Volume 2013, Issue 683

Abstract. We present a simple unified proof of the classification of discrete amenable group actions on injective factors. Our argument does not depend on the types of factors, and is based on the technique of Evans and Kishimoto.

Abstract. Upper bounds for the eigenvalues of the Laplace–Beltrami operator on a hypersurface bounding a domain in some ambient Riemannian manifold are given in terms of the isoperimetric ratio of the domain. These results are applied to the extrinsic geometry of isometric embeddings.

Abstract. On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. The two identities are intimately connected with the classical circle and divisor problems , respectively. For each of Ramanujan's identities, there are three possible interpretations for the double series. In two earlier papers, the authors proved the two identities under each of two possible interpretations. Weighted (or twisted) divisor sums are central to the proofs. The ideas that the authors used in the second paper are extended here to derive analogous Bessel series identities for finite sums of products of two trigonometric (sine–sine, cosine–cosine, sine–cosine) functions.

Abstract. The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold M with boundary . We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on M . We propose a new boundary condition for the flow and prove two short-time existence results.

Abstract. Let R be a normal, equi-codimensional Cohen–Macaulay ring of dimension with a canonical module . We give a sufficient criterion that establishes a derived equivalence between the noncommutative crepant resolutions of R . When , this criterion is always satisfied and so all noncommutative crepant resolutions of R are derived equivalent. Our method is based on cluster tilting theory for commutative algebras, developed by Iyama and Wemyss (2010).

Abstract. An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As a consequence, we find many new examples of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type. We apply our conceptual approach to derive explicit classifications of cohomogeneity one actions on some symmetric spaces.

Abstract. The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on “exotic” 4-spheres (if they exist) and the Gromoll–Meyer sphere. In particular, as one of the main results we prove: there exists no properly transnormal function on any exotic 4-sphere. Furthermore, by projecting an S 3 -invariant isoparametric function on , we construct a properly transnormal but not an isoparametric function on the Gromoll–Meyer sphere with two points as the focal varieties.

Abstract. The moduli spaces of trigonal curves of odd genus are proven to be rational.

Abstract. We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by Andruskiewitsch and Schneider states that any finite-dimensional pointed Hopf algebra over an algebraically closed field of characteristic zero is generated as an algebra by its group-like and skew-primitive elements. As an application of our main result, we prove the conjecture when the group of group-like elements is abelian.