# Volume 11 (2007) No. 2

## Volume 11 (2007) No. 2

### Hodge structures in non-commutative geometry

Maxim Kontsevich

**Abstract:**

XI Solomon Lefschetz Memorial Lecture Series, Department of Mathematics, CINVESTAV, 2005. Notes by Ernesto Lupercio.

Traditionally, Hodge structures are associated with complex projective varieties. In my expository lectures I discussed a non-commutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry.

### A stringy product on twisted orbifold $K$-theory

Alejandro Adem, Yongbin Ruan, and Bin Zhang

**Abstract:**

In this paper we define an associative stringy product for the twisted orbifold $K$-theory of a compact, almost complex orbifold $\mathcal{X}$. This product is defined on the twisted $K$-theory $K_{orb}(\wedge \mathcal{X})$ of the inertia orbifold $\wedge \mathcal{X}$, where the twisting gerbe is assumed to be in the image of the inverse transgression $H^4(B \mathcal{X}, \mathbb{Z}) \to H^3 (B \wedge \mathcal{X}, \mathbb{Z})$.