This paper gives an extension of the classical Zariski–van Kampen theorem describing the fundamental groups of the complements of plane singular curves by generators and relations. It provides a procedure for computation of the first non-trivial higher-homotopy groups of the complements of singular projective hypersurfaces in terms of the homotopy variation operators introduced here.

AMS 2000 Mathematics subject classification: Primary 14F35; 14D05; 32S20; 32S50. Secondary 14J70; 32S25; 55P99

We establish a transfer of (twisted) orbital integrals in the context of twisted endoscopy between a real reductive algebraic group $G$ and a reductive quasi-split real group $H_1$ associated to an endoscopic datum.

AMS 2000 Mathematics subject classification: Primary 22E45; 11F70

We consider a transitive uniformly quasi-conformal Anosov diffeomorphism $f$ of a compact manifold $\mathcal{M}$. We prove that if the stable and unstable distributions have dimensions greater than two, then $f$ is $C^\infty$ conjugate to an affine Anosov automorphism of a finite factor of a torus. If the dimensions are at least two, the same conclusion holds under the additional assumption that $\mathcal{M}$ is an infranilmanifold. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and a small perturbation.

AMS 2000 Mathematics subject classification: Primary 37C; 37D

For a variety over a local field, we show that the alternating sum of the traces of the composition of the actions of an element of the Weil group and an algebraic correspondence on the $\ell$-adic etale cohomology is independent of $\ell$. We prove the independence by establishing basic properties of weight spectral sequences.

AMS 2000 Mathematics subject classification: Primary 14G20. Secondary 11G25