We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology over fields of characteristic zero. The key ingredient is the construction of a degree-one Hochschild cohomology class on a Floer $A_{\infty }$-algebra associated to the $(k,k)$-nilpotent slice ${\mathcal{Y}}_k$ obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification ${\overline{\mathcal{Y}}}_k$. The space ${\overline{\mathcal{Y}}}_k$ is obtained as the Hilbert scheme of a partial compactification of the $A_{2k-1}$-Milnor fiber. A sequel to this paper will prove formality of the symplectic cup and cap bimodules and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.

We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov operator associated to the rotation component of the isometries has spectral gap. We also prove that certain self-similar measures are absolutely continuous with smooth densities. These families of self-similar measures give higher-dimensional analogues of Bernoulli convolutions on which absolute continuity can be established for contraction ratios in an open set.

We construct time quasiperiodic solutions to the energy supercritical nonlinear Schrödinger equations on the torus in arbitrary dimensions. This introduces a new approach, which could have general applicability.