Volume 2007, Issue 608

It is shown that the Gromov-Hausdorff limit of a subanalytic 1-parameter family of compact connected sets (endowed with the inner metric) exists. If the family is semialgebraic, then the limit space can be identified with a semialgebraic set over some real closed field. Different notions of tangent cones (pointed Gromov-Hausdorff limits, blow-ups and Alexandrov cones) for a closed connected subanalytic set are studied and shown to be naturally equivalent. It is shown that geodesics have well-defined Euclidean directions at each point.

Second derivative pinching estimates are proved for a class of parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the author. The result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives.

We study certain canonical automorphic forms associated to irreducible characters of N = 1 superconformal minimal models in both the Neveu-Schwarz and Ramond sector by using representation theoretic methods. By extending the techniques from [ A. Milas , Virasoro algebra, Dedekind eta-function and Specialized Macdonald's identities, Transf. Groups 9 (2004), 273–288.] to the setting of N = 1 superconformal algebras we obtain a series of modular identities involving certain Wronskian determinants in both Neveu-Schwarz (untwisted) and Ramond (twisted) sector. Three Weber modular functions play a fundamental role here as well as the Dedekind η-function. In the most interesting case of minimal series, we obtain a derivation and generalizations of several classical modular q -series identities (e.g., Jacobi's Four Square Theorem, a Carlitz' identity, specialized Macdonald's identities for B series, etc.)

We study 3-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the 3-dimensional geometries is given, in particular, non-gradient solitons are found on Nil and Sol.

We prove a q -analogue of the Carter-Payne theorem for the two special cases corresponding to moving an arbitrary number of nodes between adjacent rows, or moving one node between an arbitrary number of rows. As a consequence, we show that these homomorphism spaces are one dimensional when q ≠ −1. We apply these results to complete the classification of the reducible Specht modules for the Hecke algebras of the symmetric groups when q ≠ −1. Our methods can also be used to determine certain other pairs of Specht modules between which there is a homomorphism. In particular, we describe the homomorphism space for an arbitrary partition μ .

Given a global field K and a polynomial defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K -rational pre-periodic points of is bounded in terms of only the degree of K and the degree of . In 1997, for quadratic polynomials over K = ℚ, Call and Goldstine proved a bound which was exponential in s , the number of primes of bad reduction of . By careful analysis of the filled Julia sets at each prime, we present an improved bound on the order of s log s . Our bound applies to polynomials of any degree (at least two) over any global field K .

There are well-known analogues of the prime number theorem and Mertens' Theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections. The proof of the dynamical analogue of Mertens' Theorem uses transcendence theory and Dirichlet characters.

We prove that there are long intervals containing fewer prime numbers than the average for intervals of such length.

We use pluri-potential theory to study the bifurcations of holomorphic families { ƒ λ } λ∈ X of rational maps on or endomorphisms of . To this purpose we establish some formulas for L ( ƒ λ ) and dd c L ( ƒ λ ) where L ( ƒ λ ) is the sum of the Lyapunov exponents of ƒ λ with respect to the maximal entropy measure. We show that the bifurcation current dd c L ( ƒ λ ) both detects the instability of repulsive cycles and the interaction between critical and Julia sets. For families of rational maps of degree d , we introduce a bifurcation measure defined by ( dd c L ( ƒ λ )) 2 d −2 add study its first properties. In particular, we show that the support of this measure is contained in the closure of the set of rational maps having 2 d − 2 distinct Cremer-Cycles. This approach yields to a purely potential-theoretic proof of Mañé-Sad-Sullivan theorem and, moreover, allows us to extend it.