Given a 4-manifold, we build a non-empty {C}^{1} ${C}^{1}$-open set of vector fields having a (chain transitive) attractor containing singularities of different indices. Then, we begin the study of the hyperbolic properties of such a robust singular attractor.

In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant K$K$-theory with respect to a compact torus G$G$ of various spaces associated to a linear action of G$G$ in a vector space M$M$ can both be described using some vector spaces of distributions, on the dual of the group G$G$ or on the dual of its Lie algebra \mathfrak{g}$\mathfrak{g}$. The morphism from K$K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a G$G$-transversally elliptic operator on M$M$ are determined using the infinitesimal index of the symbol.

We prove that the Brauer–Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic p$p$. More precisely, we show that the only obstructions come from étale covers of exponent p$p$ or, alternatively, from flat covers coming from torsors under connected group schemes of exponent p$p$.

We prove a new global stability estimate for the Gel’fand–Calderón inverse problem on a two-dimensional bounded domain. Specifically, the inverse boundary value problem for the equation {- }\Delta \psi + v\hspace{0.167em} \psi = 0${- }\Delta \psi + v\hspace{0.167em} \psi = 0$ on D$D$ is analysed, where v$v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain D$D$. The main feature of this estimate is that it shows that the smoother a potential is, the more stable its reconstruction is. Furthermore, the stability is proven to depend exponentially on the smoothness, in a sense to be made precise. The same techniques yield a similar estimate for the Calderón problem for electrical impedance tomography.

In our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces X$X$ attached to real orthogonal groups of type (p, q)$(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in X$X$ with coefficients.

In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification \overline{X} $\overline{X}$ of X$X$. However, for the \mathbb{Q} $\mathbb{Q}$-split case for signature (p, p)$(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of \overline{X} $\overline{X}$ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.

As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of X$X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for {L}^{2} ${L}^{2}$-cohomology.

We show improved local energy decay for the wave equation on asymptotically Euclidean manifolds in odd dimensions in the short range case. The precise decay rate depends on the decay of the metric towards the Euclidean metric. We also give estimates of powers of the resolvent of the wave propagator between weighted spaces.

We describe a probability distribution on isomorphism classes of principally quasi-polarized p$p$-divisible groups over a finite field k$k$ of characteristic p$p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics (p$p$-corank, a$a$-number, etc.) of p$p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the p$p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over k$k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for \text{char~} k\not = p$\text{char~} k\not = p$, in which case the random p$p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over {\mathbf{F} }_{3} ${\mathbf{F} }_{3}$ appear substantially less likely to be ordinary than hyperelliptic curves over {\mathbf{F} }_{3} ${\mathbf{F} }_{3}$.