• 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 conference: November 26 - 30, 2018

Seminars in week 1:  schedule (pdf) November 6, 2018, 10:30 a.m.: Hermann Schulz-Baldes, "Monopole insertion and spectral flow in topological insulator" November 7, 2018, 10:30 a.m.: Lashi Bandara, "First-order elliptic boundary value problems beyond self-adjoint induced boundary operators" November 8, 2018, 10:30 a.m.: Siegfried Echterhoff, "The minimal exact crossed product and the Baum-Connes conjecture" November 9, 2018: 10:30 a.m.: Branimir Cacic, "Noncommutative principal bundles in unbounded KK-Theory"
Seminars in week 2:  schedule (pdf) November 13, 2018, 10:30 a.m.: Paul Baum, "Twisted K-homology" November 14, 2018, 10:30 a.m.: Joachim Cuntz, "Linear functionals on C*-algebras and their commutative subalgebras" November 15, 2018, 10:30 a.m.: Ryszard Nest,  "Group cocycles and algebraic K-theory" November 16, 2018: 10:30 a.m.: Giovanni Landi,  "Line bundles over noncommutative spaces"
Masterclass, November 19 - 23, 2018: schedule (pdf), Nigel Higson Abstract (pdf) Conference, November 26 - 30, 2018: schedule (pdf)

 

Interested PhD students are asked to have their supervisors send a letter of recommendation for this programme to Walter van Suijlekom and to register via the following website: http://www.noncommutativegeometry.nl/esi2018/registration/

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.