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Manama: Although she has been "unemployed" for three years now, Saudi pilot Yasmeen Mohammad Al Maimani has high hopes for the opportunity to fly a plane for Saudia, the kingdom's national carrier.
Yasmeen made history by becoming in 2014 the second Saudi woman to receive a commercial pilot license from the General Authority of Civil Aviation (GACA).
The family went along and they supported Yasmeen in all ways as she headed to Jordan where she joined a private aviation academy.
Driven by her dream and determined to make it come true, she continued with her studies and training.
Yasmeen said that she received an offer from a US flight academy to be their representative in the Arabian Gulf.
Around 500 Saudi women are employed by Saudia, mainly in the financial and IT departments and reservation section. However, none of them are pilots.
Last week, Saudia denied online reports that it was planning to send Saudi women abroad to study to become pilots.


Raj Chakraborty is a most successful Tollywood director, and producer.
He was born into the middle-class family among 3 siblings.
During his school days, he used to participate in theatre group plays.
After completing his graduation, he left his home and moved to Kolkata to make his career in acting.
Despite the struggle, he did not get relevant roles, so he decided to change his career from acting to directing.
In 2000, he started working as an assistant director under director Arindam Dey (Mamdo Da).
After 6 years of continuous and hard struggle, he directed his own telefilm 'Nawder Chaand' and by the huge success of the telefilm, he directed almost 17 telefilms, talk shows and non-fiction reality shows like 'Mirakkel' and 'Dance Bangla Dance'.
His direction contributed on a large scale to the Bangla TV.
He has directed and produced more than 20 Bangla films.
He has his own production namely 'Raj Chakraborty Productions'.
He promoted a lot of Bengali celebrities like Dev, Subhashree Ganguly, Soham Chakraborty, Mimi Chakraborty, Paayel Sarkar, Saayoni Ghosh, Koushani Mukherjee, etc.
In 2012, he started his acting career.
On March 2018, he married his longtime girlfriend and Bengali actress Subhashree Ganguly.


text:
\section{Introduction}
The strong notions of smoothly or Lipschitz
converging manifolds have proven to be exceptionally useful when
studying manifolds with curvature and volume bounds,
Einstein manifolds,
isospectral manifolds of low dimensions,
conformally equivalent manifolds, Ricci flow and
the Poincare conjecture, and even some questions in general relativity.
{\em However
many open questions require weaker forms of convergence that do not
produce limit spaces that are manifolds themselves.}  Weaker notions
of convergence and new notions of limits have proven necessary in
the study of
manifolds with no curvature bounds or only lower bounds on Ricci or scalar
curvature,
isospectral manifolds of higher dimension,
Ricci flow through singularities,
and general relativity.
Here we survey a variety of weaker notions of  convergence and the corresponding
limit spaces covering
both well established concepts, newly discovered ones and speculations.

We begin with the
the convergence of submanifolds of Euclidean space as there is a wealth of
different weak kinds of convergence that mathematicians have been applying
for almost a century.
Section~\ref{Sect-1.1} covers the {\em Hausdorff convergence of sets},
Section~\ref{Sect-1.2} covers Federer-Fleming's {\em flat convergence of integral currents}, and
Section~\ref{Sect-1.3} covers Almgren's {\em weak convergence of varifolds}.  Each
has its own kind of limits and preserves different properties.  Each is useful for
exploring different kinds of problems.  While Hausdorff convergence is most well
known in the study of convex sets, flat convergence in the study minimal surfaces
and varifold convergence in the study of mean curvature flow, each has appeared
in diverse applications.   Keep in mind that these are extrinsic notions of convergence
which depend upon how a manifold is located within the extrinsic space.

It is natural to believe that each should have
a corresponding intrinsic notion of convergence which should prove
useful for studying corresponding intrinsic questions about Riemannian
manifolds which do not lie in a common ambient space.
The second section vaguely describes some of these
intrinsic notions of convergence with pictures illustrating key examples.
Section~\ref{Sect-2.1} covers  \em{Gromov-Hausdorff convergence of metric spaces}
which is an intrinsic Hausdorff distance,
Section~\ref{Sect-2.2} covers  various notions of \em{convergence
of metric measure spaces},
Section~\ref{Sect-2.3} covers the \em{intrinsic flat convergence on integral current spaces}, and
Section~\ref{Sect-2.4} covers the  weakest notion of all: \em{ultralimits of metric spaces}.
Illustrated definitions and key examples will be given for each kind
of convergence.
This survey is not meant to provide a thorough rigorous definition of any of these
forms of convergence but rather to provide the flavor of each notion and direct
the reader to further resources.

The survey closes with speculations on new notions of convergence:
Section~\ref{Sect-3.1} describes difficulties arising when attempting to
define an intrinsic varifold convergence.
Section~\ref{Sect-3.3}  describes the possible notion of area convergence and area spaces.
Section~\ref{Sect-3.2} discusses the importance of weaker notions of convergence
for Lorentzian manifolds.
The author attempts to include all key citations of initial work in these directions.

The author apologizes for the necessarily
incomplete bibliography and encourages the reader to consult mathscinet and
the arxiv for the most recent results in each area.  For some of the older
forms of convergence entire textbooks have been written focussing on
one application alone.

\section{Converging Submanifolds of Euclidean Space}

There are numerous textbooks written covering the three notions of
extrinsic convergence we describe in this section.  Textbooks covering
all three notions include \cite{Morgan-text} and \cite{Lin-Yang-GMT-text}.
Morgan's text providing pictures and overviews of proofs with precise references
to theorems and proofs in Federer's classic text \cite{Federer}.  The focus
is on minimal surface theory.
Another source with many pictures
and intuition regarding Plateau's problem, is Almgren's classic textbook
\cite{Almgren-Plateau}.
Lin-Yang's textbook \cite{Lin-Yang-GMT-text}
covers a wider variety of applications and provides a more modern
perspective incorporating recent work of Ambrosio-Kirchheim.
An excellent resource on varifolds is Brakke's book \cite{Brakke}
which is freely available on his webpage.  Simon's classic
text \cite{Simon-text} is another indispensable resource.

\subsection{Hausdorff Convergence of Sets} \label{Sect-1.1}

The notion of Hausdorff distance dates back to the early 20th century.
Here we define it on an arbitrary metric space, $(Z, d_Z)$, although initially
it was defined on Euclidean space.
The Hausdorff distance between subsets  $A_1,A_2 \subset Z$ is
 \begin{equation} \label{eqn-Hausdorff-dist}
 d^Z_H(A_1, A_2) :=\inf\Big\{\, R \,{\bf :}
 \, \, A_1 \subset T_R(A_2), \,\, A_2 \subset T_R(A_1) \Big\},
 \end{equation}
where $T_r(A):=\{y: \, \exists \, x\in A\,\,s.t.\,\, d_Z(x,y)<r\}$.
We write $A_j \stackrel { \textrm{H}}{\longrightarrow}  A$ iff $d^Z_H(A_j,A) \to 0$.

\begin{figure}[h]
   \centering
   \includegraphics[width=4.6in]{2010-Hconv.jpg}
   \caption{Hausdorff Convergence}
   \label{fig-Hconv}
\end{figure}

In Figure~\ref{fig-Hconv}, we see
two famous sequences of submanifolds converging in the Hausdorff sense.
The sequence $A_i$ are tori which converge to a circle $A$, depicting
how a sequence may lose both topology and dimension
in the limit.  Sequence $B_i$ of jagged paths in the $yz$ plane
converges to a straight segment, $B$, in the $y$ axis.
Notice that $L(B_i)=\sqrt{2}$ while $L(B)=1$.  In addition to a sudden loss
of length under a limit, we lose all information about the derivative
(much like a $C_0$ limit).  In fact one may construct a sequence of jagged
curves that converge in the Hausdorff sense to the nowhere differentiable
Weierstrass function.

One property which is preserved by Hausdorff convergence is convexity.
For this reason, Hausdorff convergence has often been applied in the
study of convex subsets of Euclidean space.

In 1916, Blaschke proved that if a sequence of nonempty compact sets
$K_i$ lie in a ball in Euclidean space, then a subsequence converges in
the Hausdorff sense to a nonempty compact set \cite{Blaschke-1916}.  If
the sets are connected, then their limit is connected.  Path connectedness,
however, is not preserved as can be seen by a sequence approaching
the well known set:
\begin{equation}
\{(x,sin(1/x)): \, x\in (0,1]\}\cap \{(0,y): \, y\in [-1,1]\}.
\end{equation}

Dimension and measure are also not well controlled.
The $k$ dimensional Hausdorff measure of a set $X$
is defined by covering $X$ with countable collections of
sets $C_i$ of small diameter:
\begin{equation*}
\mathcal{H}_k(X):=\lim_{r\to 0} \inf\left\{\sum_{i=1}^\infty \alpha(k)
\left(\frac{\textrm{diam}(C_i)}{2}\right)^k \,:\,\, X\subset \bigcup_{i=1}^\infty C_i, \,\, \textrm{diam}(C_i)<r\right\}
\end{equation*}
where $\alpha(k)$ is the volume of a unit ball of dimension $k$ in
Euclidean space.  For a beautiful exposition of this notion see \cite{Morgan-text}.
When $X$ is a submanifold of dimension $k$, then
$\mathcal{H}_k(X)$ is just the $k$ dimensional Lebesgue measure: when $k=1$
it is the length, when $k=2$ it is the area and so on.

The Hausdorff dimension of a set, $X$, is
\begin{equation} \label{eqn-def-Hausdorff-dimension}
H_{\textrm{dim}}(X):=\inf\{k \in(0,\infty): \mathcal{H}_k(X)=0\}.
\end{equation}
A compact k-dimensional submanifold, $M^k$, has $H_{\textrm{dim}}(M)=k$.
Notice that in Figure~\ref{fig-Hconv}, $H_{\textrm{dim}}(A_j)=2$ while
$H_{\textrm{dim}}(A)=1$.  It is also possible for the dimension to
go up in the limit.  In Figure~\ref{fig-splines}, $H_{\textrm{dim}}(Y_j)=2$ while
$H_{\textrm{dim}}(Y)=3$.  It is also possible for a sequence of $1$
dimensional submanifolds, to converge to a space of fractional
dimension like the von Koch curve.
Such a sequence of curves must have length diverging to infinity.

In fact, if $X_j$ have uniformly bounded
length, $\mathcal{H}_1(X_j) < C$, and $X_j$ converge to $X$ in the
Hausdorff sense, then
\begin{equation} \label{eqn-H-length}
\mathcal{H}_1(X) \le \liminf_{j\to\infty}\mathcal{H}_1(X_j).
\end{equation}
Note that the sequence $B_j$ depicted in Figure~\ref{fig-Hconv} has
$\mathcal{H}_1(B_j)=\sqrt{2}$ for all $j$, but $\mathcal{H}_1(B)=1$,
so lower semicontinuity is the best one can do.
This is not true for higher dimensions as can be seen in Figure~\ref{fig-splines},
where $\lim_{j\to\infty} \mathcal{H}_2(Y_j)=1$ yet $\mathcal{H}_2(Y)=\infty$.

\begin{figure}[h]
   \centering
   \includegraphics[width=4.6in]{2010-splines.jpg}
\caption{Disk of Many Splines}
   \label{fig-splines}
\end{figure}

Figure~\ref{fig-splines} depicts a famous example of ``a disk with splines" described in Almgren's text on Plateau's problem \cite{Almgren-Plateau}.  In Plateau's problem one is given a closed curve and is asked to find the surface of smallest area spanning that curve.
In this example, the sequence of smooth surfaces, $Y_j$,
have a common boundary, $\partial Y_j=\{(x,y,0):\, x^2+y^2=1\}$
which is a given closed curve, and the area of the $Y_j$ is approaching the minimal area
filling that circle.  However the sequence does not converge in the Hausdorff
sense to the flat disk, $D^2$, which is the solution to the Plateau problem for a circle.
Instead, due to the many splines, the sequence converges in the Hausdorff
sense to the solid cylinder.

The disk with many splines example convinced mathematicians that the Hausdorff
distance was not useful in the study of minimal surfaces.  New notions of
convergence had to be defined.

\subsection{Flat Convergence of Integral Currents} \label{Sect-1.2}

Federer and Fleming introduced the notion of an integral current and
the flat convergence of integral currents to deal with examples
like the disk with many splines depicted in Figure~\ref{fig-splines}.
When viewed as integral currents, the submanifolds depicted in
that figure converge in the flat sense to the disk.  All the splines
disappear in the limit.

A current, $T$, is a linear functional on smooth $k$ forms.
Any compact oriented submanifold, $M$, of dimension $k$ with a smooth
compact boundary, may be viewed an a $k$ dimensional
current, $T$, defined by
\begin{equation}
T(\omega):=\int_M  \omega.
\end{equation}
Notice that in Figure~\ref{fig-splines}, we have
\begin{equation}
\lim_{j\to\infty}\int_{Y_j}\omega = \int_{D^2} \omega
\end{equation}
for any smooth differentiable $3$ form, $\omega$.  So, viewed as currents,
$Y_j$ converge weakly to the flat disk $D^2$.

Federer-Fleming proved that any sequence of compact $k$ dimensional oriented
submanifolds, $M_j$, in a disk in Euclidean space, with a uniform
upper bound on $\mathcal{H}^k(M_j)\le V_0$ and a uniform
upper bound on $\mathcal{H}^{k-1}(\partial M_j) \le A_0$, has a subsequence
which converges when viewed as currents in the weak sense.
The limit is an ``integral current".

An integral current, $T$, is a current with a canonical set, $R$, and a
multiplicity function $\theta$.  The canonical set $R$ is countably
$\mathcal{H}^k$ rectifiable,
which means it is contained in the image of a countable collection of Lipschitz
maps, $\varphi_i:E_i\to R$ from Borel subsets, $E_i$, of $k$ dimensional Euclidean space.  The multiplicity
function $\theta$ is an integer valued Borel function.  We define
\begin{equation} \label{eqn-def-integral-current}
T(\omega) \,:= \,\int_R \theta \, \omega \,=\, \sum_{i=1}^\infty \,\int_{E_i} \theta\circ\varphi_i \,\,\,
\varphi_i^*\omega.
\end{equation}
It is further required that an integral current have finite mass
\begin{equation} \label{eqn-def-mass}
\textrm{M}(T) := \int_R \theta \, d\mathcal{H}^k
\end{equation}
and that the boundary, $\partial T$, defined by
$
\partial T (\omega) := T( d\omega)
$
have finite mass, $\textrm{M} (\partial T)<\infty$.  Note that Federer-Fleming
and Ambrosio-Kirchheim proved
this implies that
$\partial T$ also has a rectifiable canonical set although it is
one dimension lower than the dimension of $T$.   \footnote{See \cite{Morgan-text} and \cite{Lin-Yang-GMT-text} for more details.
Note there are slight differences in the definition as \cite{Morgan-text} follows
Federer-Fleming \cite{FF}, while \cite{Lin-Yang-GMT-text} follows the newer
version introduced by Ambrosio-Kirchheim \cite{AK}.}

When a submanifold
$M$ is viewed as an integral current $T$, then $M$ itself is the
canonical set, it has multiplicity $1$, the boundary, $\partial T$,
is just $\partial M$ viewed as an integral current and the mass,
$\textrm{M}(T)$, is just the volume of $M$.

Also included as a $k$ dimensional integral current is the
$0$ current.  In Figure~\ref{fig-Hconv}, the sequence $A_j$ may be viewed as integral
currents.  They converge in the weak sense as integral currents to
the ${ 0}$ current.  More generally, if $\mathcal{H}^k(M_j)$ decreases to
zero, then the weak limit of the $k$ dimensional submanifolds $M_j$
viewed as integral currents is also the $0$ current.   In fact, whenever
$M_j$ have a uniform upper bound on $\mathcal{H}^k(M_j)$ and
$M_j=\partial N_j$ where $\mathcal{H}^{k+1}(N_j) \to 0$, then
$M_j$ viewed as integral currents also converge weakly to the $0$ current.
The sequence $A_j$ depicted in Figure~\ref{fig-Hconv} are the
boundaries of solid tori whose volumes decrease to $0$.
It was not actually necessary that their areas decrease to $0$.
This idea of filling in the manifold to assess where it converges makes
it much easier to see the limits of integral currents and leads naturally
to the following definition.

The flat distance between two $k$ dimensional integral currents,
$T_1$ and $T_2$, is defined by
  \begin{equation} \label{eqn-def-flat}
 d_{\mathcal{F}}(T_1,T_2) =\inf \Big\{\textrm{M}(A) + \textrm{M}(B)
 \,{\bf {:}} \,\, \textrm{\em{int curr }}A,B \,\,s.t.\,\,  A +\partial B=T_1-T_2\Big\},
 \end{equation}
where the infimum is taken over all $k$ dimensional integral currents, $A$,
and all $k+1$ dimensional integral currents, $B$.  \footnote{See prior footnote.}

\begin{wrapfigure}{r}{2.2in}
   \begin{center}
   \includegraphics[width=2in]{2010-flat-distance.jpg}
\end{center}
\end{wrapfigure}

In the figure to the right
we see a choice of $A$ with small area
that looks like a catenoid, and then we fill in the space between the flat disk
$T_2$ and the disk with many splines, $T_1$, to define $B$.  If the
surface with the splines actually shares the same boundary with the disk
(as it does in Figure~\ref{fig-splines}), then we can take $A$ to be
the $0$ current and $B$ just to be a filling (the region in between them).
It is easy to see that the sequence of manifolds $Y_j$ depicted in
Figure~\ref{fig-splines} converges to the disk $D^2$ using the flat
distance.

In Figure~\ref{fig-Hconv} the sequence $B_j$ viewed as $1$ dimensional
integral currents, $T_j$, converges to the Hausdorff limit $B$ viewed as a one
dimensional current, $T$ (as long as they are given the same orientation
left to right).  Here $T_j-T$ is again a cycle and we can find surfaces
viewed as $2$ dimensional currents, $S_j$,
such that $\partial S_j=T_j-T$ whose areas, $\textrm{M}(S_j)$, converge to $0$.

Federer-Fleming proved that when a sequence of integral currents
has a uniform upper bound on mass and on the mass of their boundaries,
then they converge weakly iff they converge with respect to the flat distance.
Their compactness theorem can now be restated as follows: {\em if $T_j$
is a sequence of $k$ dimensional integral currents supported in a compact subset
with $\textrm{M}(T_j)\le V_0$
and $\textrm{M}(\partial T_j)\le A_0$ then a subsequence converges in the
flat sense to an integral current space.}

One of the beauties of this theorem is that the limit space is
rectifiable with finite mass,
and in fact the mass is lower semicontinuous.  This makes flat convergence
an ideal notion when studying Plateau's problem.  The limit integral current
has the same boundary as the sequence and minimal area.

In addition integral currents have a notion of an approximate tangent plane,
which exists almost everywhere and is a subspace of the same
dimension as the current.  When $M$ is a submanifold, this approximate
tangent plane is the usual tangent plane and exists everywhere inside
$M$.  In 1966 Almgren was able to apply the strong control on the approximate
tangent planes to obtain even stronger regularity results for the limits
achieved when working on Plateau's problem \cite{Almgren-1966}.
The limit space ends up having multiplicity $1$.  In general the limit
space may have higher multiplicity or regions with higher multiplicity
[Figure~\ref{fig-cancels}], which we will discuss in more detail later.

In 1999, Ambrosio-Kirchheim extended the notion of an integral current
from Euclidean space to arbitrary metric spaces, proving the
existence of a solution to Plateau's problem on Banach spaces \cite{AK}.
A key difficulty was that on a metric space, $Z$, there  is no notion
of a differential form.   They applied a notion of DeGiorgi \cite{DeGiorgi}, replacing
a $k$ dimensional differential form like
$\omega=fdx_1\wedge dx_2\wedge\cdots\wedge dx_k$ with a $k+1$
tuple, $\omega=(f,x_1, x_2, ...x_k)$, of Lipschitz functions
satisfying a few rules, including
$
d\omega:\,\,=\,\,(1,f,x_1,x_2,...x_k).$

An integral current, $T$, is defined as a linear functional on $k+1$ tuples
using a rectifiable set, $R$, and a multiplicity function, $\theta$, so that
as in (\ref{eqn-def-integral-current}), we have
\begin{equation}
T(\omega) \,\,: = \,\,\sum_{i=1}^\infty \int_{E_i} \theta\circ\varphi_i \,\,\,
\varphi_i^*\omega.
\end{equation}
where $\varphi_i^*\omega \,:= \,f\circ \varphi \,\,d(x_1\circ \varphi)\wedge d(x_2\circ\varphi)
\wedge \cdots \wedge d(x_k\circ \varphi)$ is defined almost everywhere
by Rademacher's Theorem.
They can then define mass exactly as in (\ref{eqn-def-mass}) and
boundary exactly as before as well.   They require integral currents
to have bounded mass and their boundaries to have bounded mass.\footnote{See \cite{AK} or \cite{SorWen2} for the precise definition.}

Ambrosio-Kirchheim prove that if $T_j$ are integral currents in a compact metric
space and $\textrm{M}(T_j) \le V_0$ and $\textrm{M}(\partial T_j)\le A_0$ then a
subsequence converges in the weak sense to an integral current.  Furthermore
the mass is lower semicontinuous.  There is also a notion of an approximate
tangent plane, however, here the approximate tangent plane is a normed
space.  The norm is defined using the metric differential described in earlier work by
Korevaar-Schoen on harmonic maps and also by Kirchheim
on rectifiable space \cite{Korevaar-Schoen}\cite{Kirchheim}.

Wenger extended the class of metric spaces $Z$ which have a solution
to Plateau's problem and  defined a flat distance exactly as in (\ref{eqn-def-flat}).
We will use the notation $d_{\mathcal{F}}^Z$ for the flat distance in $Z$ to be
consistent with $d_H^Z$ denoting the Hausdorff distance in $Z$.
He proved that on this larger class of spaces, which includes Banach spaces,
when $\textrm{M}(T_j) \le V_0$ and $\textrm{M}(\partial T_j)\le A_0$ then
$T_j$ converge weakly to $T$ iff $T_j$ converges in the flat sense to $T$
\cite{Wenger-Plateau}\cite{Wenger-flat}.
Recall that by Rademacher's Theorem, any separable metric space, $Z$,
isometrically embeds into a Banach space, $\textrm{W}$, so we can always study
the convergence of integral currents in $Z$ using the flat distance
of the push forwards of the currents in Banach space, $d_{\mathcal{F}}^{\textrm{W}}$.
If $\varphi:Z \to \textrm{W}$, then the push forward of a current $T$ in $Z$, is a current
$\varphi_\#T$ in $\textrm{W}$ defined by
$
\varphi_\# T \, (f,x_1,...x_k) \,\, := \,\, T \, (f\circ \varphi, x_1\circ\varphi, ... x_k \circ \varphi).
$

Another extension of the notion of integral currents is that of the
${\mathbb Z}_n$ integral currents where the mutiplicity function takes values
in ${\mathbb Z}_n$.  This was introduced by Fleming \cite{Fleming-finite}
and has been extended to arbitrary metric spaces by Ambrosio and
Wenger \cite{Ambrosio-Wenger}.  One application for this notion
is the study of minimal graphs (c.f. \cite{Almgren-Plateau}).
To understand this better we will go over an example using Federer-Fleming's notation.

In Figure~\ref{fig-cancels}, the sequence of embedded curves $C_i$
converges in $C^1$ to a limit curve $C$ which is not embedded.
Viewed as $1$ dimensional currents, $T_i$, they converge in the flat sense to $T$.
It is perhaps easier to understand Figure~\ref{fig-cancels} using weak
convergence rather than flat convergence.  Note that
\begin{equation*}
T_i(\,f_1(x,y)dx+ f_2(x,y) dy\,)= \int_0^1 (f_1\circ C_i)(t)\,
d(x\circ C_i) + (f_2\circ C_i)(t)\,  d(y\circ C_i)
\end{equation*}
converges to
\begin{equation*}
T(\,f_1(x,y)dx+ f_2(x,y) dy\,)\,\,= \,\,\int_0^1 (f_1\circ C)(t)\,
d(x\circ C) + (f_2\circ C)(t)\,  d(y\circ C)
\end{equation*}
Notice that $T$ has multiplicity $1$
where $C$ does not overlap itself.
It has multiplicity $2$ on the segment $C$ covers twice in the same
direction.  The segment depicted with dashes where $C$ passes
in opposite directions is not part of the canonical set of $T$ because
the integration cancels there.
 If one uses ${\mathbb Z}_n$ valued coefficients
we get the same limit space for $n>2$, but for $n=2$ the doubled up
segment disappears as well as the cancelled dashed segment.

\begin{figure}[h]
   \centering
   \includegraphics[width=4.6in]{2010-cancels.jpg}
   \caption{Doubling and Cancellation of Flat Limits}
   \label{fig-cancels}
\end{figure}

The cancellation depicted in Figure~\ref{fig-cancels} is problematic for
some applications.  Note that the same effect can occur in higher
dimensions, by extending the curve to a sheet in ${\mathbb R}^3$.
While often it is useful for thin splines to disappear
in the hopes of preserving dimension, canceling sheets which overlap
with opposing orientations is not necessary to obtain a rectifiable limit
space.  This is seen using the notion of a varifold.

\subsection{Weak Convergence of Varifolds}  \label{Sect-1.3}

Varifolds were introduced by Almgren in 1964 \cite{Almgren-notes}.
Significant further work was completed by Allard in 1972 \cite{Allard-1972}.
Almgren's goal was to define
a new notion of convergence for submanifolds which had rectifiable limit
spaces and a notion of tangent planes for those limit spaces, but did
not have the kind of cancellation that occurs when taking flat limits of
integral currents.  Under varifold convergence, the sequence of
$1$ dimensional manifolds depicted in Figure~\ref{fig-cancels}, converges
to the rectifiable set with weight $1$ everywhere including the dashed
segment.  There is no notion of orientation on the limit, but there
are tangent planes almost everywhere.

A $k$ dimensional varifold is a Radon measure on ${\mathbb R}^N \times \Gamma(k,N)$ where
$\Gamma(k,N)$ is the space of $k$ subspaces of Euclidean space, ${\mathbb R}^N$.
\footnote{In some books $\Gamma(k,N)$ is written as $GL(k,N)$}
A submanifold $M^k\subset {\mathbb R}^n$, may be viewed as the varifold, $V$,
defined on any $W\subset {\mathbb R}^N \times \Gamma(k,N)$ as follows:
\begin{equation} \label{eqn-def-varifold}
V(W) := \mathcal{H}^k \left( \, W\,  \cap\, \left\{ (x, T_xM): \, x\in M \subset {\mathbb R}^m\right\} \right),
\end{equation}
where $T_xM$ is the tangent space to $x$ translated to the origin.

A sequence of varifolds, $V_j$, is said to converge weakly to a varifold, $V$,
\begin{equation} \label{eqn-def-varifold-convergence}
\int f \, dV_j\,\, \to \,\,\int f \, dV \qquad \forall \, f \, \in \, C_0({\mathbb R}^N\times \Gamma(k,N)).
\end{equation}

For example, let us examine the sequence of curves $C_j$ which converge $C^1$
to the curve $C$ in Figure~\ref{fig-cancels}.
They may be viewed as integral currents $V_j$,
\begin{equation} \label{eqn-ex-varifold-1}
V_j(W)=\mathcal{H}^1\left( W \, \cap \,
\left\{ \left( C_j(t), \pm C_j'(t)/|C_j'(t)| \right): \, t\in [0,1]\right\} \, \right),
\end{equation}
where we view points in $\Gamma(1,2)$ as $\pm v$ where $v$ in a unit
$2$ vector.  Then
\begin{equation} \label{eqn-ex-varifold-2}
\int f(x,v) dV_j \, = \, \int_0^1 f(C_j(t), \pm C_j'(t)/|C_j'(t)|) \, dt
\end{equation}
converges to $\int_0^1 f(C(t), \pm C(t)/|C(t)|) dt =\int f(x.v)\, dV$
for the varifold $V$ defined by
\begin{equation} \label{eqn-ex-varifold-3}
V(W) := \int _0^1 \chi_W(C(t), \pm C'(t)/|C'(t)|) \,dt
\end{equation}
where $\chi_W$ is the indicator function of $W$.  This varifold corresponds
to viewing the limit curve $C$ as having weight $1$ on the segments where
it doesn't overlap itself and weight $2$ on the segments where it does overlap.
There is no cancellation.  The dashed segment has weight $2$.

Not all limit varifolds end up with tangent planes that align well with the
rectifiable set as they do in (\ref{eqn-ex-varifold-3}).  If we examine instead
the sequence $B_j$ in Figure~\ref{fig-Hconv}, as a sequence of curves $C_j$
converging to a curve $C$, we see that the limit varifolds, $V$, has the form
\begin{equation} \label{eqn-ex-varifold-4}
V(W) := \,\,a \int _0^1 \chi_W(C(t), \pm av_1) \,dt \,\,+\,\,a \int _0^1 \chi_W(C(t), \pm av_2) \,dt
\end{equation}
where $v_1=(1,1)$, $v_2=(-1,1)$ and $a =\sqrt{2}/2$ because $C_i'(t)$ is always
in the direction of $v_1$ or $v_2$.
This effect was observed by Young
and is discussed at length in \cite{Lin-Yang-GMT-text}.   Varifolds like
$V$ are of importance but are not considered to be integral varifolds
because $C'(t)$ is unrelated to $v_1$ and $v_2$.

An integral varifold, $V$, is defined as a positive integer weighted countable sum
of varifolds, $V_j$, which are
defined by embedded submanifolds $M_j$ as in (\ref{eqn-def-varifold}).
While $V$ is a measure on ${\mathbb R}^N\times \Gamma(N,k)$, it
defines a natural measure $||V||$ on ${\mathbb R}^N$,
\begin{equation} \label{eqn-def-varifold-density}
||V||(A) := V(A \times \Gamma(k,N)).
\end{equation}
When $V$ is defined by a submanifold, $M$, then $||V||(A)=\mathcal{H}^k(A \cap M)$.

A $k$ dimensional varifold $V$  is said to have a tangent plane, $T\in \Gamma(k,N)$ with
multiplicity $\theta\in (0,\infty)$ at a point  $x\in {\mathbb R}^N$ if a sequence of rescalings
of $V$ about the point $x$  converges to $\theta(x) T$. \footnote{See \cite{Lin-Yang-GMT-text}
for a more precise definition.}
An integral varifold, $V$,
has a tangent space $||V||$ almost everywhere on ${\mathbb R}^N$.

Varifolds do not have a notion of boundary.  Instead a varifold, $V$, has a notion of
first variation, $\delta V$, which is a functional that maps functions
$f\in C_0^\infty({\mathbb R}^N,{\mathbb R}^N)$
to ${\mathbb R}$.  There is no room here to set up the background for
a precise definition \footnote{See \cite{Lin-Yang-GMT-text} 6.2 for the full definition.},
so we just describe
$\delta V$ when $V$  is defined by a $C^2$ submanifold $M$
with boundary $\partial M$.   In this case,
 $\delta V (f)$  is the first variation in the area of $M$ as it is flowed through
 a diffeomorphism defined by the vector field $f$:
 \begin{equation} \label{eqn-def-first-var}
 \delta V(f) \, = \, - \int_M \, f(x) \cdot H(x) \, d\mathcal{H}^k(x) \,\,+\,\, \int_{\partial M} \, f(x)\cdot \eta(x)\,
 d\mathcal{H}^{k-1}(x)
 \end{equation}
 where $H(x)$ is the mean curvature at $x\in M$ and $\eta(x)$ is the outward normal
 at $x \in \partial M$.

 A varifold, $V$, is said to be stationary when $\delta V=0$.

 A varifold has bounded first variation if
 \begin{equation}
 ||\delta V|| (W) := \sup \{\delta V(f): \, f\in C_0^\infty({\mathbb R}^N, {\mathbb R}^n), \, |f|\le 1,
 \, \textrm{spt}(f) \subset A \} <\infty.
 \end{equation}
 When $V$ corresponds to a submanifold, $M$, as above, then
 \begin{equation}
 ||\delta V|| (W) \, = \, \mathcal{H}^{k-1}(W \cap \partial M) \, + \,
 \int_{W \cap M} |H(x)| \, d\mathcal{H}^m(x).
 \end{equation}
 There is an isoperimetric inequality for varifolds.

 Allard proved that if a sequence of integral varifolds, $V_j$, with a uniform
 bound on  $||\delta V_j||(W)$ depending only on $W$, converges
 weakly to $V$, then $V$ is an integral varifold as well \cite{Allard-1972}.
 In particular a weakly converging sequence of minimal surfaces, $M_j$, with a uniform
 upper bound on the length of their boundaries, $\partial M_j$,
 converges to an integral varifold.  For more general submanifolds, one
 needs only uniformly control the volumes of the boundaries and the $L^1$ norms
 of the mean curvatures.

One key advantage of varifolds is that they have a notion of mean curvature
defined using $\delta V$
 and an integral similar to the one in (\ref{eqn-def-first-var}).  This notion
 is used to define the Brakke flow, a mean curvature flow past singularities \cite{Brakke}.

Recently White has set up a natural map, $F$, from integral varifolds to ${\mathbb Z}_2$
flat chains, which basically preserves the rectifiable set and takes the integer
valued weight to a ${\mathbb Z}_2$ weight.  Each ${\mathbb Z}_2$ flat chain corresponds
uniquely to a ${\mathbb Z}_2$ integral current.
He has proven that if a sequence of submanifolds,
$M_j$
viewed as varifolds converge weakly to an integral varifold $V$
and satisfy the conditions of Allard's compactness theorem,
and if further $\partial M_j$ converge as ${\mathbb Z}_2$ integral currents,
then $M_j$ viewed as ${\mathbb Z}_2$ integral currents converge to a
${\mathbb Z}_2$ integral current $T$ corresponding to $F(V)$.
 This is easily seen to be the situation for
the sequence depicted in Figure~\ref{fig-cancels}.  \cite{White-2009}

\section{Intrinsic Convergence of Riemannian Manifolds}

When studying sequences of Riemannian manifolds, $M_j$, the strongest notions
of convergence require that $M_j$ be diffeomorphic to the limit space $M$
with the metrics, $g_j$, converging smoothly:
\begin{equation}
\exists \varphi_j: M \to M_j \textrm{ such that } \varphi_{j}^* g_j \to g.
\end{equation}
  In this survey we are concerned with
sequences of Riemannian manifolds which do not converge in such a strong sense.
Here we describe a few weaker notions of convergence which allow us
to better understand sequences which do not converge strongly.  We begin
with a pair of motivating examples.

The sequence of flat tori, $M_j=S^1_\pi \times S^1_{\pi/j}$, has volume converging
to $0$: $\textrm{vol}(M_j)=2\pi(2\pi/j) \to 0$.
Sequences with this property are called collapsing sequences.  They
do not converge in a strong sense to a limit which is also a torus.  Intuitively one
would hope to define a weaker notion of convergence in which these tori
converge to a circle.   Examples like this lead to Gromov's
notion of an intrinsic Hausdorff  convergence.

If one views Figure~\ref{fig-splines} as a sequence of Riemannian disks
with splines, $M_j$, with the induced Riemannian metrics, they do not converge
in a strong sense to a flat Riemannian disk.  While they are diffeomorphic to
the disk and the volumes are converging, $\textrm{vol}(M_j)\to \textrm{vol}(D^2)$,
the metrics do not converge smoothly.  Examples like this lead to the notion
of intrinsic flat convergence.

In this section we first present Gromov-Hausdorff convergence, then
metric measure convergence, then the intrinsic flat convergence and finally,
weakest of all, the notion of an ultralimit.  We include a few key examples,
applications and further resources for each notion.

\subsection{Gromov-Hausdorff Convergence of Metric Spaces}\label{Sect-2.1}

In 1981,  Gromov
introduced an intrinsic Hausdorff convergence for sequences of
metric spaces \cite{Gromov-1981}.  A few excellent references are
Gromov's book \cite{Gromov-metric},
 the textbook of Burago-Burago-Ivanov\cite{BBI},
 Fukaya's survey \cite{Fukaya-survey} and Bridson-Haefliger's
 book \cite{BH}.

The Gromov-Hausdorff distance is
 defined between any pair of compact metric spaces,
  \begin{equation} \label{eqn-GH-dist}
 d_{GH}(M_1, M_2)= \inf \Big\{ d^Z_H\left(\varphi_1(M_1), \varphi_2(M_2)\right)
 \,{\bf:} \,\,
 isom\,\, \varphi_i: M_i \to Z\Big\}
 \end{equation}
where the infimum is taken over all metric spaces, $Z$, and all
isometric embeddings, $\varphi_i:M_i \to Z$.
An isometric embedding, $\varphi:X \to Z$
satisfies
\begin{equation} \label{eqn-isom-embed}
d_Z(\varphi(x_1), \varphi(x_2)) = d_X(x_1, x_2) \qquad \forall x_1, x_2 \in X.
\end{equation}
We write $M_i \stackrel { \textrm{GH}}{\longrightarrow}  X$ iff $d_{GH}(M_j, X) \to 0$.
See Figure~\ref{fig-GHconv}.

\begin{figure}[h]
   \centering
   \includegraphics[width=4.6in]{2010-GHconv.jpg}
   \caption{Gromov-Hausdorff Convergence}
      \label{fig-GHconv}
\end{figure}

 The sequences of Riemannian manifolds depicted in Figure~\ref{fig-GHconv}
 reveal a variety of properties that are not conserved under Gromov-Hausdorff
 convergence.   The first sequence $A_j$ are the flat tori $S^1_\pi\times S^1_{\pi/j}$
 converging to a circle $A=S^1_\pi$.  To see this one takes the common space
 $Z_j=A_j$ and isometrically embeds $A$ into $Z_j$ so that
 \begin{equation}
 d_{GH}(A_j, A) \le d_H^{Z_j}(A_j, \varphi_j(A))= \pi/(2j) \to 0.
 \end{equation}
 Here we see the topology and Hausdorff dimension may decrease in the limit.

Note that if $M_1$ and $M_2$ are compact then $d_{GH}(M_1, M_2)=0$
iff $M_1$ and $M_2$ are isometric.  The Gromov-Hausdorff distance
between $M_1$ and $M_2$ is almost $0$ iff there is an almost isometry
$f:M_1 \to M_2$ satisfying
\begin{equation} \label{eqn-def-almost-isometry}
|d_1(x,y) - d_2(f(x),f(y))|<\epsilon \textrm{ and }
M_2\subset T_\epsilon(f(M_1)).
\end{equation}
Note that an almost isometry need not be continuous.
In Figure~\ref{fig-GHconv}, it is easy to construct $\epsilon_j$
almost isometries, $\varphi_{j}: B \to B_j$, such that $\epsilon_j \to 0$
and conclude that $B_j \stackrel { \textrm{GH}}{\longrightarrow}  B$.
This example
 reveals that the Gromov-Hausdorff
 limit of a sequence of Riemannian manifolds may not be a Riemannian manifold.

 Gromov-Hausdorff limits of Riemannian manifolds are geodesic metric spaces.
 This means that the distance between any pair of points is equal to the length
 of the shortest curve between them.  The shortest curve exists and is called
 a minimal geodesic.  As in Riemannian geometry, a curve $\gamma$ is called
 a geodesic if for every $t$, there is an $\epsilon>0$ such that $\gamma$
 restricted to $[t-\epsilon, t+\epsilon]$ is a minimal geodesic.\footnote{Alexandrov
 space geometers use the term ``geodesic" to refer to
 a ``minimal geodesic" and ``local geodesic" to refer to a geodesic.}

 In the third sequence of Figure~\ref{fig-GHconv},
 $C_j$ are also the boundaries of increasingly thin
 tubular neighborhoods.  The limit, $D$, is the Hawaii Ring, a metric space of
 infinite topological type that has no universal cover.  For more about the
 topology of Gromov-Hausdorff converging sequences of manifolds
 see \cite{SorWei1} \cite{SorWei3} and \cite{ShnSor2}.

  In the last sequence of Figure~\ref{fig-GHconv},
 $D_j$ are the smoothed boundaries of increasing
 thin tubular neighborhoods of increasingly dense grids.  They converge to a
 square $D=[0,1]\times[0,1]$.
 The metric on the square is the taxicab metric (also called the
 $l_1$ metric):
 \begin{equation}
 d_C((x_1,y_1),(x_2,y_2)) := |x_1-x_2| + |y_1-y_2|.
 \end{equation}
 This is easiest to see by showing the $D_j$ are Gromov-Hausdorff
 close to their grids and that the grids converge to the taxicab square.

 Gromov proved that sequences of Riemannian manifolds, $M_j$, with uniform
 upper bounds on their diameter and on the number, $N(r)$, of disjoint balls
 of radius $r$ have subsequences which converge in the Gromov-Hausdorff
 sense to a compact geodesic space, $Y$ \cite{Gromov-1981}.
 Conversely, when $M_j$ converge to a compact $Y$, $N(r)$ is uniformly
 bounded.  Consequently, if one views the sequence of disks with splines
 depicted in Figure~\ref{fig-splines} as Riemannian manifolds (with the
 intrinsic distance), the sequence
 does not converge in the Gromov-Hausdorff sense: a ball of radius $1/2$
 about the tip of a spline does not intersect with the a ball of radius $1/2$
 about the tip of another spline.  As the number of splines approaches infinity
 so does the number of disjoint balls of radius $1/2$.

 By the Bishop-Gromov volume
 comparison theorem, sequences of manifolds, $M_j$, with uniform lower bounds
 on Ricci curvature and upper bounds on diameter satisfy these compactness criteria
 \cite{Gromov-1981}.  This includes, for example, the sequence of flat tori, $A_j$,
 in Figure~\ref{fig-GHconv} as well as any other collapsing sequence of
 manifolds with bounded sectional curvature.  This lead to a series of papers
 on the geometric properties of $M_j$ with uniformly bounded sectional curvature
 and $\textrm{vol}(M_j)\to 0$ by Cheeger-Gromov, Fukaya, Rong, Shioya-Yamaguchi and others
\cite{Fukaya-collapsing} \cite{CheegerGromovI} \cite{CheegerGromovII}
\cite{ShioyaYamaguchi-I}
\cite{ShioyaYamaguchi-II}
 (c.f. \cite{Rong-collapsing-survey} \cite{Morgan-Tian-text}).  Collapsing
 Riemannian manifolds with boundary have been studied by Alexander-Bishop and
 Cao-Ge  \cite{Cao-Ge}
 \cite{Alexander-Bishop-ThinI}
\cite{Alexander-Bishop-ThinII}.
This collapsing theory is an essential component of Perelman's
 proof of the Geometrization Conjecture using Hamilton's
 Ricci flow \cite{Morgan-Tian-text} \cite{Kleiner-Lott-perelman} \cite{Cao-Zhu-perelman}.

 In 1992,  Greene-Petersen applied the notion of Gromov's filling volume
 to find a lower bound on the volume of a ball in a Riemannian
 manifold with a uniform geometric contractibility function.   A geometric contractibility function is a function $\rho:(0,r_0] \to (0,\infty)$  with $\lim_{r\to 0}\rho(r)=0$ such that
 any ball $B_p(r)\subset M$ is contractible in $B_p(\rho(r))\subset M$.
Applying Gromov's compactness theorem, one may conclude that
a sequence $M_j$ with a uniform geometric contractibility
function and a uniform upper bound on volume, has a subsequence that
converges in the Gromov-Hausdorff sense to a compact metric space.
Note that in this setting, volume is uniformly
bounded below so the sequence is not collapsing \cite{Greene-Petersen}.
The limits of such sequences of spaces have been studied by Ferry,
Ferry-Okun, Schul-Wenger and Sormani-Wenger \cite{Ferry}
\cite{Ferry-Okun}\cite{SorWen1} and \cite{SorWen2}.

Noncollapsing sequences of Riemannian manifolds with Ricci
curvature bounded from below were studied by Colding and
Cheeger-Colding in their work on almost rigidity in which they
weakened the conditions of well known rigidity theorems.  One example
of a rigidity theorem is the fact that
 if $M^m$ has $\textrm{Ricci} \ge (m-1)$ and $\textrm{vol}(M^m)=\textrm{vol}(S^m)$
 then $M^m$ is isometric to $S^m$.  Colding
 proved a corresponding almost rigidity  theorem
 which states that if $M^m$ has $\textrm{Ricci} \ge (m-1)$ then
 $\forall \epsilon>0$, $\exists \delta_{m,\epsilon}>0$ such that
 $\textrm{vol}(M^m)\ge \textrm{vol}(S^m)-\delta_{m,\epsilon}$ implies $d_{GH}(M^m, S^m)< \epsilon$.
 For a survey of such almost rigidity theorems see \cite{Colding-survey},
 \cite{ChCo-almost-rigidity} and Section 8 of \cite{Sor-length}.
 The proofs generally involve an explicit construction of an almost
 isometry using distance functions and solutions to elliptic
 equations on the Riemannian manifolds.

When a sequence of $M_j$ with uniform lower bounds
on Ricci curvature does not collapse Cheeger-Colding proved
the volume is continuous and the Laplace spectrum converges.
Fukaya observed that when a sequence of Riemannian manifolds
with uniform lower bounds on Ricci curvature is collapsing, then
the Laplace spectrum does not converge.  In Figure~\ref{fig-Fukaya}
there are two sequences of Riemannian manifolds $A_i$ and
$B_i$ with nonnegative Ricci curvature converging to the same limit space
$A=B=[0,1]$ with the standard metric.  The eigenvalues converge
to real numbers,
\begin{equation}
\lambda^A_i:=\lim_{j\to\infty} \lambda_i(A_j) \textrm{ and }
\lambda^B_i:=\lim_{j\to\infty} \lambda_i(B_j)
\end{equation}
but $\lambda^A_i \neq \lambda^B_i$.  While $\lambda^B_i$
made sense as eigenvalues for $[0,1]$, the other collection
of numbers $\lambda^A_i$ did not.  The alternating
sequence $\{A_1, B_1, A_2, B_2...\}$ also converges in the
Gromov-Hausdorff sense  but the eigenvalues do not converge
at all. \cite{Fukaya-87}.

This example disturbed Fukaya in light of the successful isospectral
compactness theorems of Osgood-Phillips-Sarnak,
Brooks-Perry-Petersen and Chang-Yang \cite{Osgood-Phillips-Sarnak}
\cite{Brooks-Perry-Petersen} \cite{Chang-Yang-isospec-1990}
all of which imposed stronger conditions on the manifolds and
involved smooth convergence or even conformal convergence
of the sequences.  This lead to Fukaya's notion of metric measure
convergence.

\subsection{Metric Measure Convergence of Metric Measure Spaces} \label{Sect-2.2}
In 1987, Fukaya introduced the first notion of metric-measure convergence.
A sequence of metric measure spaces $(X_j, d_j, \mu_j)$
converge in the metric measure sense to a metric measure space
$(X, d,\mu)$ if there is a sequence of $1/j$ almost isometries,
$f_j: X_j \to X$, such that push forwards of the measures, $f_{j*}\mu_j$,
converge weakly to $\mu$ on $X$.  Recall
that $\varphi_\#\mu(A):=\mu(\varphi^{-1}(A))$.

In Figure~\ref{fig-Fukaya},  $A_j$ and $B_j$ are given
probability measures, $\mu_{A_j}$ and
$\mu_{B_j}$ proportional to $\mathcal{H}^2$.   Then
$B_j$ converge in the metric measure sense to $[0,1]$
with the standard metric and $\mu_B(W)=\mathcal{H}^1(W)$.
Meanwhile $A_j$ converge in the metric measure sense to $[0,1]$ with the
measure
\begin{equation}
\mu_A(W)\,:=\, \int_W (2-|2-4x|)\,dx.
\end{equation}
Defining the Laplacian
with respect to these measures, the spectrum for
$(A, \mu_A)$ is $\{\lambda^A_i\}$ and the spectrum for $(B, \mu_B)$
is $\{\lambda^B_i\}$.\footnote{See \cite{Fukaya-1987} or \cite{ChCo-PartIII} for
more details.}

\begin{figure}[h]
   \centering
   \includegraphics[width=4.6in]{2010-Fukaya.jpg}
\caption{Metric Measure Convergence}
   \label{fig-Fukaya}
\end{figure}

Cheeger-Colding then proved that any sequence of compact Riemannian
manifolds with uniform lower bounds on Ricci curvature endowed
with a probability measure proportional to the Hausdorff measure,
has a subsequence which converges in the metric measure sense.
The measures on the limit space satisfy the Bishop-Gromov comparison
theorem and are therefore doubling \cite{ChCo-PartI}.
Sturm and Lott-Villani extended the notion
 of a Ricci curvature bound to general metric measure spaces using mass transport
 \cite{Sturm-06}\cite{Lott-Villani-09} (c.f. \cite{Villani-text}).
 Recently Topping and others have been developing a notion of Ricci flow on
 this larger class of spaces in hopes of defining Ricci flow through a singularity
 \cite{Topping-new}.

Cheeger-Colding also prove a Poincare inequality
on these limit spaces and deduced that the eigenvalues converge
\cite{ChCo-Partii} \cite{ChCo-PartIII}.  They proved that the
limit spaces of a sequence of Riemannian manifolds
with uniform lower bounds on Ricci curvature also have a notion
of tangent plane almost everywhere.  The tangent plane
at a point is found by rescaling the space $Y$ outward
and taking a pointed Gromov-Hausdorff limit.  At many points, one
does not get a unique limit under rescaling and the limits are
not necessarily planes or cones.  However they do exist at every point
and are called tangent cones.  Cheeger-Colding proved
tangent cones are Euclidean planes almost everywhere.  In fact
they prove $Y$ is a countably $\mathcal{H}^m$
rectifiable space and is a $C^{1,\alpha}$ manifold
away from the singularities.
Key steps in the proof involve the Splitting
Theorem and the Poincare inequality (c.f. \cite{Cheeger-survey}).
There is an example where the limit space has infinite topological type
so it isn't a $C^{1,\alpha}$ manifold \cite{Menguy-inf-top-type}.
However, at least the manifold has a universal cover
unlike the Hawaii Ring depicted in Figure~\ref{fig-GHconv} \cite{SorWei1}.

It is also natural to study metric measure convergence without
Ricci curvature bounds on the sequence of manifolds.  As long as
the measure is doubling one can apply Gromov's compactness
theorem.  Sometimes the measure on the limit space is supported
on a smaller set.  This occurs for example for the sequence $B_j$
of Figure~\ref{fig-GHconv}.  Viewed as a metric measure limit
space, the limit space $B$ has a measure which is supported on the
two spheres.

In 1981, Gromov introduced $\square_\lambda$ convergence to handle
this issue \cite{Gromov-metric}.  When $\lambda=1$, the limit of the sequence of $B_j$ is
just the two spheres with the line segment removed.  Basically, if $(Y,d,\mu)$
is the metric measure limit of a sequence, and if there is a ball such that
$\mu(B_y(r))=0$ then $y$ is removed from the space.  This leaves us
with a new, smaller limit space, $(\bar{Y}, d_Y, \mu)$, which may no longer
be connected.  This process may be called ``reduction of measure".
The spaces are then no longer close in the Gromov-Hausdorff topology.

In 2005, Sturm introduced a new distance between metric measure spaces
which leads to such limits more naturally and also interacts well with
mass transport notions mentioned above \cite{Sturm-2006-I}.  He uses the
Wasserstein distance, $W_p$, to measure the distance between measures
in a set $Z$ and defines an intrinsic Wasserstein distance using an infimum
over all metric spaces $Z$ and all isometric embeddings $\varphi_i: X_i \to Z$:
\begin{equation} \label{eqn-measure}
d_{W_p}((X_1,d_1, \mu_1), (X_2, \mu_2, d_2)) :=
\inf\{ d_{W_p}(\varphi_{1,\#}\mu_1, \varphi_{2\#}\mu_2)\}.
\end{equation}
Sturm proved convergence with respect to this distance is equivalent to
Gromov's $\square_1$ convergence \cite{Sturm-2006-I}.

Villani recently defined an intrinsic Prokhorov
distance replacing the Wasserstein distance of order $p$ in (\ref{eqn-measure})
with the Prokhorov
distance between measures.  As convergence with respect to the Wasserstein
distance and with respect to the Prokhorov distance  agree with weak convergence,
the intrinsic Prokhorov and intrinsic Wasserstein limits agree as well.  He refers
to these kinds of convergence as ``measure convergence"
Two metric measure spaces are a zero distance apart with respect to these
intrinsic measure distances  iff there is a measure preserving isometry between them
\cite{Villani-text}.

Villani also described  ``metric measure distances".  The Gromov-Hausdorff
Wasserstein distance is defined: $ d_{W_p}((X_1,d_1, \mu_1),(X_2, d_2, \mu_2)) :=$
\begin{equation}
=\inf\{ d_H(\varphi_1(X_1), \varphi_2(X_2)) +
d_{W_p}(\varphi_{1\#}\mu_1, \varphi_{2\#}\mu_2)\},
\end{equation}
where the infimum is taken over all metric spaces $Z$
and all isometric embeddings $\varphi_i:X_i\to Z$.
The Gromov-Hausdorff Prokhorov distance has the same formula replacing the
Wasserstein distance  by the Prokhorov distance between measures.
Convergence with respect to these metric measure distances agrees with
Fukaya's metric measure convergence. The advantage of
having multiple distances is that one may be easier estimate
than another \cite{Villani-text}.

A sequence of metric measure spaces with doubling measures has a
converging subsequence and the
metric measure limit  agrees with the measure limit
of a sequence \cite{Villani-text}.  Any sequence of Riemannian manifolds
with lower bounds on their Ricci curvature satisfies the doubling condition
by the Bishop-Gromov volume comparison theorem.  The sequence of
$B_j$ in Figure~\ref{fig-GHconv} does not, as the metric measure and
measure limits do not agree.

Despite the immense success in applying these definitions of convergence to
 study manifolds with Ricci curvature bounds, there has been a need to introduce
 a weaker form of convergence to study sequences of manifolds which do not
 satisfy these strong conditions.  Mathematicians studying manifolds with scalar
 curvature bounds and those interested only in sequences with an upper bound
 on volume and diameter without curvature bounds, need a weaker version of
 convergence.  In particular, geometric analysis related to cosmology and the
 study of the spacelike universe requires a weaker form of convergence.

An important example introduced by Ilmanen may be called
the $3$-sphere of many splines.
It is a sequence of three dimensional spheres with positive scalar curvature whose
volume converges to the volume of the standard three sphere, but has an increasingly
dense set of thinner and thinner splines of ``length" $1$.  Cosmologically, one may
think of these splines as deep gravity wells.
The sequence does not converge in the
Gromov-Hausdorff sense because balls of radius $1/2$
centered on the tip of each spline are disjoint and the number of such disjoint
balls approaches infinity.  This example naturally lead Sormani and Wenger to develop
a notion of intrinsic flat convergence.

\subsection{Intrinsic Flat Convergence of Integral Current Spaces} \label{Sect-2.3}
In 2008 Sormani and Wenger introduced the intrinsic flat distance
between compact oriented Riemannian manifolds \cite{SorWen1}\cite{SorWen2}:
 \begin{equation}
d_{\mathcal{F}}(M_1, M_2):=\inf \Big\{ d^Z_{\mathcal{F}}(\varphi_{1\#}T_1, \varphi_{2\#}T_2)
 \,{\bf :}
 \,\, isom\,\, \varphi_i:M_i \to Z\Big\}
 \end{equation}
 where the flat distance in $Z$ is defined as in (\ref{eqn-def-flat}),
 the $\varphi_i:M_i\to Z_i$ are isometric embeddings as in (\ref{eqn-isom-embed})
 and where $T_i$ are defined by integration over $M_i$ so that
 $\varphi_{i\#}T_i (f, x_1,...x_k) = $
 \begin{equation} \label{eqn-push-forward}
 =
 T_i (f\circ \varphi_i, x_1\circ\varphi_i ,... x_k\circ\varphi_i)
 = \int_{M_i} f\circ \varphi_i \,\, d(x_1\circ \varphi_i) \wedge \cdots \wedge d(x_k\circ \varphi_i).
 \end{equation}

 One may immediately note that
 $d_{\mathcal{F}}(M_1, M_2) \le \textrm{vol}(M_1)+\textrm{vol}(M_2)<\infty$
 as we may always take the integral current $A$ in (\ref{eqn-def-flat})
 to be $\varphi_{1\#}T_1-\varphi_{2\#}T_2$ and $B=0$.
 The intrinsic flat distance is a distance between compact oriented
 Riemannian manifolds in the sense that $d_{\mathcal{F}}(M_1,M_2)=0$
 iff there is an orientation preserving isometry between $M_1$ and $M_2$ \cite{SorWen2}.

 Note that in practice it is often possible to estimate the intrinsic flat distance
 using only notions from Riemannian geometry.  That is, if $M^k_1$ and $M^k_2$
 isometrically embed into a Riemannian manifold $N^{k+1}$ such that
 $
 \partial N^{k+1}=\varphi_1(M^k_1) \cup \varphi_2(M^k_2)
 $
 and the manifolds have been given an orientation consistent
 with Stoke's theorem on $N^{k+1}$, then
 $
 d_{\mathcal{F}}(M_1, M_2) \le \textrm{vol}(N^{k+1})
 $
 This viewpoint makes it quite easy to see that Ilmanen's $3$ sphere of many splines
 example describes a sequence converging to the standard sphere.  Note that
 one cannot just embed the sequence into four dimensional Euclidean space and
 take $N^{4}$ to be the flat region lying between the two spheres because such
 $\varphi_i$ would not be isometric embeddings.
 Instead, one rotates each spline into half a thin $4$ dimensional
 spline and glues it smoothly to a short $S^3 \times [0,\epsilon_j]$ where
 $\epsilon_j$ is ``thinness" of the spline.  This produces a four dimensional
 manifold $N^4$ with $S^3$ isometrically embedded as one boundary and
 the $3$ sphere of many splines isometrically embedded as the other boundary.
 So the intrinsic flat distance between these two spaces is $\le \textrm{vol}(N^4)$
 which is approximately $\epsilon_j \textrm{vol}(S^3)$ plus the sum of the volumes
 of the four dimensional splines each of which is approximately
 $ \epsilon_j^3$.  See \cite{SorWen2} for this example and many others.

 The limit spaces are called integral
 current spaces, and are oriented weighted countably $\mathcal{H}^k$
 rectifiable metric spaces.  They have the same dimension as the
 sequence.  They have a set of
Lipschitz charts as well as a notion of an approximate tangent space.
The approximate tangent space is a normed space whose
norm is defined by the metric differential.  The notion of
the metric differential was developed in work of
Korevaar-Schoen \cite{Korevaar-Schoen} and Kirchheim \cite{Kirchheim}.
 See \cite{SorWen2} for
 a number of examples of limit spaces.

 Integral current spaces are said to have a ``current structure", $T$,
 defined by integration using the orientation and weight.  One writes
 $(X,d,T)$.  The current structure defines a mass measure $||T||$.
 The points in $x$ all have positive density with respect to this measure.
 These spaces have a notion of boundary coming
 from this current structure.  The boundary is also an integral current space.
  \footnote{See \cite{SorWen2} for more details about the relationship
  between $X$ and $T$ and how to find the boundary.}

    An important integral current space is the $0$ current
 space, and collapsing sequences of manifolds,
 $\textrm{vol}(M_j)\to 0$, converge to the $\bf{0}$ space.
 Notice that in Figure~\ref{fig-GHconv}, the sequences $A_j$, $C_j$
 and $D_j$ all converge to $0$ in the intrinsic flat sense.
 The sequence $B_j$ converges to a pair of spheres in the intrinsic
 flat sense where the limit does not include the line segment.

Most results about integral current spaces and intrinsic flat convergence
are proven by finding a way to isometrically embed the entire sequence
into a common metric space, $Z$, and apply the theorems for integral
currents in $Z$.  The rectifiablility, dimension and boundary properties of
the limits follows from Ambrosio-Kirchheim's corresponding results on
integral currents.  So does the slicing theorem and the lower semicontinuity
of mass.  Combining Gromov's compactness theorem with Ambrosio-Kirchheim's
compactness theorem, one sees that a sequence of manifolds which converge
in the Gromov-Hausdorff sense that have a uniform upper bound on volume,
also converge in the intrinsic flat sense \cite{SorWen2}.

In general the intrinsic flat and Gromov-Hausdorff limits
do not agree: the intrinsic flat limit may be a strict subset of the Gromov-Hausdorff
limit.  If the Gromov Hausdorff limit is lower dimensional than the sequence,
then the intrinsic flat limit is the ${\bf{0}}$ space.  Any regions in the Gromov-Hausdorff
limit that are lower dimensional disappear from the intrinsic
flat limit  \cite{SorWen2}.
Note that this is in contrast with
metric measure limits, which do not lose regions of lower dimension that
still have positive limit measures.  Such a situation can occur, for example, if
a thin region collapsing to lower dimension is very bumpy and has a uniform
lower bound on volume.

Sequences of Riemannian manifolds may also disappear due to cancellation.
Recall that in Euclidean space, a submanifold which curves in on itself as in
Figure~\ref{fig-cancels} will have cancellation in the limit due to the opposing
orientation.  An example described in \cite{SorWen1} is a sequence of
three dimensional manifolds with positive scalar curvature created by taking a
pair of standard three spheres and joining them by increasingly dense and
increasingly thin and short tunnels.  The Gromov-Hausdorff limit is a 3-sphere
while the intrinsic flat limit is the $0$ current space.  If each tunnel is twisted,
then the intrinsic flat limit is the standard three sphere with multiplicity two.

In some
cases the intrinsic flat limits and Gromov-Hausdorff limits agree giving
new insight into the rectifiability of the Gromov-Hausdorff limits.
They agree for
noncollapsing sequences of manifolds with nonnegative Ricci curvature and also for
sequences of manifolds with uniform linear contractibility
functions and uniform upper bounds on volume \cite{SorWen1}.  As a consequence the
Gromov-Hausdorff limits are countably $\mathcal{H}^m$ rectifiable metric spaces.
This was already shown by Cheeger-Colding for the sequences with
bounded Ricci curvature  but is
a new result for the sequences with the linear geometric contractibility hypothesis.
With only uniform geometric contractibility functions that are not linear,
Schul and Wenger have shown the limits need not be so rectifiable \cite{SorWen1}
and when there is no upper bound on volume, Ferry has shown the limit
spaces need not even be finite dimensional \cite{Ferry}.
Recall that Cheeger-Colding have proven limits of manifolds with nonnegative
Ricci curvature have Euclidean tangent cones almost everywhere.  In contrast,
there is a sequence of Riemannian manifolds with uniform
linear contractibility functions that converge to the taxicab space \cite{SorWen2}.

The key ingrediant in the proofs of these noncancellation results is an estimate
on the filling volumes of small spheres in the spaces.  The filling volumes of
the spheres are continuous with respect to the intrinsic flat distance and can
be more easily applied to control sequence than the mass (or volume).
It is conjectured that a sequence of three dimensional Riemannian manifolds
with positive scalar curvature, no interior minimal surfaces and a uniform upper
bound on volume and on the area of the boundary converges in the intrinsic
flat sense without cancellation.  Such manifolds are important in the study of
general relativity. \cite{SorWen1}

In fact, Wenger has proven that a sequence of
oriented Riemannian manifolds with a uniform upper bound on diameter and
on volume has a subsequence which converges in the intrinsic flat sense
to an integral current space \cite{Wenger-compactness}.  The proof involves
an even weaker notion of convergence: the Ultralimit.

\subsection{Ultralimits} \label{Sect-2.4}

The
notion of an ultralimit was introduced by van den Dries and Wilkie in 1984
\cite{DW-ultralimit}
and developed by Gromov in \cite{Gromov-asympt}.  An ultralimit of a sequence
of metric spaces is defined using Cartan's 1937 notion of a nonprincipal ultrafilter:
a finitely
additive probability measure $\omega$ such that all subsets $S \subset {\mathbb N}$
are $\omega$ measureable, $\omega(S)\in \{0,1\}$ and
$\omega(S)=0$ whenever $S$ is finite.  Given an ultrafilter, $\omega$, and a bounded
sequence of $a_j \in {\mathbb R}$, there exists an ultralimit, $L=\omega-\lim_{j\to\infty} a_j$,
such that $\forall \epsilon >0$, $\omega\{j: |a_j-L|<\epsilon \}=1$.

Given a sequence of compact metric spaces, $(X_j, d_j)$, with a uniform bound
on diameter, and given an ultrafilter, $\omega$, there is an ultralimit $(X,d)$
which is a metric space that may no longer be compact, but is at least complete.
The space, $X$,
is constructed as equivalence classes of sequences $\{x_j\}$ where $x_j \in X_j$
and the metric of $X$ is defined by taking ultralimits:
\begin{equation}
d(\{x_j\}, \{y_j\}) := \omega - \lim_{j\to\infty} d_j(x_j,y_j).
\end{equation}
Two sequences are equivalent when the distance between them is zero.  Notice
that one does not need a subsequence to find a limit and that in general the
ultralimit depends on $\omega$.

If $X_j$ has a Gromov-Hausdorff limit, then the ultralimit is the Gromov-Hausdorff
limit and there is no dependence on $\omega$.  If one has two
sequences $X_j \stackrel { \textrm{GH}}{\longrightarrow}  X$ and $Y_j\stackrel { \textrm{GH}}{\longrightarrow}  Y$, then the ultralimit
of the alternating sequence $\{X_1, Y_1, X_2, Y_2, X_3, ...\}$ is $Y$ iff
$\exists N $ such that $\forall n>N$ we have
$\omega\{2n, 2n+2, 2n+4, 2n+6,...\}=1$.  Otherwise the ultralimit is $X$.

It is often useful to compute the ultralimits of sequences
which do not have Gromov-Hausdorff limits.   For example,
the three sphere of many splines sequence, $\{M_j\}$,
converges to a standard three sphere
with countably many unit line segments attached at various points on the sphere.
To see this, one may view each $M_j$ as a union of regions $W_j \cup U_{1,j}\cup U_{2,j}\cup...\cup U_{N_j,j}$,
where each $U_{i,j}$ covers a spline and $W_j$ covers the spherical
portion.  Then $W_j \stackrel { \textrm{GH}}{\longrightarrow}  S^3$, while $U_{i_j,j} \stackrel { \textrm{GH}}{\longrightarrow}  [0,1]$.  Thus any
ultralimit of these $M_j$ is built by connecting countably many intervals to a three
sphere.     The locations where
the intervals are attached to the sphere may depend on the ultrafilter, $\omega$.

If $X_j$ are geodesic spaces,
then the ultralimit is a geodesic space.
If the $X_j$ are $CAT(0)$ spaces then the ultralimit is as well.
Ultralimits have been applied extensively in the study of CAT(0) spaces
and also Lie Groups.  See for example \cite{BH} and \cite{Kapovich-text} for
more details as well as applications.

\section{Speculation}
While the methods of intrinsic convergence defined above have all
proven to be useful in a variety of settings, and should in fact have
more applications that have not yet been discovered or fully explored,
it is clear that each has its disadvantages.  Gromov-Hausdorff
convergence, like Hausdorff convergence, provides no rectifiability
for its limits.  The intrinsic flat convergence, like Federer-Fleming's flat
convergence, has difficulty with cancellation in the limit.  Ultraconvergence
has the same difficulties as Gromov-Hausdorff convergence and the
additional concern that the spaces in the sequence are not particularly
close to the limit space in a measurable way using a notion of distance.

There are important problems both related to General Relativity
and Ricci flow which have not yet been addressed using the above methods.
While intrinsic flat convergence may prove useful as it is further explored,
other methods of convergence may be necessary to address all the problems
that arise.

\subsection{Intrinsic Varifold Convergence}\label{Sect-3.1}
For some time, mathematicians have been exploring possible methods of extending Ricci
flow through singularities.  Mean curvature flow behaves a lot like Ricci flow
and the Brakke flow uses varifolds to extend this notion to the nonsmooth
setting. The key missing ingredient that has kept people from directly extending the work
on Brakke flow to better understand Ricci flow is that there has never been
a notion of an intrinsic varifold space and intrinsic varifold convergence.

In the prior section we saw how mathematicians have repeatedly applied
Gromov's trick of isometrically embedding metric spaces into a common space,
measuring the distance between them in that common space, and then taking
an infimum.  Here, however,
there is no metric measuring the distance between varifolds, just the
natural notion of weak convergence against test functions.
One might try to define a notion of distances between varifolds
in Euclidean space which defines a convergence that agrees with the standard
varifold convergence and then apply Gromov's trick.

Alternately, one might venture to say that a sequence of Riemannian manifolds, $M_j$,
converges to a space, $M$, as varifolds if and only if there is a sequence of isometric
embeddings from, $M_j$, into a common space, $Z$, such that the images of
$M_j$ converge as varifolds in $Z$ to an image of $M$.  However, there is as yet
no notion of convergence as varifolds in a metric space.  Recall that if
$Z$ is Euclidean space, convergence as varifolds, means weak convergence
as measures on ${\mathbb R}^N \times \Gamma(N,k)$.  We would need some sort
of notion of a $\Gamma(Z,k)$ perhaps representing all possible ``tangent spaces"
of $Z$ at a point.  Perhaps this might be done by first isometrically embedding
$Z$ into a Banach space.

\subsection{Area Convergence} \label{Sect-3.3}

One may view Gromov-Hausdorff convergence as a notion of convergence
defined by the fact that lengths of minimizing geodesics converge.
Measure convergence is defined by the fact that volumes converge.
It becomes natural to wonder, particularly in the case of three dimensional
manifolds, whether there is a notion of convergence defined by the fact that
areas, or areas of minimal surfaces, converge.

Note that Ilmanen's example of the three sphere of many splines, $\{M_j\}$,
converges in some area sense to the standard three sphere.  There are
radial projections $f_j: M_j \to S^3$ which are almost area preserving,
one-to-one and onto.  Such a map seems perhaps to extend the notion
of an almost isometry without involving a distance.  However requiring
a diffeomorphism to define area convergence seems too strong for any
applications.

Burago, Ivanov and Sormani spent a few years investigating possible notions
of area convergence and the area distance between spaces.   They
observed that two distinct metric spaces could easily have a surjection between
them that is area preserving.  One space could, for example, be the unit
square, $X=[0,1]\times[0,1]$, with the Euclidean metric.  The other space
could be the unit square with a ``pulled thread": $Y=[0,1]\times[0,1]/ \sim$ where
$(x_1,x_2)\sim (y_1,y_2)$ iff $x_1=y_1=0$ with the metric:
\begin{equation}
d_Y((x_1,x_2), (y_1,y_2)) =\min \{ \sqrt{(x_1-y_1)^2+(x_2-y_2)^2}, |x_1|+|y_1| \}.
\end{equation}
This seemed to indicate the need for a notion of ``area space" defined as
an equivalence class of metric spaces.
It was essential before even beginning the project
to verify that each equivalence class would only include one Riemannian
manifold.

Another difficulty arose in that one would not expect to always find an
almost area preserving surjection between spaces which should intuitively
be close in the area sense.  In an almost isometry, there are two requirements:
almost distance preserving and almost onto.  Both requirements involve distance
and one can prove that if there is an almost isometry from a metric space
$X$ to a metric space $Y$ then one can find an almost isometry in the reverse
direction (although the errors change slightly in reverse) \cite{Gromov-metric}.
To take advantage of Gromov's ideas, it was decided that one needs to convert
the notion of area into a notion of distance.

Given a Riemannian manifold, $M$, one may examine the loop
space, $\Omega(M)$.  The flat distance between loops is defined using
the notion of area.  Recall that the flat distance defined in (\ref{eqn-def-flat})
involves both area and length, however, in the setting of closed
loops one may choose $A=0$ and only take an infimum over all
possible fillings by surfaces $B$.  Perhaps an area convergence of
$M_j$ to some sort of area space $M$ could be defined by taking
pointed Gromov-Hausdorff limits or  ultralimits of the $\Omega(M_j)$.

Preliminary work in this direction was completed by Burago-Ivanov.  They
proved that if a pair of three dimensional Riemannian manifolds, $M^3$
and $N^3$, have isometric loop spaces, then $M^3$ and $N^3$ are isometric
\cite{Burago-Ivanov-area}.  This step alone was very difficult because $M^3$
is not isometrically embedded into its loop space, $\Omega(M)$, so one must
localize points and planes in $TM$ using sequences of loops without controlling
the lengths of the loops.

Now one may move forward and investigate
whether there is a natural compactness theorem for some notion
of area convergence.   It would not be surprising if
positive scalar curvature on $M_j$ control areas of minimal
surfaces well enough to control $\Omega(M_j)$.  Or perhaps one
could skip compactness theorems, and apply ultralimits to the
$\Omega(M_j)$ and see what properties are conserved under
such a limit.  Ultimately one does not just want to take a limit of
$\Omega(M_j)$ but find an area space $X$, such that $\Omega(X)$
is the limit of the $\Omega(M_j)$.  Then one could say $X$ is
the area limit of $M_j$.

\subsection{Convergence of Lorentzian manifolds} \label{Sect-3.2}
Another direction of research that is fundamental to General Relativity
is the development of weak forms of convergence for Lorentzian manifolds.
While Gromov-Hausdorff convergence has proven useful in the
study of the stability of the spacelike universe \cite{Sor-Cosmos}, one needs
to extend Gromov-Hausdorff convergence to the Lorentzian setting
to study the stability of the space-time universe.
The techniques described above immediately fail for these spaces
because they do not isometrically embed into metric spaces.

Noldus has developed a Gromov-Hausdorff distance between Lorentzian
spaces \cite{Noldus-Lorentzian}.
Further work in this direction including a notion of the arising limit
spaces appears in \cite{Noldus-Limit}
and compactness theorems and an examination of causality in the limit spaces are
proven with Bombelli in \cite{Bombelli-Noldus}.   This work has not yet been applied to
prove stability questions
arising in general relativity.

Perhaps similar methods might be applied to extend the notion of the intrinsic
flat distance to Lorentzian spaces.

\subsection{Acknowledgements}
The author is indebted to Jeff Cheeger for all the excellent advise he has given
over the past fifteen years.  His intuitive explanations of the beautiful geometry
of converging Riemannian manifolds in his many excellent talks have been
invaluable to the mathematics community.   The author would also like to thank
the editors for soliciting this article and William Wylie for discussions regarding the exposition.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Q: Custom UIView changes after second call my first question here :D
I've a Swift based iOS App. Everything is under a TabBarController.
So i call Tab 1 and there is a TableView. I click on a cell and everything is fine except this:

I try to round the Top corners of a UIView. Now it gets important. I stay on this detail page and change to another Tab.
Now i click on Tab 1 again and:

It's perfect. But why on the second call?
The rounded function is called in viewWillAppear. So it should load on the first call.
Anyone an idea?
Many thanks in advance
Florian

A: If the frame of this view is being calculated programatically based on the size of other views that comes from nib, it won't work if you call it on viewWillAppear because the frames of the views are not setted yet. Try to put it on viewDidAppear method. But if the view comes from storyboard or nib try on the view the methods layoutIfNeeded() or in the superview layoutSubviews(). Hope that helps :)

A: This is one of the common mistake one does, if you want to change something in your UI just before its appearance on the screen, viewWillAppear is not right place to do, you should always do those stuffs in viewDidLayoutSubviews.
Just go through viewController's life cycle -
viewDidLoad - Gets Called when the controller's view is loaded into memory.
viewWillAppear - Gets called when the view controller's view is about to be added to a view hierarchy. (Note -the view is yet to be added to the view hierarchy).
viewDidLayoutSubviews - Gets called when view controller just laid out its subviews. (Note - This is the place if you want to modify your UI just before appearing on screen.)
viewDidAppear - Gets called when view controller's view was added to a view hierarchy

redpajama_set_name: RedPajamaStackExchange


Cheap beer prices? What cheap beer... where?
A BID to boost beer prices to bring an end to 'cheap beer' through higher taxes has been dismissed as "misguided, lazy and flawed" by Australia's brewers.
"The reality is beer is not cheap in Australia," Brewers Association of Australia CEO Brett Heffernan explained today.
"Any Aussie venturing overseas who has slaked their thirst with a cleansing ale knows we pay a premium for beer in Australia. Why? Great question.
"Australians pay amongst the highest excise on beer in the world. Then we pay a further 10% in GST on top of that. Last year, taxes on beer drinkers alone netted the Australian Government almost $2.4 billion.
"In fact, research from the University of Adelaide in 2014 shows that, in Australia, beer is taxed at more than twice the OECD average.
"Comparing our $0.58 excise on the $2.00 wholesale pre-tax beer price, Australians pay over seven times more than Argentina, Belgium, Chile and Poland ($0.08); over six times more than Austria, Hungary, the Netherlands and South Africa ($0.09); almost five times more than Italy and Greece ($0.12); double the beer excise paid in the US ($0.29) and almost double that of New Zealand ($0.33)** .
"Of the $47.99 retail price of a typical 24-carton of full-strength beer, $15.08 of that price is tax. It means the single most expensive ingredient in beer is Australian Government tax. Tax accounts for almost one-third (31%) of the total price of a carton of full-strength beer.
"But wait, there's more. The excise on beer in Australia is automatically increased by the Australian Government every six months, so the price of beer is always going up and up and up.
"Calls for higher taxes or minimum floor prices on beer are misguided.
"Firstly, when it comes to cheap alcohol products, beer is not one of them.
**Source: Excise Taxes on Wines, Beers and Spirits: An Updated International Comparison, Table 4, University of Adelaide, September 2014.


New release for the Qpid Advanced Message Queuing Protocol implementation.
Apache Qpid version 0.8 is now available. Apache Qpid is a cross platform Enterprise Messaging solution implementing the Advanced Message Queuing Protocol. This release adds support for the C++ and Python clients, and improves 0-10 support for the Java broker. A new hybrid SQL-CLFS persistence module for the C++ broker on Windows has also been added.


text:
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More Man for all seasons
The 2019 Festival of Speed will honour Sir Jackie Stewart, who this summer celebrates both his 80th birthday and the 50-year anniversary of winning his first Formula 1 World Championship. The Duke of Richmond talked to the motor racing icon about his glittering career, early experiences of Goodwood and enduring love of motorsport.
Words by Duke of Richmond
DUKE OF RICHMOND: Tell me about your earliest memories of Goodwood.
JACKIE STEWART: I first went to Goodwood with my brother Jimmy for the Nine Hour race in the 1950s. Later on I was there with Jimmy when he was racing for David Murray's Ecurie Ecosse team. We stayed at the Ship Hotel in Chichester, and David took us to some nice pubs and restaurants. Jimmy won his race in a C-Type Jaguar. He wore a visor rather than goggles, I remember that. In 1962 I won at Goodwood in a Cooper T49 with Ecurie Ecosse – that was a big moment. Then in September 1963 I won there again in their Cooper-Monaco, and set a new lap record. The circuit manager Robin Mackay was watching and told Ken Tyrrell he should "have a look at this Jackie Stewart", and that's how I came to sign for Ken's Formula 3 team after a test on a cold winter's day early in 1964. I knew the circuit but had never driven the car, a new Cooper T72-BMC, and Bruce McLaren was there to set a lap time before I went out. Ken told me to take my time, but I was soon beating Bruce's times, and John Cooper came running down from Madgwick Corner and I heard him tell Ken: "You gotta sign him, you gotta sign him. Now!" So that day at Goodwood was an incredibly important moment for me. We went on to win the F3 Championship that year, and then of course my three World Championships.
Jackie Stewart in his Team Surtees Lola T70
DR: You came back to race at Goodwood with BRM in your first year of Formula 1.
JS: Yes, I'd signed for BRM, alongside Graham Hill, and started from pole in the Sunday Mirror Trophy on Easter Monday in 1965. In the race I shared a new lap record of 1:20.4 with my friend Jimmy Clark in the Lotus, who did an identical time, and that remains in your pre-1966 Goodwood record books as the official lap record. Sadly the car broke down and Jimmy won the race. Nobody knew much about me in those days. I was just a young lad starting out, really, so I don't remember going to your grandfather's cocktail parties in Goodwood House. I met him, of course, testing and racing there, and he was a lovely man. I know your family have always been great hosts, but neither Ken (Tyrrell) nor I were party animals back then.
Goodwood is a driver's circuit, you can make a real difference.
Jackie Stewart at Goodwood in 1965
DR: Tell me about the Goodwood Motor Circuit – why is it a challenge for drivers?
JS: It's a demanding track, not an easy one. The first corner at Madgwick, for example, the double apex right-hander, you have to know where to put the car and where to brake. You have to carry the speed out of the corner onto the straight towards St Mary's and then into Lavant. These are demanding corners in a really quick car. In a Can-Am car, for example, it's mighty – even a wee bit scary. Goodwood is a driver's circuit, you can make a real difference. It looks easy, but it's not – it's so fast. If you go off the road you're going to hit the bank, and back in my day the chicane was made of bricks… To lap Goodwood at the limit has never been easy, but it's a wonderful circuit.
Jackie Stewart at Goodwood in 2008 in his record-beating BRM P261
DR: Why did you never compete after you retired in 1973? Many drivers have come back to do historics.
JS: I made a commitment to myself and my family that I would never race again. I was happy to do demo runs, so the Festival of Speed and Revival have given me so many opportunities to be reunited with lots of the cars I raced. These are such great events, not only because of getting back behind the wheel, but also because of all the people, the drivers and the fans. My sons Paul and Mark and I all drove Tyrrells for a display at Revival and that was such a lovely moment for me. My old mechanics ran the cars for us: Roy Topp, Roger Hill, Roland Law. That was just great – the same guys who prepared my cars in that period. I just wish Ken (Tyrrell) could have been there, too. He was so good at choosing the very best mechanics. That's what I love about Goodwood, those reunions, seeing the people you remember. All our yesterdays are there again. I sign hundreds of autographs, though the selfie is the big thing these days. I think an autograph is better, much more personal, something to keep for ever. I still have my autograph book from when I was a wee boy.
DR: Is motor-racing still an important part of your life?
JS: Oh, yes, my love of the sport is as deep as it's ever been. I go to at least ten Grands Prix every year, through my work with Rolex and Heineken. The sport changed everything for me, and has given me and my whole family such a wonderful life. I went to the Members' Meeting for the first time this year and really enjoyed it, being back at the circuit where that test day launched my career. We must remember how dangerous it was in my day – I lost so many friends. The work we did to make F1 so much safer is perhaps the most important thing we did, the achievement of which I'm most proud, along with my World Championships.
When I raced, I never let emotions distract me but reunions with cars, and the people, bring a smile to my face and maybe, just maybe, a tear to my eye.
DR: You will celebrate your career at the Festival of Speed this year. An emotional milestone for you?
JS: Yes, I'm looking forward to being there. When I slip down into the cockpits of the cars I raced, it's like putting on a pair of favourite old gloves, especially the Tyrrells and the Matra. It's 50 years since I won my first World Championship and you know, the Tyrrell has the very same seat that I used all those decades ago. When I raced, I never let emotions distract me but reunions with cars, and the people, bring a smile to my face and maybe, just maybe, a tear to my eye.
Goodwood will celebrate Sir Jackie Stewart's 80th birthday, and the 50th anniversary of his first Formula 1 World Championship triumph, by featuring some of his most famous cars at this year's Festival of Speed (which takes place July 4 – 7). The list includes the first car Stewart ever raced – the Marcos Xylon Special – and the BRM P261 in which he took his first Grand Prix victory.
The world of Jackie Stewart
Jackie Stewart made an impact well beyond the world of motor racing. His friends and admirers have included royalty, TV personalities, pop stars and opera singers…
JACKIE ON PRINCESS ANNE
"We've been friends since the early 1970s when we were both voted BBC Sports Personality of the Year. When you're in sport there's an understanding and a respect for each other. I was certainly impressed by Anne; she did something no member of the Royal Family had ever done before: compete in the Olympics. It's a friendship that's lasted all these years and [my wife] Helen is godmother to Zara. The thing about good friends is, you don't need to see each other all the time. We know we'll always be there for each other."
Jackie Stewart and Princess Anne
JACKIE ON GEORGE HARRISON
"I first met George Harrison when the Beatles turned up at the Monaco Grand Prix in 1969. George enjoyed motor racing. We shared many wonderful times. On one occasion, when he was staying with us in Switzerland, we went to the French Grand Prix in Dijon. It was a clear, warm evening when we arrived home. Helen had prepared a barbecue, and [my son] Paul, who was 12 at the time, brought out his guitar. George started to play, running through all the great Beatles hits, singing parts of the songs, explaining what the lyrics meant to various members of the band. I remember sitting there thinking this had to be one of the greatest privileges anybody could have. George had a great soul. His instinct was to forgive ratherthan to condemn and when people behaved badly he would make excuses for them. I learnt so much from him."
Jackie Stewart and George Harrison
DAVID COULTHARD ON JACKIE
"Sir Jackie is to my mind one of the greatest ever professional sportspeople. He defined a brand ambassador is, which has benefitted all professional sports, and his contribution to our sport's safety is immeasurable. I have no doubt that his role in my early career enabled me to race for 15 seasons in F1 while enjoying a second career in broadcasting today."
He can talk the leg off an iron pot and argue on any subject known to man.
DAME KIRI TE KANAWA ON JACKIE
"Jackie has been a dear friend for many years. Always the consummate gentleman and perfectionist. A very caring, generous man with a great sense of humour."
SIR MICHAEL PARKINSON ON JACKIE
"I've known Jackie Stewart for 50 years or more – long enough to appreciate the fact that of all the people I've met over a lifetime of meeting remarkable people, he would be among the four or five most remarkable – not only in conquering the world of Formula 1, but in using his fame and celebrity to make the sport much safer. Successive generations of F1 drivers have good reason to consider him their patron saint. He can talk the leg off an iron pot and argue on any subject known to man. His marriage to Helen has been a rare love story; none better demonstrated over recent years than through his devoted care. He is also a very good golfer and an excellent shot. Proud to call him a friend."
This article was taken from the Summer 2019 edition of the Goodwood Magazine.
The breakthrough

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text:
After Charleston, South revisits its many Confederate images
Opponents point to state flags, license plates and statues of Civil War politicians and generals.
After Charleston, South revisits its many Confederate images Opponents point to state flags, license plates and statues of Civil War politicians and generals. Check out this story on detroitnews.com: http://detne.ws/1N823ss
AP Published 9:08 p.m. ET June 23, 2015
The state flag of Mississippi is unfurled against the front of the Governor's Mansion in Jackson, Miss. Republican Lt. Gov. Tate Reeves said Tuesday, that Mississippi voters, not lawmakers, should decide whether to remove the Confederate battle emblem from the state flag.(Photo: Rogelio V. Solis / AP)
ATLANTA (AP) — Calls to remove Confederate imagery from public places multiplied rapidly across the South and beyond Tuesday, with opponents eyeing state flags, license plates and statues of Civil War politicians and generals.
The startling movement, driven by the killing of nine black churchgoers in an apparent racist attack in Charleston, South Carolina, has made converts of politicians who have long supported or stood silent on such symbols. Many of the efforts appear to have the muscle to succeed.
Statehouse displays such as the Confederate battle flag flying in South Carolina are coming under the heaviest fire. But the familiar banner, with its star-studded blue 'X' overlaying a field of red, is just one of scores, if not hundreds, of state-sanctioned displays that honor the vanquished Confederacy and the era of Jim Crow segregation that lasted for more than a century after the end of the Civil War.
The homages — from veterans' memorials and statues of politicians to counties, streets, government buildings and public schools named for Confederate figures and subsequent white supremacists — haven't always generated the same political and social tensions as the battle flag, and Confederate heritage groups say the outcry is misplaced, despite widely seen images of what appears to be the church shooting suspect, 21-year-old Dylann Roof, holding the rebel flag.
At the least, however, the flag and other tributes remain a constant reminder of the nation's perpetual struggle with race, and of some Southerners' defiance of the federal government's efforts on civil rights.
"Statues and monuments aren't history," said Stan Deaton, a historian for the Georgia Historical Society. "They are what we choose to tell future generations about the past. … It's a very delicate subject, and let's not kid ourselves: So much of it has to do with race."
In Kentucky, the Republican candidate for governor, Matt Bevin, and U.S. Sen. Mitch McConnell said Tuesday that a statue of Confederate President Jefferson Davis should be removed from the Capitol rotunda, where it sits just feet from a statue of Abraham Lincoln, whose election spurred the South's secession. Both men were born in Kentucky, a border state during the Civil War.
Virginia Gov. Terry McAuliffe, a Democrat, said he wanted the state to stop issuing a vanity license plate for the Sons of Confederate Veterans that includes the battle flag. "Even its display on state-issued license tags, in my view, is unnecessarily divisive and hurtful to too many of our people," McAuliffe said, speaking in Richmond, the second and final capital of the Confederacy.
In Tennessee, Republican Gov. Bill Haslam and lawmakers of both parties called for removing a Capitol grounds bust of Nathan Bedford Forrest, a Confederate general and early Ku Klux Klan leader. The longest-serving black legislator in Alabama said he plans to introduce a resolution that would remove the Confederate flags that fly outside the Alabama Capitol next to a towering monument to Confederate soldiers.
New Orleans Mayor Mitch Landrieu's office said the city should consider changes to several monuments, including a prominent statue of Confederate Gen. Robert E. Lee near downtown, as the city prepares to celebrate the 300th anniversary of its founding.
Top Mississippi Republicans appear divided over the state's flag, the last of the 50 state banners to include a specific image of the battle flag. House Speaker Philip Gunn said Monday that the image, which appears in the top left corner of the Mississippi flag, is offensive and should be removed. Lt. Gov. Tate Reeves responded Tuesday that the decision should be up to Mississippians, who voted 2-to-1 in 2001 to keep the flag. Gov. Phil Bryant, also a Republican, said he supports that referendum result.
Chris McDaniel, a state senator and tea party hero who nearly unseated U.S. Sen. Thad Cochran last year, decried Gunn's call. "A cultural or historical cleansing of all things potentially offensive will do nothing to alleviate the problems caused by racism," he said.
Deaton, the Georgia historian, said McDaniel misses the point. "Symbols do matter and naming practices do matter," he said, arguing that the Confederate monuments across the region, placed mostly by Confederate veterans and women's groups in the late 19th and early 20th centuries, offer "a very narrow interpretation of the past."
There were signs that the tension was spilling beyond the political realm. Vandals have tagged several monuments in recent days, including a Charleston statue of John C. Calhoun, a strong defender of slavery and secession before the war. It and other targets in Maryland and Texas were spray-painted with phrases such as "Black Lives Matter," a slogan rooted in recent police shootings of black men.
The leader of a national Confederate heritage organization argues that Roof's actions should not reflect on American citizens who identify with the Confederacy.
"First it's the flags, then the monuments, then the streets names, then the holidays. I feel like it's open season on anything Confederate," said Kelly Barrow, commander in chief of the Sons of Confederate Veterans, adding that the vandalism scares him.
"Is someone going to be attacked because they have an S.C.V. sticker on their car? We're open targets, battle flag or not," he said.
Barrow, based south of Atlanta, said the discussion over the monuments should at least wait until after the church victims' funerals.
"Bury the dead, then we can sit down and talk about all this," he said.
Najee Washington, whose grandmother Ethel Lance was among the nine slain last week during Bible study at Emanuel African Methodist Episcopal Church in Charleston, said it would be meaningful to her family to remove the flag: "It's just a part of the past that we don't need to be reminded of every day."
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To speak another language fluently absolutely blows my mind. For those of you who are privileged to have accomplished this, I envy you!
In the past I taught English as a Second Language in a program dedicated to helping Spanish-speaking mothers who had children in the public school system in California. There was an understanding that in order to help the children excel in their new environment, the moms needed to be able to communicate and understand English, as well. I taught those ladies without any understanding of their situation, but how times have changed.
We don't have a school aged kid just yet, but now I am the mom who needs to grasp a language that is so foreign to my ears. This week I learned a new Thai word. Not from our private lessons or a book, but from Charlie! He doesn't say or understand it perfectly, but for a 20 month old he is communicating well. He always points to the lights or a light switch as says, "fais" (pronounced fies). So I soon learned that "fais" was his word for light and I knew then that he wanted to be picked up so he could push the light switch.
Well, as his communication skills increase daily (he is a jabber box) it finally dawned on me that I should ask what the Thai word for light is. Knowing that he often adds an "s" to everything, which derives from his familiarity with English, I asked if "fai" can mean light and sure enough, it does. Thanks Charlie, I think we might learn more from you than from a book.
Speaking of learning Thai, we have been in private lessons for 13 weeks now (just 2 hours a week) and apparently are quite a bit behind the average student…learning the Thai word for failure this week was not a good sign. Haha. But, despite that, we have loved learning and actually feel like we've progressed a lot as this was our first formal introduction. Honestly, we have all kinds of excuses for why we don't practice more and all I'm going to say is that if you are ever given the opportunity for intensive language study take it!
Keep at the language study! We're proud of ya'll! G$ is talking a good bit too now.