# Bivariant K-theory in Geometry and Physics

- Weeks one and two will focus on seminars and collaborative research amongst the participants.
**Week three**will take the form of a**masterclass**at the postdoctoral level on Kasparov theory and its applications. There will be a particular focus on the unbounded Kasparov product and on open problems in the theory and the techniques required in the applications. There will be between 12 and 15 lectures directed at this audience.- Week four will be a research
**workshop: November 26 - 30, 2017**

**Objectives**

There is currently a great deal of research activity in bi-variant K-theory and various application areas. The organizers are looking forward to the end of 2018 when they expect substantial new literature to have appeared detailing progress in the topic. A meeting to review progress and to outline future directions will be very timely and this is the objective of the programme.

The recent and ongoing rigorous development of the unbounded model for bi-variant K-theory has paved the way for direct computational applications that were formerly not available. The new methods are being applied in the more developed ones of index theory and gauge theory and also in emerging applications in condensed matter theory and to recent questions about non-commutative approaches to Lorentzian spaces and manifolds with boundary. In addition to the homological (index-theoretic) content, more refined spectral information may also be extracted. The use of these new methods thus surpasses index theory per se, and gives access to finer analytic invariants.

The applications would be of interest to mathematical physicists working in condensed matter theory, string theory and gauge theory. All of these fields have seen huge influence from classical K-theory. However, quantum theory is non-commutative and it is only very recently that the added power of fully non-commutative methods in the form of bi-variant K-theory (often referred to as Kasparov or KK-theory) became available. We are especially concerned with the constructive form of the theory, which adds a new computational dimension to K-theory. In addition, some of the refinements that make the bi-variant theory attractive to mathematical physics are explicit refinements to the geometric and computational parts of the theory.