Title: Noncommutative gauge theories, deformation quantization and formality. (2 lectures)

Abstract: An elementary introduction to noncommutative gauge theory of the type that arises in string theory with background B-field is given. We discuss the mathematics of gauge fields from the point of view of Kontsevich's deformation quantization and the related notions of a noncommutative line bundle and of a noncommutative gerbe.

Speaker: Brano Jurco

Title: Nonabelian bundle gerbes

Abstract: Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential. As an application the anomaly of M5-branes is discussed.

Speaker: F.A.Sukochev

Title: Dixmier traces and their applications. (2 lectures)

Abstract: A Dixmier trace is a non-normal singular trace vanishing on all finite rank operators. It was discovered by J.Dixmier in 1966 as an example of a non-normal trace on the algebra B(H) of all bounded linear operators on an infinite dimensional Hilbert space. Later, Alain Connes discovered that although Dixmier trace is not normal, nevertheless it has many important applications to non-commutative geometry. In fact, a Dixmier trace considered on the algebra of pseudodifferential operators on a compact smooth manifold can be viewed as a non-commutative integral in the sense of differential geometry, whereas the ordinary trace on B(H) is a non-commutative integral in the sense of measure theory. Furthermore, Dixmier traces are strongly related to the so-called Wodzicki residue of pseudo-differential operators, and to the Chern-Connes character.

Speaker: Hisham Sati

Title: The M-theory partition function and topology.

Abstract: Witten has shown that the topological part of the M-theory partition function is encoded in an index of an E8 bundle in eleven dimensions. Diaconescu-Moore-Witten related this to the K-theoretic partition function of type IIA string theory obtained via dimensional reduction, and later Mathai and I generalized part of the construction to twisted K-theory. In this talk, after reviewing the above, I report on my recent work with Igor Kriz on the appearance of elliptic cohomology in this context. I will try to focus on the general idea rather than on the technical construction.