We present the first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph---which undergoes both edge insertions and deletions---in polylogarithmic time. Our algorithm is randomized and, per update, takes $O(log^2 Delta cdot log^2 n)$ expected time. Furthermore, the algorithm can be adjusted to have $O(log^2 Delta cdot log^4 n)$ worst-case update-time with high probability. Here, $n$ denotes the number of vertices and $Delta$ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC18] who presented an algorithm with $O(m^{3/4})$ update-time (and thus broke the natural $Omega(m)$ barrier) where $m$ denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had $widetilde{O}(min{sqrt{n}, m^{1/3}})$ update-time [Assadi et al. SODA19]. Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time.
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