Volume 2008, Issue 624

We find a quartic example of a smooth embedded negatively curved surface in ℝ 3 homeomorphic to a doubly punctured torus. This constitutes an explicit solution to Hadamard's problem of constructing complete surfaces with negative curvature and Euler characteristic in ℝ 3 . Further we show that our solution has the optimal degree of algebraic complexity via a topological classification for smooth cubic surfaces with a negatively curved component in ℝ 3 : any such component must either be topologically a plane or an annulus. In particular we prove that there exists no cubic solutions to Hadamard's problem.

Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let be the first non-negative eigenvalue of the Dirac operator on ( M,g,χ ). We set where the infimum runs over all metrics g of volume 1 in a conformal class [ g 0 ] on M and where the supremum runs over all conformal classes [ g 0 ] on M . Let be obtained from ( M ,χ) by 0-dimensional surgery. We prove that . As a corollary we can calculate τ( M ,χ) for any Riemann surface M .

In the paper we first establish the local well-posedness for a family of nonlinear dispersive equations, the so called b -equation. Then we describe the precise blow-up scenario. Moreover, we prove that for the b -equation we do have the coexistence of global in time solutions and blow-up phenomena: Depending on the initial data solutions may exist for ever, while other data force the solution to produce a singularity in finite time. Finally, we prove the uniqueness and existence of global weak solution to the equation provided the initial data satisfy certain sign conditions.

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk U such that f | ∂ U is a homeomorphism. We show that U is bounded. Using a result of Herman, we deduce that if additionally the rotation number of U is Diophantine, then ∂ U contains a critical point of f . Suppose furthermore that all singular values of f lie in the Julia set. We prove that, if f has a Siegel disk U whose boundary contains no singular values, then the condition that f : ∂ U → ∂ U is a homeomorphism is automatically satisfied. We also investigate landing properties of periodic dynamic rays by similar methods.

Let K be any field, let L n denote the Leavitt algebra of type (1, n – 1) having coefficients in K , and let M d ( L n ) denote the ring of d × d matrices over L n . In our main result, we show that M d ( L n ) ≅ L n if and only if d and n – 1 are coprime. We use this isomorphism to answer a question posed in [ W. Paschke and N. Salinas , Matrix algebras over , Michigan Math. J. 26 (1979), 3–12.] regarding isomorphisms between various C*-algebras. Furthermore, our result demonstrates that data about the K 0 structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple K -algebras.

We denote by spt( n ) the total number of appearances of the smallest part in each integer partition of n . We shall relate spt( n ) to the Atkin-Garvan moments of ranks, and we shall prove that 5 | spt(5 n + 4), 7 | spt(7 n + 5) and 13 | spt(13 n + 6).

In 1878, Jordan [ C. Jordan , Mémoire sur les equations différentielle linéaire à intégrale algébrique, J. reine angew. Math. 84 (1878), 89–215.] showed that a finite subgroup of GL ( n , ℂ) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds [ M. J. Collins , On Jordan's theorem for complex linear groups, J. Group Th. 10 (2007), 411–423.]. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic ℓ. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic ℓ, and we show that generically the bound is similar to that in characteristic 0—being ( n + 1)!, or ( n + 2)! when ℓ divides n + 2—given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees n and every characteristic ℓ.

A classical conjecture predicts how often a polynomial in takes prime values. The natural analogous conjecture for prime values of a polynomial f ( T ) ∈ k [ u ][ T ], where k is a finite field, is false. The conjecture over k [ u ] was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the Möbius function at the values of f on k [ u ] when f ∈ k [ u ][ T p ], where p is the characteristic of k . In this paper, for we extend the Möbius periodicity results for k [ u ]—the affine k -line—to the case when f has coefficients in the coordinate ring A of any higher-genus smooth affine k -curve with one geometric point at infinity. The basic strategy is to pull up results from the genus-0 case by means of well-chosen projections to the affine line. Our techniques can also be used to prove nontrivial properties of a correction factor in the conjecture on primality statistics for values of f ∈ A [ T p ] on A , even as f and k vary.

Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the filling inequalities depend only on dimension and orientability, not on the manifold itself. This contrasts with the analogous situation for the optimal systolic inequality, which does depend on the manifold.