We compute quantum cohomology rings of elliptic ${\mathbb{P}}^1$ orbifolds via orbicurve counting. The main technique is the classification theorem which relates holomorphic orbicurves with certain orbifold coverings. The countings of orbicurves are related to the integer solutions of Diophantine equations. This reproduces the computation of Satake and Takahashi in the case of ${\mathbb{P}}_{3,3,3}^1$ via a different method.

We construct spherically symmetric solutions to the Einstein–Euler equations, which give models of gaseous stars in the framework of the general theory of relativity. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein–Euler equations are given by the Tolman–Oppenheimer–Volkoff equations, and time-periodic solutions around the equilibrium of the linearized equations can be considered. Our aim is to find true solutions near these time-periodic approximations. Solutions satisfying a so-called physical boundary condition at the free boundary with the vacuum will be constructed using the Nash–Moser theorem. This work also can be considered as a touchstone in order to estimate the universality of the method which was originally developed for the nonrelativistic Euler–Poisson equations.

We consider equivariant solutions for the Schrödinger map problem from ${\mathbb{R}}^{2+1}$ to ${\mathbb{H}}^2$ with finite energy and show that they are global in time and scatter.

In this article, we study the rigidity properties of deformation parameters of the natural action of a discontinuous subgroup $\Gamma \subset G$, on a homogeneous space $G/H$, where $H$ stands for a closed subgroup of a Euclidean motion group $G:={\mathrm{O}}_n\left(\mathbb{R}\right)⋉{\mathbb{R}}^n$. That is, we prove the following local (and global) rigidity theorem: the parameter space admits a rigid (equivalently a locally rigid) point if and only if $\Gamma$ is finite. Remarkably, it turns out that $H$ is compact whenever $\Gamma$ is infinite, which makes accessible the study of the corresponding parameter and deformation spaces and their topological and local geometrical features. This shows that the Calabi–Markus phenomenon occurs in this setting. That is, if $H$ is a closed noncompact subgroup of $G$, then $G/H$ does not admit a compact Clifford–Klein form, unless $G/H$ itself is compact. We also answer a question posed by T. Kobayashi. That is, no homogeneous space $G/H$ admits a noncommutative free group as a discontinuous group.

Twisted spectral triples are a twisting of the notion of spectral triples aimed at dealing with some type III geometric situations. In the first part of the article, we give a geometric construction of the index map of a twisted spectral triple in terms of $\sigma$-connections on finitely generated projective modules. This clarifies the analogy with the indices of Dirac operators with coefficients in vector bundles. In the second part, we give a direct construction of the Connes–Chern character of a twisted spectral triple, in both the invertible and the noninvertible cases. Combining these two parts we obtain an analogue of the Atiyah–Singer index formula for twisted spectral triples.

We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors $\mathcal{O}{\text{'}}_C^{*}\left({\mathbb{R}}^d\right)$ for tempered ultradistributions are analyzed via the duality with respect to the test function spaces ${\mathcal{O}}_C^{*}\left({\mathbb{R}}^d\right)$ introduced in this article. We also study ultradistribution spaces associated to translation-invariant Banach spaces of tempered ultradistributions and use their properties to provide a full characterization of the general convolution of Roumieu ultradistributions via the space of integrable ultradistributions. We show that the convolution of two Roumieu ultradistributions $T,S\in \mathcal{D}{\text{'}}^{\left\{M_p\right\}}\left({\mathbb{R}}^d\right)$ exists if and only if $\left(\phi \check{S}\right)T\in \mathcal{D}{\text{'}}_{L^1}^{\left\{M_p\right\}}\left({\mathbb{R}}^d\right)$ for every $\phi \in {\mathcal{D}}^{\left\{M_p\right\}}\left({\mathbb{R}}^d\right)$.

The purpose of this article is to give explicit descriptions for stability groups of real rigid hypersurfaces of infinite type in ${\mathbb{C}}^2$. The decompositions of infinitesimal CR automorphisms are also given.