# Non-commutative Geometry and its Applications

Hausdorff Trimester Program

September 1 - December 19, 2014

Organisers: Alan L. Carey, Victor Gayral, Matthias Lesch, Walter van Suijlekom, Raimar Wulkenhaar

Overview

Non-commutative geometry, in the sense of this program, is the exploration of geometric concepts through operator-algebraic methods, as initiated and outlined by Alain Connes. Among the topics and techniques central to non-commutative geometry are Hochschild and cyclic cohomology; bivariant K-theory and index theory; the measure-theoretic and topological analysis of operator algebras; and spectral geometry. Many of these methods arose from classical problems in global analysis, topology and representation theory. Collectively they have become powerful and effective tools that bring geometric ideas to bear on the analysis of a wide range of non-standard spaces arising in mathematics and physics.

Other topics in mathematics and mathematical physics such as non-commutative algebraic geometry, foliations, groupoids, stacks, gerbes, deformations and quantization have increasingly drawn on the perspectives of non-commutative geometry, with index theory often featuring prominently.

Non-commutative spaces also arise in number theory and arithmetic geometry; and in applications to topics in physics such as quantum field theory, renormalization, gauge theory, string theory, cosmology, gravity, mirror symmetry, condensed matter physics and statistical mechanics.

Topics

A primary objective was to focus on applications of NCG with a critical appraisal of their effectiveness. Both established and potential applications were explored and the organisers included a breadth of topics in the program. A summary of the scope is given by the following list:

- the NCG approach to the standard model,
- non-commutative variational methods,
- exactly solved models,
- spectral geometry,
- non-compact (non-unital) geometries,
- dynamical systems,
- number theory,
- KMS theory,
- twisted K-theory,
- condensed matter physics applications,
- Hopf algebras and quantum groups,
- foliations.

Activities

There were four workshops during the trimester:

- September 15-19, Non-commutative geometry's interactions with mathematics.
- September 22-26, Quantum physics and non-commutative geometry.
- November 24-28, Number theory and non-commutative geometry.
- December 15-18, Future directions for non-commutative geometry.

There was a summer school directed at beginning PhD students:

There was a series of lecture courses aimed at postgraduate students and postdoctoral level researchers:

There was also a weekly seminar series on current research topics and a working seminar within that part of the program aimed at junior researchers.