Volume 2010, Issue 641

For the q -deformation G q , 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G . Our quantum Dirac operator D q is a unitary twist of D considered as an element of U 𝔤 ⊗ Cl(𝔤). The commutator of D q with a regular function on G q consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D . We show that in the case of the Drinfeld associator the latter commutator is also bounded.

For each positive integer Q , let ℱ Q denote the Farey sequence of order Q . We prove the existence of the pair correlation measure associated to the sum ℱ Q + ℱ Q modulo 1, as Q tends to infinity, and compute the corresponding limiting pair correlation function.

In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a theorem due to De Lellis and Müller.

Let X be an infinite compact metric space with finite covering dimension and let h : X → X be a minimal homeomorphism. We show that the associated crossed product C*-algebra A = C *(ℤ, X , h ) has tracial rank zero whenever the image of K 0 ( A ) in Aff( T ( A )) is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the classification result for nuclear simple C*-algebras with tracial rank zero, we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered K-theory.

It is shown that every norm-closed face of the closed unit ball A 1 in a JB*-triple A is norm-semi-exposed, thereby completing the description of the facial structure of A 1 .

In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using characteristic p methods. When X is a normal Gorenstein closed subvariety of a smooth complex variety A , we formulate a restriction property of the adjoint ideal sheaf adj X ( A ) of A along X involving the l.c.i. ideal sheaf 𝒟 X of X . The proof relies on a modification of generalized test ideals of Hara and Yoshida [Trans. Amer. Math. Soc. 355: 3143–3174, 2003].

Given a minimal surface S equipped with a generically finite map to an Abelian variety and C ⊂ S a rational or an elliptic curve, we show that the canonical degree of C is bounded by four times the self-intersection of the canonical divisor of S . As a corollary, we obtain the finiteness of rational and elliptic curves with an optimal uniform bound on their canonical degrees on any surface of general type with two linearly independent regular one forms.

This article is concerned with the p -basic set existence problem in the representation theory of finite groups. We show that, for any odd prime p , the alternating group 𝔄 n has a p -basic set. More precisely, we prove that the symmetric group 𝔖 n has a p -basic set with some additional properties, allowing us to deduce a p -basic set for 𝔄 n . Our main tool is the concept of generalized perfect isometries introduced by Külshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition numbers of 𝔄 n .

We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length .