government is one in which the people make the laws they live under. But, second, they cannot make just any law. The laws they make must be in accord with a more fundamental law which they do not make but is known by tradition. Third, the task of the republic is to preserve and perfect the character of that inherited tradition. And finally, the republic must be small. It must be small because self-government and rule of law is not possible unless citizens know the character of their rulers directly or through those they trust.
The Greeks created a brilliant civilization that was entirely decentralized. It was composed of 1,500 tiny independent republics strung out from Naples to the Black Sea. Most were under 10,000. One of the largest was Athens with around 200 thousand people. For over two thousand years, up to the French Revolution, republics seldom went beyond 200-300 thousand people, and the great majority were considerably smaller.
In contrast, a modern state is supposed to be large. Thomas Hobbess, published in 1651 the first systematic theory of the modern state. He titled the book "Leviathan,' a large sea monster. It contains a central government endowed with irresistible and indivisible power over individuals in a territory. Unlike republicanism, it does not require, self-government or tradition. Nor does it require the rule of law since the central authority itself can make law. Its purpose is to contain anarchy by enabling autonomous individuals to pursue their own ends in a condition of enlightened self-interest called "civil association." Such a regime is compatible with an association of strangers, as in a regime of traffic regulations.
Since the only goal of the modern state is "civil association," there is no internal limit to its size. In fact, the larger the better because outside the realm of civil association lies anarchy or its ever present threat. The logical extension of this is global government or as close an approximation as possible. Although a modern state may expand in size indefinitely, its territory cannot be divided by secession because if one set of individuals could lawfully secede, so could any other set, and so on within each set, to the unraveling of all government.
Here we have two incompatible models of government. The small classical republic and the indefinitely large modern state. But there is a third model to consider. Medieval civilization was also decentralized, and it was vast in scale. It was a mosaic of thousands of independent and quasi-independent political units: kingdoms, principalities, dukedoms, bishoprics, papal states, republics, free cities, and tens of thousands of titled manors.
The medieval contribution to politics is the idea of a federated polity where various independent political units are held together in a larger realm by compacts and traditional hierarchies. As we will see shortly, it is through the logic of the medieval federation that the Southern tradition sought to bring together the best aspects of the small republic with those of the large modern state.
The modern state system begins in the 17th century with the rise of "absolute monarchies"–'absolute,' meaning irresistible and indivisible centralized power. Modern monarchs sought to crush the medieval mosaic of independent social authorities they had inherited into larger and more centralized states. And they were successful.
In the mid-1850s Tocqueville left us a melancholy description of what two centuries of monarchical centralization had done: "The old localized authorities disappear without either revival or replacement, and everywhere the central government succeeds them in the direction of affairs. The whole of Germany, even the whole of Europe … presents the same picture. Everywhere men are leaving behind the liberty of the Middle Ages, not to enter into a modern brand of liberty but to return to the ancient despotism; for centralization is nothing else than an up-to-date version of the administration seen in the Roman Empire."
But just as absolute monarchy was emerging in the 17th century, demanding a large scale state, there was also a revived interest in classical republicanism which demanded small scale. This latter sparked a Cato-like resistance to modern state consolidation which ran throughout the centralized monarchies of Europe. But one thinker requires special mention, namely Johannes Althusius (1563-1638). He was a German Calvinist philosopher who proposed a federation of small polities in a state larger than the classical republic, but smaller than a European monarchy. He called it a federation of "medium" size–about the size of Switzerland which is half the territory of South Carolina.
To prevent the central government from consolidating the smaller polities into a unitary modern state, Althusius introduces a constitutional right of secession from the federation. If a federation grew too large, it could always be brought back to a republican scale by secession.
The language of republicanism was perverted by the French Revolution which declared itself to be a "republic one and indivisible." But with 26 million people, France was too large to be a republic. Moreover, the new regime retained the kings' coercive mechanism of centralization So the new France was, in fact, an absolute monarchy pretending to be a republic. And since it was said to be "one and indivisible," secession was ruled out absolutely.
To speak of the French "republic" is an oxymoron. Yet this muddled union of the modern state with republican connotations would spread around the globe. Today, France with 65 million people, the US with 325 million, and China with one billion, 300 million, all describe themselves, without embarrassment, as "republics.".
This was not true, however, of the American founders whose republican thought was established before the French Revolution and which acknowledged the classical republican requirement of human scale and a limit to size as such. But this meant Americans were immediately confronted with an uphill challenge. The republican tradition told them that the extensive territory they acquired from Britain was too large for republican government. How they tried to solve it is best illustrated with the case of Virginia.
Virginia conquered the vast Northwest territory which includes the present States of Illinois, Indiana, Ohio, Wisconsin, and parts of Michigan and Minnesota. Virginians told themselves that this was a temptation they must resist; that they could not both enjoy republicanism and rule such a vast territory. So the territory was ceded to the Confederation.
The important point here is that republican political liberty was preserved by reducing the size of Virginia. But there is more. The State itself was deemed too large, and was reduced further in size when a number of western counties seceded and formed the State of Kentucky. Even so, Jefferson thought the State was still too large, and he urged that its counties be divided into small sovereign states (similar to the Swiss cantons) which he called "ward republics." In 1810, he wrote that: "these little republics would be the main strength of the great one." And fourteen years later he said of this federation of small republics : "the wit of man cannot devise a more solid basis for a free, durable well-administered republic."
The country closest to Jefferson's vision for Virginia is Switzerland which has twenty six sovereign states, the smallest of which has 15,974 people and the largest a little over a million. The average size is 300 thousand. Fourteen states have less than that, and eight are under 100,000. Switzerland is so decentralized that its central government has no original taxing power. Its power to tax requires a constitutional amendment approved by a majority of the cantons, each of which has one vote, and a majority of individuals. And the military is in the hands of the canton militias. Switzerland is regularly ranked by the UN's World Happiness Report in the top ten happiest countries in the world. The top ten are usually always small states. The U.S. has yet to make the top ten.
Yet Jefferson himself put more pressure on republican liberty with the Louisiana Purchase which doubled the size of the Union. New Englanders threatened secession, claiming that Virginia sought to become the "Austria of America." Jefferson agreed that America would indeed be an empire, but not a centralized one. It would be what he called an "empire of liberty." Like Althusius, he would do this by tying republican political liberty to division of territory through secession.
How so? As people moved West, new States would be formed. As population increased these might be divided through secession as Kentucky seceded from Virginia–Tennessee from North Carolina, and Maine from Massachusetts. And if Jefferson had his way, these States would be further divided into "ward republics" in the manner of Switzerland. In time, there might be too many States for representative government. In that case, States would secede from the mother Union and form a federation of their own. Jefferson imagined a future America composed of three countries. The old Atlantic federation, a Mississippi federation, and a Pacific federation.
Just as Virginia had resisted the temptation to become a centralized modern state by reducing its size and ceding the Northwest territory to the Confederation, and by allowing Kentucky to secede, so the original Union should generously allow itself to be divided for the sake of its own republican life (since it had grown too large) and the republican life of other Americans.
So when eleven Southern States seceded in 1861, they were simply enacting the rights embedded in Jefferson's "empire of liberty." If you want to know why the South seceded, read the Confederate Constitution it is merely the US Constitution reformed to bring it into closer accord with the Jeffersonian vision of reconciling republicanism with extensive territory. By seceding, the South actually strengthened republican political liberty in both the South and in the North. Lincoln subverted republicanism in both by creating what would become a monster modern state.
Lincoln's invasion of the Confederate States was America's French Revolution as it derailed Jeffersonian America's on-going and successful effort to reconcile republicanism with large size–in favor of yet another dime a dozen modern European state. In this, Lincoln was right on schedule with the centralizing trend in Europe that so terrorized Tocqueville in the 1850s. Lord Acton viewed the the Confederacy as a continuation of the dissenting decentralist tradition in Europe. With its defeat no counter force existed and would-be modern states would appear everywhere around the globe, all claiming to be "one and indivisible." They would have different ideologies: liberal, libertarian, socialist, communist, democratic, fascist. But all would seek to have the same form, a central government with plenary power over individuals in a territory.
The 17th century modern state promised peaceful civil association on an extensive–and possibly–a global scale. But its devotees failed to read the fine print. No practical way was found to check its disposition to centralize power. Instead of pursuing mere "civil association," it turned to nation and empire building. Moreover, these centralized monster states were allergic to each other. The result would be the Napoleonic wars, World Wars I and II, and the Cold War. World War II alone left 60 million dead, mostly civilians.
But even worse, having eliminated, or drawn the teeth, of its historic independent social authorities (the church, the nobility, free cities, provinces, small states within), the state met no corporate resistance in turning on its own people. R. J. Rummel has studied the phenomenon of the state killing people within its jurisdiction. He calculates that nearly four times as many people have been killed by their own governments as have been killed in all the wars, domestic and foreign, fought around the globe in the twentieth century. Killing on this scale would not be possible without the subversion of independent social authorities caused by massive centralization. If so, the greatest threat to human life in the twentieth century has not been war but the massive centralization of power in modern states. Rummel says, its as if nuclear war occurred, and no one noticed. 1
Southerners were the most clear eyed students of the Jeffersonian republican vision of America and the need for checks on centralization. Over and over they warned what would happen if fallen man should ever acquire such power. Alexander Stephens, former vice president of the Confederacy, surveying the wreckage of Jeffersonian America caused by Lincoln's war gave this warning to posterity in 1870: "Depend upon it …. there is no difference between Consolidation and Empire …. If the worst is to befall us; if our most serious apprehensions and gloomiest forebodings as the future … are to be realized; if Centralism is ultimately to prevail; if our entire system of free Institutions as established by our common ancestors is to be subverted, and an Empire is to be established in their stead; if that is to be the last scene in the great tragic drama now being enacted; then, be assured, that we of "the South" will be acquitted … by the judgment of mankind, of all responsibility for so terrible a catastrophe, and from all the guilt of so great a crime against humanity." 2
What Stephens calls "empire" is nothing but the modern European state which Lincoln established in America with a writ of fire and sword. It was resisted in Europe by decentralists such as Althusius, David Hume, Goethe, Tocqueville, Lord Acton, Proudhon, and many others. Lord Acton called Lincoln's invasion of the South an "awful crime." In its current form, the modern state has dominated for two centuries, beginning with the French Revolution. But a dramatic change occurred in 1991 when fifteen states seceded from the "one and indivisible" Soviet Union. This was the greatest bloodless revolution in history, something American elites have yet fully to appreciate. Since then numerous peaceful secessions have occurred in Europe and a great many more decentralist movements have sprung up demanding self-government.
Nor has the United States been spared this decentralist discontent as evidenced by the Tenth Amendment movement. Even secession, long thought to have been buried, is again topical in the United States. Polls are now regularly taken to register secession opinion. A 2014 Reuter's poll found that 25 percent of Americans favored secession of their State. That is 80 million people. 3 A 2015 Gallup Poll found that millenials were the most supportive of a Palestinian state and an independent Scotland. American millenials strongly supported Britain's secession from the EU by 42 percent to 17, and 37 percent favored secession of Texas from the United States. 4
What has happened in the last 25 years is a paradigm shift. The modern state, dominant for two centuries, no longer commands the authority it once had. Many Americans no longer believe the United States is "one nation indivisible." And they are right, it never really was. But paradigm shifts at their beginning are confusing, and it is not clear how to think about our current condition as we try to penetrate the muddle caused by uniting republican discourse with that of the modern state. The part of the American tradition from which we have the most to learn, practically, theoretically, and historically, is the Southern political tradition which since 1865 has not been ignored so much as suppressed.
1. R. J. Rummel, Death by Government (New Brunswick, NJ: Transaction, 1997).
2. Quoted in Alexander Stephens, A Constitutional View of the War Between the States, 2 vols. (Harrisonburg, VA: Sprinkle Publications, 1994), vol. 2, p. 669.
3. http://blogs.reuters.com/jamesrgaines/2014/09/19/one-in-four-americans-want-their-state-to-secede-from-the-u-s-but-why/
4. http://redalertpolitics.com/2016/06/30/poll-american-millennials-support-brexit-37-support-texas-secession/
Donald Livingston
Donald Livingston is the founder of the Abbeville Institute and retired Professor of Philosophy at Emory University. He has been a National Endowment Independent Studies fellow and a fellow for the Institute of Advanced Studies in the humanities at the University of Edinburgh. His books include Hume's Philosophy of Common Life and Philosophical Melancholy and Delirium, Hume's Pathology of Philosophy.
Donald LivingstonNullificationSecessionSouthern Political TraditionSouthern Tradition
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AdTech Industry, Data & Privacy, MarTech Industry
FAQs About Safari's Intelligent Tracking Prevention (ITP)
Updated on June 7, 2022 by Michael Sweeney, Paulina Zawiślak
Since the release of version 1.0 in September 2017, Webkit has released 5 new versions of its Intelligent Tracking Prevention (ITP) to strengthen user privacy for Safari users.
We've put together an FAQ post to answer some questions you may have about what ITP is, how it works, and what impact it has on AdTech and MarTech companies.
What Is Intelligent Tracking Prevention?
How Does Intelligent Tracking Prevention Work?
What Restricts Does ITP Place on Cookies and Other Web Browser Storage?
How Has ITP Evolved?
Which Companies or Services Are Affected by ITP?
AdTech Vendors
MarTech Vendors
What Are Possible ITP Workarounds and Solutions for AdTech Companies?
Possible Workarounds for Analytics and MarTech Companies?
How Does ITP Impact Walled-Garden Advertising Ecosystems?
How Much of the Browser Market Share Worldwide Does Safari Have?
Can Safari Users Still See Ads?
Intelligent Tracking Prevention (ITP) is a feature of Apple's Safari web browser. It's designed to improve the privacy for Safari users by blocking third-party cookies (aka trackers) that identify and track users across different websites; a process known as cross-site tracking. It also restricts the lifespan of other types of web browser storage that can be used to identify users, such as first-party cookies and local storage.
Intelligent Tracking Prevention is built by Webit, which is an open-source web browser engine that powers the Safari browser, among others.
Intelligent Tracking Prevention uses a machine-learning model called the Machine Learning Classifier to identify which domains can be used for cross-site identification and tracking.
Once the Machine Learning Classifier identifies a domain as having cross-site tracking capabilities, certain limitations are placed on cookies created by those domains.
You can read more about this and see some scenarios in one of our previous blog posts.
Here's an overview:
Third-party cookies are blocked by default.
First-party cookies created by JavaScript's document.cookie will expire in 7 days. If the cookies are accessed within those 7 days, then their expiration date will be extended by 7 days.
First-party cookies created by JavaScript's document.cookie AND are created by a tracking domain AND use link decoration will expire in 24 hours. If the cookies are accessed within 24 hours, then their expiration date will be extended by another 24 hours.
All non-cookie data stored in Safari, such as in local storage, will expire in 7 days. If the data in local storage is accessed within those 7 days, then its expiration date will be extended by 7 days.
Companies like Google and Facebook will have to get consent from users via the Storage Access API to set cookies and access data for widgets, like those used to log in to accounts.
Intelligent Tracking Prevention was announced in June 2017 and released with Safari 12 and iOS 11 in September 2017.
Since its release, several new versions have been published. Each new release has brought with it more restrictions on cross-site tracking.
To view the full details about each ITP release, read our blog post.
Not being able to identify users across different websites means AdTech companies can't run cross-site ad targeting, perform frequency capping, and measure the performance of campaigns across different publishers and advertisers.
Because advertisers aren't able to identify their audience, they will pay less for a publisher's ad inventory. This means lower CPMs and ad revenue.
Although ITP doesn't affect advertisers in the same way as publishers or AdTech companies as they can simply adjust their target criteria to exclude traffic from Safari browsers, the limitations mean that it's harder for advertisers to reach their target audience.
Most MarTech companies weren't impacted by ITP initially as the main restrictions were on first-party cookies originating from known trackers (most MarTech platforms don't do this), but ITP 2.1 now deletes first-party cookies set via JavaScripts document.cookie within 7 days.
While most AdTech processes that rely on third-party cookies (e.g. cross-site identification) will be restricted, it's still possible to run some types of attribution, such as click-through attribution.
Click-through attribution can be done by passing a unique click ID (or visitor ID) in the URL to the destination website. Then, the destination site can use a JavaScript tag to capture and store the ID and notify the ad server about the click ID.
It may also be possible to measure conversions, but the attribution window will be limited to 24 hours as per ITP 2.2 if the referring domain is classified as a cross-site tracking domain, which is highly probable with a majority of AdTech platforms.
Also, second-party data (or data co-operative) partnerships may also implement a workaround to ITP by exchanging identifiers between sites, for instance, between partner sites or domains owned by the same company. To do this, the websites would pass the IDs via URLs and then store them in a first-party cookie.
One of the workarounds that some AdTech companies have also created is to store IDs in the browser's LocalStorage.
However, ITP 2.3 deletes all non-cookie data from a web visitor's Safari browser after 7 days if the referring domain has been classed as a tracking domain and the referring URL contains a link decoration.
Because ITP 2.1 includes a seven-day expiry for first-party cookies created via JavaScript's document.cookie, web analytics and MarTech vendors will need to implement a workaround to maintain accurate data.
Simon Ahava provides some ITP 2.1 workarounds for analytics tools in his blog post, such as setting first-party cookies server side via the HTTP response, storing the first-party cookie in LocalStorage, and creating a reverse proxy server.
Because of this restriction on first-party cookies, the metrics and reports on users in Safari browsers will be inaccurate if their last visit was more than seven days ago, so web analytics and MarTech tools will need to implement a workaround to measure activity outside of this 7-day window.
While tech giants like Facebook, Google and Amazon weren't affected too much by ITP 1.0 and 1.1, the introduction of ITP 2.0, 2.1, and 2.2 and 2.3 has changed this.
Before companies like Facebook and Google can set cookies via their widgets (i.e. the 'Sign in with your Facebook or Google account' ones), users will first have to consent via the Storage Access API.
Also, all conversion attribution carried out via link decoration and first-party cookies will be limited to 24 hours as part of ITP 2.2's release, or 7 days as per ITP 2.3.
As of May 2020, Safari has a 18% global market share.
Source: StatCounter Global Stats – Browser Market Share
Yes. AdTech companies will still be able to display ads in Safari, but they won't be able to create third-party cookies.
However, if a Safari user has an ad blocker plugin installed, then they won't see ads because ad blockers stop the ad tags from loading, meaning ads can't be displayed.
Cookieless AdTech
The end of third-party cookies doesn't mean the end of AdTech, but companies will need to change how their technology works to survive a world without them.
Schedule a call with one of our AdTech development teams and find out how we can help you.
Prepare your AdTech platforms for a world without third-party cookies
Fill in the form below and find out how we can help you
Intelligent Tracking Prevention (ITP)
The True Cost of Hiring AdTech and MarTech Developers [infographic]
November 2, 2022 by Michael Sweeney, Paulina Zawiślak
The Challenges and Opportunities Facing Influencer Marketing Platforms
December 6, 2021 by Michael Sweeney
Apple's Changes to IDFA in iOS 14: FAQs and the Impact on Mobile Advertising
Intelligent Tracking Prevention (ITP): The Impact on Cookies, AdTech, and MarTech
October 3, 2019 by Maciej Zawadziński, Michael Sweeney
Privacy in AdTech FAQ: Google Chrome's Privacy Sandbox, Safari ITP, Firefox, GDPR
What Is Intelligent Tracking Prevention and How Does It Work? [versions 1.0 - 2.3]
September 27, 2017 by Maciej Zawadziński


Stuck on the sidelines: The most immediate effect of the long government shutdown for hundreds of thousands of federal workers and contractors is, of course, missed paychecks. But Eric Wolff of Politico notes how many of the watchdog functions of the government are stalled as well. Hundreds of environmental inspections at power plants, manufacturing sites and oil wells have been missed. Financial fraud investigations and tax audits have slowed. The Federal Trade Commission's investigation into Facebook's use of personal data is on pause. The National Transportation Safety Board has been unable to investigate at least a dozen crashes, including a plane crash that killed four people and a tractor-trailer collision with a school bus that injured 15, Wolff reports. And the list goes on. Meanwhile, The Washington Post reports that this shutdown may be achieving exactly what some White House allies had hoped for – not a big, beautiful wall but a smaller government.
Also: The airline industry is feeling the effects of the shutdown, The Hill reports. –– Many inspections of food producers aren't happening during the shutdown, but Dan Charles of NPR explains why the effect on food safety may be minimal. –– Government offices may be closed, but rather than save money the shutdown will ultimately cost the country more, writes Jim Tankersley. –– Data collection related to climate change, preparing for the next hurricane season, and evaluating the devastating effects of the California wildfires has been stymied by the shutdown too, InsideClimate News reports.
Gun-toting teachers: Ever since the mass shooting at a school in Parkland, Florida, last February left 17 people dead, at least 215 school districts in the U.S. have adopted policies to arm teachers, bringing the total to 466 districts or more, according to reporting by Tess Owen of Vice News. It's difficult to get an exact figure, Owen writes, because school districts often aren't required to tell state authorities–or parents–that they've decided to arm teachers. Meanwhile, the nation's largest teachers union says that having teachers carry guns is an "ill-conceived, preposterous, and dangerous" idea.
Also: Among young people ages 14 to 29, school shootings are the most important issue facing the U.S. today, according to a SocialSphere poll reported on by Axios. –– Echoing work done on President Trump's 2016 campaign, the National Rifle Association appears to have coordinated advertising with Republicans in important Senate races in 2016 and 2018, in violation of federal election law, according to The Trace and Mother Jones. — The Tampa Bay Times reports that schools in Florida are having a hard time hiring armed guardians to protect students and staff.
A grim milestone: A person's risk of dying of an opioid overdose is now greater than the likelihood of dying in a traffic accident, according to a new report from the National Safety Council. The skyrocketing overdose numbers are driven largely by the powerful synthetic opioid fentanyl. The odds of a person's death resulting from an overdose were 1 in 96 as of 2017. From a motor vehicle crash: 1 in 103. The overdose deaths are part of an overall rise in deaths from preventable, unintentional injury, Ian Stewart of NPR reports. "It is impacting our workforce, it is impacting our fathers and mothers who are still raising their children," said Ken Kolosh, manager of statistics at the National Safety Council.
Also: The governor of Pennsylvania pledged that his coal-mining state would speed carbon emissions reductions, cutting 26 percent off of 2005 levels by 2025. But he'll need Republican lawmakers' help to do it, InsideClimate News reports. –– Even with a drop in coal-fired power production, U.S. carbon emissions rose last year by an estimated 3.4 percent, due in part to a cold winter and a surging economy.
A life-saving smoke screen for Juul?: The dominant player in an e-cigarette industry that reversed decades of success in lowering youth nicotine use is now "billing itself as a public-health crusader," Kevin Roose of The New York Times writes. Juul, under pressure for fueling an epidemic of nicotine addiction among teens, has launched a $10 million ad campaign to persuade adult cigarette smokers to switch to vaping and lower the impact on their health. Juul may want people to focus on its "life-saving mission," as one magazine story put it, but Roose writes that the company's record, including its aggressive marketing to young people, shows a different side. Just a few years ago, a development engineer said in an interview that Juul was "not trying to design a cessation product at all," adding "anything about health is not on our mind." Another cause of the skepticism aimed at Juul's public health image: Altria, the largest U.S. cigarette manufacturer and parent of Marlboro maker Philip Morris USA, bought a 35 percent stake in Juul late last year.
Also: New York Gov. Andrew Cuomo wants to raise the legal age for cigarette and e-cigarette sales to 21 as part of a broader plan to curb youth nicotine use. Several states, including California, Massachusetts and New Jersey, have already set the age for tobacco sales at 21.
PG&E bankruptcy: Facing potential liabilities of $30 billion or more for the deadly Camp Fire last year and the 2017 wine country fires in California, Pacific Gas and Electric Co. said Monday it planned to file for bankruptcy protection. Chief executive Geisha Williams announced her resignation over the weekend and received a severance package of about $2.5 million. The company is accused of contributing to the fires by failing to properly maintain their lines and not adequately responding once they started. The company faces hundreds of lawsuits from people affected by the fires, including several cases that seek to be certified as class action suits, James F. Peltz of the Los Angeles Times reports. It's too soon to tell just how bankruptcy will affect fire victims, but some lawmakers said they will be watching the proceedings closely and could intervene. The Natural Resources Defense Council said the bankruptcy also threatens the state's clean energy efforts, in which PG&E is the largest investor. PG&E held just $1.4 billion in wildfire insurance coverage in 2018, The Washington Post reports.
Single-stream dreams: What's not to love about single-stream recycling? It's easy. Everything goes in one bin and out to the curb. And, for years, it's been cost effective. But single-stream recycling has brought an era of more significant contamination, reports Maggie Koerth-Baker at FiveThirtyEight. About a quarter of all recycling is so contaminated –think diapers and shards of glass mixed in with the paper–that it goes right to the landfill. That's up from 7 percent a decade ago. The Chinese market for some contaminated recyclables has disappeared. That means the cost of recycling for many municipalities is going up, causing some to ask residents to start sorting again.
Mining deaths down: The U.S. Mine Safety and Health Administration recorded 27 mining fatalities in 2018, the second-lowest number of deaths ever recorded for the industry. A Department of Labor press release does not mention the decline in coal power and coal mining overall.
Also: Two utility contractors working in Wisconsin, Bear Communications of Kansas and subcontractor VC Tech of Michigan, face proposed penalties of $12,834 each after investigators found they failed to determine the location of an underground gas line that exploded when struck by excavating equipment. A volunteer firefighter was fatally injured after responding to the scene. –– An employee of RKM Utility Services died after being exposed to hydrogen sulfide while working in a trench in Dallas. The company faces penalties of $422,006 for failing to properly train workers on the hazard or how to avoid it. –– Investigators have again cited the U.S. Postal Service for failing to protect workers from extreme heat after a mail carrier died while on the job in Woodland Hills in Southern California, delivering mail on a July day when the temperature reached 117 degrees. The agency was also cited for improper record keeping of heat stress incidents. The proposed penalties total $149,664.


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Police Look for Hit-and-Run Driver in Los Angeles
blog home Hit-and-Run Accident Police Look for Hit-and-Run Driver in Los Angeles
By Brian Chase on February 27, 2017 - No comments
Police in Los Angeles are looking for a hit-and-run driver who struck and killed a woman, identified as Alicia Bello, as she carried her 2-year-old son across the street in El Sereno. According to a CBS Los Angeles news report, the fatal incident occurred the evening of February 22, 2017 on Alhambra Avenue. Police say Bello, 32, was standing in the middle of the roadway when a westbound vehicle struck her. The impact hurled her into the eastbound lanes where she was hit by a pickup truck.
Fatal Hit-and-Run Crash
Her son was thrown 50 feet down the road. Bello was pronounced dead at the scene and the toddler was hospitalized for treatment of lacerations to the head. The driver fled the scene, but the pickup driver remained at the scene. He was questioned as released. The hit and run vehicle may be a Honda SUV with left side damage according to police. A reward of up to $50,000 is available to anyone who can provide a tip leading to the suspect's identification.
Bello's husband told CBS that the couple has two other children. He added that he was shocked that the driver didn't stop even after seeing a young child fly off his mother's hands. The little boy, Romeo, is expected to survive. Anyone with information is asked to contact LAPD at 213-833-3713.
Our heartfelt condolences go out to Bello's family, especially her husband and their three young children. Please keep them in your thoughts and prayers.
Justice for Victims' Families
Los Angeles is notoriously known as the "hit-and-run capital of the U.S." Nearly half of the fatal traffic accidents that occur in Los Angeles involve hit-and-run drivers, statistics show. There may be a number of reasons why drivers choose to leave the scene of the crash including but not drunken driving, driving without a valid license or sheer panic. However, none of these are good enough reasons for leaving the scene of an injury or fatal crash.
Under California Vehicle Code Section 20001 (a), it is illegal to leave the scene of a car accident is illegal to leave the scene of a crash. Motorists are required to remain at the scene regardless of whose fault the accident was. They are also required to exchange information with other parties involved and most importantly, to help the injured victims.
Our heart goes out to the deceased victim's family members who must be struggling to get a sense of closure with the driver still at large. We hope the hit-and-run driver in this case is apprehended and brought to justice, and that the victim's family receives the justice and closure they rightfully deserve.
Posted in: Hit-and-Run Accident
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