Delocalized Chern character for stringy orbifold K-theory

In this paper, we define a stringy product on K^*_{orb}(\XX) \otimes \C , the orbifold K-theory of any almost complex presentable orbifold \XX. We establish that under this stringy product, the de-locaized Chern character ch_{deloc} : K^*_{orb}(\XX) \otimes \C \longrightarrow H^*_{CR}(\XX), after a canonical modification, is a ring isomorphism. Here H^*_{CR}(\XX) is the Chen-Ruan cohomology of \XX. The proof relies on an intrinsic description of the obstruction bundles in the construction of Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K-theory $K^*_G(G)$ of a finite group $G$ with the conjugation action. It turns out that the stringy product is different from the Pontryajin product (the latter is also called the fusion product in string theory).Comment: 34 pages. Final version to appear in Trans. of AMS. Improve the expositions and Change of the title thanks the referee

A KP-mKP Hierarchy via Operators with Two Derivations

By using pseudo-differential operators containing two derivations, we extend the Kadomtsev-Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we obtain its B\"{a}cklund transformations, bilinear equations of Baker-Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked