Speakers and Organization

An Introduction to Noncommutative Topology

C∗–algebras provide an elegant setting for many problems in mathematics and physics. In view of Gel’fand duality, their study is often referred to as noncommutative topology: general noncommutative C∗–algebras are interpreted as non- commutative spaces. These form an established research field within mathematics, with applications to quantum theory and other areas where deformations play a role. Many classical geometric and topological concepts can be translated into operator algebraic terms, leading to the so-called noncommutative geometry (NCG) dictionary.  These lectures aim to provide the participants with the tools to understand, consult and use the NCG dictionary, covering the basic theory of C∗-algebra and their modules. Focus will be given on examples, especially those that come from deformation theory and quantisation.

Lecture Notes

Website of Francesca Arici

Deformation Quantization and Symmetries

After a brief introduction to deformation quantization, we shall describe some of the achievements of formal deformation quantization (in particular concerning existence, classification, and representation theory), dwelling on points linked to group actions (invariant star products, universal deformations, quantum moment map, reduction).  We shall end up by considering the convergence question in deformation quantization.

Website of Simone Gutt

The Yang-Baxter equation, operator algebras, and braid group characters

The motivating problem of these lectures is the challenge to understand the solutions of the Yang-Baxter equation (YBE), a polynomial equation for matrices that is important in surprisingly many fields in mathematics and physics, such as quantum mechanics, Hopf algebras, statistical mechanics, knot theory, quantum field theory, subfactors, braid groups, and quantum information theory. Despite its finite-dimensional and simple appearance, the YBE is best studied with advanced mathematical tools from operator algebras, and its solutions are far from being completely understood.

Guided by this concrete motivation, these lectures will introduce abstract ideas and concepts of operator algebras that are helpful to study the concrete solutions of the YBE ("R-matrices") and also of prominent importance in many other areas of modern mathematics and mathematical physics. In particular, we will look at C*-algebras and von Neumann algebras as well as their inclusions and endomorphisms. Indeed, any R-matrix induces an endomorphism of the Cuntz algebra and the structure of this endomorphism contains relevant information about the R-matrix. The reduction of such endomorphisms (decomposition into irreducibles) uses ideas from algebraic quantum field theory that have been used in the analysis of the statistics of low-dimensional field theories, and sheds new light onto the solutions of the YBE. There is a also a prominent to link to the infinite braid group and its characters that will be explained. The abstract setting will be accompanied by concrete examples such as the two-eigenvalue solutions of the YBE that  can be classified and turn out to be elastic in the selfadjoint direction but quite rigid in the unitary direction.

Website of Gandalf Lechner

Deformations of Poisson structures

The main difficultly in studying deformations of Poisson structures comes from the lack of ellipticity of the Poisson-Lichnerowics complex, which controls infinitesimal deformations. Therefore only few general results on deformations of Poisson structures are known. In this mini-course I intend to discuss the following topics: the geometric interpretation of Poisson cohomology, deformations of log-symplectic structures, a class of rigid Poisson structures, Conn's theorem, and deformations of so-called "source-compact Poisson manifolds".

Website of Ioan Marcut

Noncommutative Geometry and Differential Calculus

Website of Boris Tsygan