  In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an
undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs
$(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$
pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices
$i\in\{1,\ldots,k\}$. We present a $2^{O(k^2)}n$-time algorithm for the
\textsf{Planar Disjoint Paths} problem. This improves the two previously
best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied
Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].
