We show that every continuous action of a finite group on a smooth three-manifold is a uniform limit of smooth actions.

We completely determine which simply connected rational homology 5-spheres admit Sasaki–Einstein metrics.

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation

$${\mathit{u}}_{\mathit{t}}-\mathrm{\Delta }\mathit{u}={\mathit{u}}^{\mathit{p}}\text{in}{\mathbb{R}}^{\mathit{n}}\times \mathbb{R},\mathit{p}\mathrm{>}1,$$

the nonexistence of positive classical solutions in the subcritical range $\mathit{p}(\mathit{n}-2)\mathrm{<}\mathit{n}+2$ has been conjectured for a long time, but all known results require either a more restrictive assumption on p or deal with a special class of solutions (time-independent or radially symmetric or satisfying suitable decay conditions). We solve this open problem and—by using the same arguments—we also prove optimal Liouville theorems for a class of superlinear parabolic systems. In the case of the nonlinear heat equation, straightforward applications of our Liouville theorem solve several related long-standing problems. For example, they guarantee an optimal Liouville theorem for ancient solutions, optimal decay estimates for global solutions of the corresponding Cauchy problem, optimal blowup rate estimate for solutions in nonconvex domains, and optimal universal estimates for solutions of the corresponding initial-boundary value problems. The proof of our main result is based on refined energy estimates for suitably rescaled solutions.

We prove a general theorem for constructing integral quantum cluster algebras over $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$, namely, that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that, for every symmetrizable Kac–Moody algebra $\mathfrak{g}$ and Weyl group element w, the dual canonical form ${\mathit{A}}_{\mathit{q}}{({\mathfrak{n}}_{+}(\mathit{w}))}_{\mathbb{Z}[{\mathit{q}}^{\pm 1}]}$ of the corresponding quantum unipotent cell has the property that ${\mathit{A}}_{\mathit{q}}{({\mathfrak{n}}_{+}(\mathit{w}))}_{\mathbb{Z}[{\mathit{q}}^{\pm 1}]}{\otimes }_{\mathbb{Z}[{\mathit{q}}^{\pm 1}]}\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$ is isomorphic to a quantum cluster algebra over $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$ and to the corresponding upper quantum cluster algebra over $\mathbb{Z}[{\mathit{q}}^{\pm 1/2}]$.

Our main theme here is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension $2\mathit{n}+1$ any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least $\mathit{n}+1$ simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension 5 or greater, we construct examples of antipodally symmetric, dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has nonempty ${\mathit{C}}^1$-interior.