Volume 2016, Issue 718

We compute the Ozsváth–Szabó contact invariants for all tight contact structures on the manifolds -Σ⁢(2,3,6⁢n-1){-\Sigma(2,3,6n-1)} using twisted coefficients and a previous computation by the first author and Ko Honda. This computation completes the classification of the tight contact structures in this family of 3-manifolds.

We construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group N , we construct quasiisometries from N to itself that is not at finite distance from any map that is a composition of left translations and automorphisms. We also construct bi-Lipschitz maps of the Heisenberg groups that send vertical lines to curves that are not vertical lines.

We show that the span of special cycles in the r -th Chow group of a Shimura variety of orthogonal type is finite dimensional, if r<5{r<5}. As our main tool, we develop the theory of Jacobi forms with rational index M∈MatN⁢(ℚ){M\in\mathrm{Mat}_{N}(\mathbb{Q})}.

We prove that the braided Thompson’s groups Vbr{V_{\mathrm{br}}} and Fbr{F_{\mathrm{br}}} are of type F∞{\rm F}_{\infty}, confirming a conjecture by John Meier. The proof involves showing that matching complexes of arcs on surfaces are highly connected. In an appendix, Zaremsky uses these connectivity results to exhibit families of subgroups of the pure braid group that are highly generating, in the sense of Abels and Holz.

Nous étudions l’anneau des fonctions rationnelles qui se prolongent par continuité sur ℝn\mathbb{R}^{n}. Nous établissons plusieurs propriétés algébriques de cet anneau dont un Nullstellensatz fort. Nous étudions les propriétés schématiques associées et montrons une version régulue des théorèmes A et B de Cartan. Nous caractérisons géométriquement les idéaux premiers de cet anneau à travers leurs lieux d’annulation et montrons que les fermés régulus coïncident avec les fermés algébriquement constructibles. We study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstellensatz. We study the scheme theoretic properties and prove regulous versions of Theorems A and B of Cartan. We also give a geometrical characterization of prime ideals of this ring in terms of their zero-locus and relate them to euclidean closed Zariski-constructible sets.

We show that the Dixmier–Douady theory of continuous fields of C*C^{*}-algebras with compact operators 𝕂{\mathbb{K}} as fibers extends significantly to a more general theory of fields with fibers A ⊗\otimes𝕂\mathbb{K} where A is a strongly self-absorbing C*C^{*}-algebra. The classification of the corresponding locally trivial fields involves a generalized cohomology theory which is computable via the Atiyah–Hirzebruch spectral sequence. An important feature of the general theory is the appearance of characteristic classes in higher dimensions. We also give a necessary and sufficient K -theoretical condition for local triviality of these continuous fields over spaces of finite covering dimension.

Using Mellin–Barnes integrals we give a method to compute elements of the monodromy group of an A -hypergeometric system of differential equations. The method works under the assumption that the A -hypergeometric system has a basis of solutions consisting of Mellin–Barnes integrals. Hopefully these elements generate the full monodromy group, but this has only been verified in some special cases.

We study three related topics in representation theory of classical Lie superalgebras. The first one is classification of primitive ideals, i.e. annihilator ideals of simple modules, and inclusions between them. The second topic concerns Arkhipov’s twisting functors on the BGG category 𝒪{\mathcal{O}}. The third topic addresses deformed orbits of the Weyl group. These take over the role of the usual Weyl group orbits for Lie algebras, in the study of primitive ideals and twisting functors for Lie superalgebras.