A noncommutative sigma-model
We begin to study a sigma-model in which both the space-time manifold and the two-dimensional string world-sheet are made noncommutative. The most precise results apply to the case where both the space-time manifold and the two-dimensional string world-sheet are replaced by noncommutative tori. In this situation, we are able to determine when maps between such noncommutative tori exist, to derive the Euler-Lagrange equations, to classify many of the critical points of the Lagrangian, and to study the associated partition function.

title: D-branes, KK-theory and duality on noncommutative spaces
abstract: We present a new categorical classification framework for D-brane charges on noncommutative manifolds using methods of bivariant K-theory. We describe several applications including an explicit formula for D-brane charge in cyclic homology, a refinement of open string T-duality, and a general criterion for cancellation of global worldsheet anomalies.

title: Dualities in Field Theories and the Role of K-Theory
abstract: It is now known (or in some cases just believed) that many quantum field theories exhibit dualities, equivalences with the same or a different theory in which things appear very different, but the overall physical implications are the same. We will discuss some of these dualities from the point of view of a mathematician, focusing on "charge conservation" and the role played by K-theory and noncommutative geometry. Some of the work described here is joint with Mathai Varghese and Stefan Mendez-Diez; the last section is new.

title: On mysteriously missing T-duals, H-flux and the T-duality group
abstract: A general formula for the topology and H-flux of the T-duals of type II string theories with H-flux on toroidal compactifications is presented here. It is known that toroidal compactifications with H-flux do not necessarily have T-duals which are themselves toroidal compactifications. A big puzzle has been to explain these mysterious ``missing T-duals'', and our paper presents a solution to this problem using noncommutative topology. We also analyze the T-duality group and its action, and illustrate these concepts with examples.

title: K-theoretic matching of brane charges in S- and U-duality
abstract: We discuss K-theoretic matching of D-brane charges in the string duality between type I on T 4 and type IIA on K3. This case is more complex than the familiar case of IIA/IIB duality, which is already well understood, but it turns out that replacing K3 by its orbifold blow-down seems largely to resolve the apparent problems with the theory.

title: Noncommutative correspondences, duality and D-branes in bivariant K-theory
abstract: We describe a categorical framework for the classification of D-branes on noncommutative spaces using techniques from bivariant K-theory of C*-algebras. We present a new description of bivariant K-theory in terms of noncommutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper. We explicitly work out our theory for a number of examples of noncommutative manifolds.

title: Design of the Chicago Air Shower Array