We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O’Hara in 1991. This class contains as a special case the Möbius energy.

For the Möbius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approach is crucially based on the invariance of the Möbius energy under Möbius transforms, which fails for all the other O’Hara energies.

We overcome this difficulty by re-interpreting the scale invariant O’Hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot parametrization. This allows us to draw a connection to the theory of (fractional) harmonic maps into spheres. Using this connection we are able to adapt the regularity theory for degenerate fractional harmonic maps in the critical dimension to prove regularity for minimizers and critical knots of the scale-invariant O’Hara knot energies.

A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $l \geq 3$ of their Levi–Tanaka algebra are rigid, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the complex tangent space. For the length $l = 3$, Beloshapka’s Conjecture was proved by Gammel and Kossovskiy in 2006. In this paper, we prove the Conjecture for arbitrary length $l \geq 3$.

As another application of our method, we construct polynomial models of length $l \geq 3$, which are not totally nondegenerate and admit large groups of point preserving nonlinear automorphisms.

We define a refined Gromov–Witten disk potential of monotone immersed Lagrangian surfaces in a symplectic 4‑manifold that are self-transverse as an element in a capped version of the Chekanov–Eliashberg dg‑algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point.

We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5‑sphere.

In an earlier paper we developed the classification of weakly symmetric pseudo–Riemannian manifolds $G/H$, where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans-Lorentzian signature $(n-2,2)$. Here we work out the classification of weakly symmetric pseudo-Riemannian nilmanifolds $G/H$ from the classification for the case $G=N \rtimes H$ with $H$ compact and $N$ nilpotent. It turns out that there is a plethora of new examples that merit further study. Starting with that Riemannian case, we see just when a given involutive automorphism of $H$ extends to an involutive automorphism of $G$, and we show that any two such extensions result in isometric pseudo-Riemannian nilmanifolds. The results are tabulated in the last two sections of the paper.