Deformation quantization in algebraic geometry

We study deformation quantizations of the structure sheaf O_X of a smooth algebraic variety X in characteristic 0. Our main result is that when X is D-affine, any formal Poisson structure on X determines a deformation quantization of O_X (canonically, up to gauge equivalence). This is an algebro-geometric analogue of Kontsevich's celebrated result.     

Average frobenius distributions for elliptic curves with 3-torsion

In this paper, we examine the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. We prove that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.     

On the dynamics of phase transitions

The dynamics of phase transitions is described in ordinary thermodynamics i.e. the bodies are considered to be homogeneous. Stability properties of phase transitions are investigated by Lyapunov's method.     

When is the union of two unit intervals a self-similar set satisfying the open set condition?

Let U _λ be the union of two unit intervals with gap λ. We show that U _λ is a self-similar set satisfying the open set condition if and only if U _λ can tile an interval by finitely many of its affine copies (admitting different dilations). Furthermore, each such λ can be characterized as the spectrum of an irreducible double word which represents a tiling pattern. Some further considerations of the set of all such λ’s, as well as the corresponding tiling patterns, are given. The first author was partially supported by the RGC grant and the direct grant in CUHK, Fok Ying Tong Education Foundation and NSFC (10571100). The second author was partially supported by NSFC (70371074) and NFSC (10571104).     

On The Span Of A Random Channel Assignment Problem

In the radio channel assignment problems considered here, we must assign a ‘channel’ from the set 1,2,... of positive integers to each of n transmitters, and we wish to minimise the span of channels used, subject to the assignment leading to an acceptable level of interference. A standard form of this problem is the ‘constraint matrix’ model. The simplest case of this model (the 0, 1 case) is essentially graph colouring. We consider here a random model for the next simplest case (with lengths 0, 1 or 2), and determine the asymptotic behaviour of the span of channels needed as n→∞. We find that there is a ‘phase change’ in this behaviour, depending on the probabilities for the different lengths.     

Non-simply connected copes

Let (W⁴,?W⁴) be a 4-manifold. Let f₁,f₂,…,f_k:(D²,?D²)→ (W⁴,?W⁴) be transverse immersions that have spherical duals $\text{α}{}_1\text{,α}{}_2\text{,…,α}{}_{\mathrm{k}}\text{:S}{}^2\text{→}\text{W}\text{?}$. Then there are open disjoint subsets V₁, V₂,…,V_k of W, such that for each 1?i?k, (a) ?V_i=V₁∩?W and ?V_i is an open regular neighborhood of f_i(?D²) in ?W, and (b) (V_i,?V_i,f_i(?D²)) is proper homotopy equivalent to (M, ?M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.     

Edge Sobolev spaces and weakly hyperbolic equations

Summary. Edge Sobolev spaces are proposed as a main new tool for the investigation of weakly hyperbolic equations. The well-posedness of the linear and semilinear Cauchy problem in the class of these edge Sobolev spaces is proved. An application to the propagation of singularities for solutions to the semilinear problem is considered. Received: October 3, 2000 Published online: December 19, 2001     

Coleman integration using the Tannakian formalism

We use a new idea to construct a theory of iterated Coleman functions on overconvergent spaces with good reduction in any dimension. A Coleman function in this theory consists of a unipotent differential equation, a functional on the underlying bundle and a solution to the equation on a residue class. The new idea is to use the theory of Tannakian categories and the action of Frobenius to analytically continue solutions of the differential equation to all residue classes. Received: 22 November 2000 / Revised version: 28 March 2001 / Published online: 24 September 2001