We give a construction of finitely generated amenable groups that do not admit any coarse $1$-Lipschitz embedding with positive compression exponent into $L_p$ for any $1\le p<\infty$, including some that are four-step solvable, answering positively a question posed by Arzhantseva, Guba, and Sapir.

We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasi-equivalence between the category of matrix factorizations and the differential graded (dg) derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toën's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry based on dg categories.

The classical principle of least action says that orbits of mechanical systems extremize action; an important subclass are those orbits that minimize action. In this paper we utilize this principle along with Aubry-Mather theory to construct (Birkhoff) regions of instability for a certain three-body problem, given by a Hamiltonian system of $2$ degrees of freedom. We believe that these methods can be applied to construct instability regions for a variety of Hamiltonian systems with $2$ degrees of freedom. The Hamiltonian model we consider describes dynamics of a Sun-Jupiter-comet system, and under some simplifying assumptions, we show the existence of instabilities for the orbit of the comet. In particular, we show that a comet which starts close to an orbit in the shape of an ellipse of eccentricity $e=0.66$ can increase in eccentricity up to $e=0.96$. In the sequels to this paper, we extend the result to beyond $e=1$ and show the existence of ejection orbits. Such orbits are initially well within the range of our solar system. This might give an indication of why most objects rotating around the Sun in our solar system have relatively low eccentricity.

A refined trilinear Strichartz estimate for solutions to the Schrödinger equation on the flat rational torus ${\mathbb{T}}^3$ is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic nonlinear Schrödinger equation in $H^s({\mathbb{T}}^3)$ for all $s\ge 1$. This is the first energy-critical global well-posedness result in the setting of compact manifolds.