Few corrections to spelling in the introductions

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

sec/part2/conclusion.tex

@@ -1,5 +1,5 @@
 From the previous analysis it seems that string theory cannot do better than field theory when the latter does not exist, at least at the perturbative level where one deals with particles.
-Moreover when spacetime becomes singular the string massive modes are not spectators anymore.
+Moreover when spacetime becomes singular, the string massive modes are not spectators anymore.
 Everything seems to suggest that issues with spacetime singularities are hidden into contact terms and interactions with massive states.
 This would explain in an intuitive way why the eikonal approach to gravitational scattering works well: it is indeed concerned with three point massless interactions.
 In fact it appears that the classical and quantum scattering on an electromagnetic wave~\cite{Jackiw:1992:ElectromagneticFieldsMassless} or gravitational wave~\cite{tHooft:1987:GravitonDominanceUltrahighenergy} in \bo and \nbo are well behaved.

sec/part2/divergences.tex

@@ -1,7 +1,6 @@
 \subsection{Motivation}
 
-Unfortunately and puzzlingly the first attempts to consider space-like~\cite{Craps:2002:StringPropagationPresence} or light-like singularities~\cite{Liu:2002:StringsTimeDependentOrbifold,Liu:2002:StringsTimeDependentOrbifolds} by means of orbifold  techniques yielded divergent four points \emph{closed string} amplitudes (see \cite{Cornalba:2004:TimedependentOrbifoldsString,Craps:2006:BigBangModels} for reviews).
-
+The first attempts to consider space-like~\cite{Craps:2002:StringPropagationPresence} or light-like singularities~\cite{Liu:2002:StringsTimeDependentOrbifold,Liu:2002:StringsTimeDependentOrbifolds} by means of orbifold  techniques yielded divergent four points \emph{closed string} amplitudes (see \cite{Cornalba:2004:TimedependentOrbifoldsString,Craps:2006:BigBangModels} for reviews).
 These singularities are commonly assumed to be connected to a large backreaction of the incoming matter into the singularity due to the exchange of a single graviton~\cite{Berkooz:2003:CommentsCosmologicalSingularities,Horowitz:2002:InstabilitySpacelikeNull}.
 This claim was already questioned in the literature where the $O$-plane orbifold was constructed.
 This orbifold should in fact be stable against the gravitational collapse but it exhibits divergences in the amplitudes (see the discussion in \cite{Cornalba:2004:TimedependentOrbifoldsString}).

sec/part2/introduction.tex

@@ -18,13 +18,13 @@ First of all we recall the formal definition of orbifold to better introduce the
 Let therefore $M$ be a topological space and $G$ a group with an action $\ccG: G \times M \to M$ defined by $\ccG(g,\, p) = g p$ for $g \in G$ and $p \in M$.
 Then the \emph{isotropy subgroup} (or \emph{stabiliser}) of $p \in M$ is $G_p = \qty{ g \in G \mid g p = p }$ such that $G_{gp} = g^{-1}\, G_p\, g$.
 Given an element $p \in M$ its \emph{orbit} is $Gp = \qty{ gp \in M \mid g \in G }$.
-The action of the group is said \emph{transitive} if $Gp = M$ and \emph{effective} if its kernel is trivial, i.e.\ $\ker{\ccG} = \qty{ \1 }$.
+The action of the group is said \emph{transitive} if $Gp = M$ and \emph{effective} if its kernel is trivial, i.e.\ $\ker{\ccG} = \qty{ \1 }_{\ccM}$.
 The \emph{orbit space} $M / G$ is the set of equivalence classes given by the orbital partitions and inherits the quotient topology from $M$.
 
 Let now $M$ be a manifold and $G$ a Lie group acting continuously and transitively on $M$.
 For every point $p \in M$ we can define a continuous bijection $\lambda_p \colon G / G_p \to Gp = M$.\footnotemark{}
 \footnotetext{%
-  For any $U \subset M$ and a given $p \in M$ then $\lambda_p^{-1}\qty( U ) = \pi_p\qty( \qty{ g \in G \mid g p \in U } )$ where $\pi_p \colon G \to G / G_p$ is the projection map.
+  For any $U \subset M$ and a given $p \in M$ then $\lambda_p^{-1}\qty( U ) = \pi_p\qty( \qty{ g \in G \mid g p \in U } )$ where $\pi_p \colon G \to G / G_p$ is a projection map.
   Thus $\lambda_p^{-1}\qty( U )$ is an open subset if $U$ is open: the bijection is continuous.
 }
 Such map is a diffeomorphism if $M$ and $G$ are locally compact spaces and $M / G$ is in turn a manifold itself.