Antony feedback: do not use 's

main.tex

@@ -144,7 +144,7 @@ As opposed to only eliminating possible values of $\chern_0(E)$ for which all
 corresponding $\chern_1^{\beta_{-}}(E)$ fail one of the inequalities (which is what
 was implicitly happening before).
 
-First, let's fix a Chern character for $F$,
+First, let us fix a Chern character for $F$,
 $\chern(F) = (R,C,D)$, and consider the possible Chern characters
 $\chern(E) = (r,c,d)$ of some semistabilizer $E$.