  The dynamic shortest paths problem on planar graphs asks us to preprocess a
planar graph $G$ such that we may support insertions and deletions of edges in
$G$ as well as distance queries between any two nodes $u,v$ subject to the
constraint that the graph remains planar at all times. This problem has been
extensively studied in both the theory and experimental communities over the
past decades and gets solved millions of times every day by companies like
Google, Microsoft, and Uber. The best known algorithm performs queries and
updates in $\tilde{O}(n^{2/3})$ time, based on ideas of a seminal paper by
Fakcharoenphol and Rao [FOCS'01]. A $(1+\varepsilon)$-approximation algorithm
of Abraham et al. [STOC'12] performs updates and queries in
$\tilde{O}(\sqrt{n})$ time. An algorithm with $O(polylog(n))$ runtime would be
a major breakthrough. However, such runtimes are only known for a
$(1+\varepsilon)$-approximation in a model where only restricted weight updates
are allowed due to Abraham et al. [SODA'16], or for easier problems like
connectivity.
  In this paper, we follow a recent and very active line of work on showing
lower bounds for polynomial time problems based on popular conjectures,
obtaining the first such results for natural problems in planar graphs. Such
results were previously out of reach due to the highly non-planar nature of
known reductions and the impossibility of "planarizing gadgets". We introduce a
new framework which is inspired by techniques from the literatures on distance
labelling schemes and on parameterized complexity.
  Using our framework, we show that no algorithm for dynamic shortest paths or
maximum weight bipartite matching in planar graphs can support both updates and
queries in amortized $O(n^{\frac{1}{2}-\varepsilon})$ time, for
$\varepsilon>0$, unless the classical APSP problem can be solved in truly
subcubic time, [...]
