Volume 2006, Issue 592

We study the Hilbert scheme of lines on hypersurfaces in the projective space. The main result is that for a smooth Fano hypersurface of degree at most 6 over an algebraically closed field of characteristic zero, the Hilbert scheme of lines has always the expected dimension.

Countable state Markov shifts are a natural generalization of the well-known subshifts of finite type. They are the subject of current research both for their own sake and as models for smooth dynamical systems. In this paper, we investigate their almost isomorphism and entropy conjugacy and obtain a complete classification for the especially important class of strongly positive recurrent Markov shifts. This gives a complete classification up to entropy-conjugacy of the natural extensions of smooth entropy-expanding maps, e.g., C ∞ smooth interval maps with non-zero topological entropy.

Let K be a number field with unit rank r > 1. In this article we show that the inhomogeneous minimum of K is attained by at least one rational point. In particular, if M ( K ) is the Euclidean minimum of K , we have . This phenomenon has consequences on the decidability of the Euclidean nature of such a field. Moreover, in case K is not a CM-field, we prove that is attained, isolated, and that the inhomogeneous minimum function takes discrete rational values.

Given a knot K in a closed orientable manifold M we define the growth rate of the tunnel number of K to be . As our main result we prove that the Heegaard genus of M is strictly less than the Heegaard genus of the knot exterior if and only if the growth rate is less than 1. In particular this shows that a non-trivial knot in S 3 is never asymptotically super additive. The main result gives conditions that imply falsehood of Morimoto's Conjecture.

In this note we give a new construction of Signature homology, and we explain how to associate to any oriented manifold M a characteristic class in Sig * ( M ) which is an integral analog of the L -class. A connection with the Novikov conjecture is explained. Further applications are in the construction of a 2-local characteristic class in the singular cohomology of a topological manifold as well as in the determination of the homotopy type of G/Top .

We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B ∞ -algebra. We construct a universal enveloping functor U 2 from nondifferential B ∞ -algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra ℋ is of the form U 2(Prim ℋ). We take advantage of the description of the free 2 as -algebra in terms of planar trees to unravel the operad associated to nondifferential B ∞ -algebras.

We know by the studies for the last decade that the norm convergence of the exponential product formula holds true even for a class of unbounded operators. As a natural development of this recent research, we study how the product formula approximates integral kernels of Schrödinger semigroups. Our emphasis is placed on the case of singular potentials. The Dirichlet Laplacian is regarded as a special case of Schrödinger operators with singular potentials. We also discuss the approximation to the heat kernel generated by the Dirichlet Laplacian through the product formula.

Let G be the k -rational points of a connected reductive k -group, where k is a p -adic field of characteristic zero. We define a notion of strongly good positive G -datum Σ, and construct a κ-type associated to such a datum, following the methods of Yu's construction of types. Suppose π is an irreducible admissible representation of G of positive depth, containing such a κ-type. Then assuming that the residual characteristic of k is sufficiently large, we prove that the character of π is Γ-asymptotic on a G -domain defined in terms of Σ, where Γ is a semisimple element naturally associated to Σ. We also obtain a domain of validity for the Shalika germ expansion around the element Γ.