memoire-m2

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -563,8 +563,24 @@ respect to which the matrices of \(\rho(\tau_{\gamma_1}), \ldots,
 \(H_1(\Sigma_g, \mathbb{C})\), concluding the classification of
 \(2g\)-dimensional representations.
 
-% TODO: Add some final comments about how the rest of the landscape of
-% representations is generally unknown and how there is a lot to study in here
 Recently, Kasahara \cite{kasahara} also classified the \((2g+1)\)-dimensional
 representations of \(\Mod(\Sigma_g^b)\) for \(g \ge 7\) in terms of certain
-twisted \(1\)-cohomology groups.
+twisted \(1\)-cohomology groups. On the other hand, the representations of
+dimension \(n > 2g + 1\) are still poorly understood, and fundamental questions
+remain unsweared. In the short and mid-terms, the works of Korkmaz and Kasahara
+lead to many follow-up questions. For example,
+\begin{enumerate}
+  \item In the \(g \ge 3\) case, Korkmaz \cite[Theorem~3]{korkmaz} established
+    the lower bound of \(3g - 3\) for the dimension of an injective linear
+    representation of \(\Mod(\Sigma_g)\) -- if one such representation exists.
+    Can we improve this lower bound?
+
+  \item What is the minimal dimension for a representation of
+    \(\Mod(\Sigma_g)\) which does not annihilate the entire kernel of the
+    symplectic representation \(\psi : \Mod(\Sigma_g) \to
+    \operatorname{Sp}_{2g}(\mathbb{Z})\)? In particular, do the \((2g +
+    1)\)-dimensional representations classified by Kasahara \cite{kasahara}
+    annihilate all of \(\ker \psi\)?
+\end{enumerate}
+
+These are some of the questions which I plan to work on during my upcoming PhD.