Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

An explicit model of default time with given survival probability

For a given filtered probability space (Ω,F,P), an F-adapted continuous increasing process Λ and a positive P-F local martingale N such that Λ₀=0 and N_te^−Λ_t≤1, we construct a probability measure Q^Z and a random time τ such that Q|_F∞=P|_F∞ and Q[τ>t|F_t]=Z_t. The probability Q^Z is linked with the well-known Cox model by an explicit density function. Various properties exist, which characterize Q^Z from others. Let G=(G_t)_t≥0 with G_t=F_t∨σ({τ≤s}:s≤t). We establish the (H^′)-property between the filtrations F and G, and we provide the enlargement of filtration formula.     

Discretization of backward semilinear stochastic evolution equations

The study on discretization and convergence of BSDEs rapidly developed in recent years. We especially mention the work of Ph. Briand, B. Delyon and J. Mémin [Donsker-type Theorem for BSDEs, Electron. Comm. Probab. 6 (2001) 1–14 (electronic)]. They got the convergence of the sequence Yⁿ$Y^n$ and pointed out that the weak convergence of filtrations was a powerful tool in this topic. In this paper, we first study the weak convergence of filtrations in Hilbert space. Using this tool, we get the convergence about discretization of backward semilinear stochastic evolution equations (BSSEEs for short).     

Some filtrations on higherK-theory and related invariants

We lift Bloch's higher Chow construction from the level of simplicial sets to the level of simplicial spaces. We construct a simplicial space that becomes isomorphic to the Bloch/Chow complex when the functor ₀ is applied in each degree. The homotopy groups of this space are theE ₂-terms in an Atiyah-Hirzebruch spectral sequence converging to algebraicK-theory. TheseE ₂-terms map nontrivially to the expected higher Chow groups. We define and compute several intermediate invariants associated to our simplicial space.     

Some examples and counterexamples of convergence of σ-algebras and filtrations

We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of σ-algebras to filtrations generated by different discretizations of a single process. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 295–306, July–September, 2000. Translated by V. Mackevičius     

模李超代数研究的若干进展

近几年，模李超代数（即素特征域上的李超代数）的研究取得了一些进展。本文分析以下6个方面介绍近几年在模李超代数研究中取得的一些新结果。1，限制李超代数，2，Cartan型模李超代数；3，深度1的Z－阶化李超代数；4，深度1的滤过李超代数。5；Cartan型模李超代数的某些内蕴性质；6，公开的问题。     

Shortening filtrations

Let C be a set of modules.We argue that there is an ordinal κ such that if a module has a filtration by modules in C,then it has a filtration of length κ by direct sums of modules in C.As an application we give another way to prove a result of Saorín and tovíek and of tovíek.     

与过程相联系的σ-代数流

讨化了σ-代数流的结构,并给出其一般性的结果;归纳出各σ-代数流的相互关系;最后介绍其两个简单应用.