Program

The schedule of the school is given on the Schedule page.

Prerequisites

In order to attend the school, some minimal knowledge are required in mathematics, as well as in physics. Mainly, here are the point we expect the participants to master:

  • Linear algebra in finite and infinite dimensions: vector spaces, Hilbert spaces…
  • Basic notions in analysis: differential calculus, integration…
  • Basic notions in differential geometry: manifolds, fiber bundles…
  • Basic notions in algebra: associative algebras, Lie algebras, groups…
  • Basic notions on operator algebras on Hilbert spaces would be useful…
  • Basic knowledge about Quantum Mechanics (the Schrödinger equation).
  • Basic knowledge about the Dirac equation in physics.
  • Basic knowledge about the geometry of gauge theories in physics, and some ideas about the Standard Model of particles physics.

Lectures

Unbounded \(KK\)-theory in Noncommutative Geometry and Physics

Bram Mesland, Institut für Analysis, Leibniz Universität Hannover, Germany.

We will give an introduction to the unbounded picture of \(KK\)-theory. The constructive Kasparov product, which can be viewed as a type of noncommutative fibration compatible with index theory, will play a central role. The course will introduce the main aspects of this product, as well as examples from hyperbolic dynamical systems, quantum groups and the factorisation of Dirac operators on principal bundles. Special attention will be given to the type III setting. This presents mathematical challenges to unify the analytic and algebraic aspects of unbounded \(KK\)-theory.

  • Lecture 1 – Dirac operator on a compact manifold, twisting by a connection
  • Lecture 2 – Spectral triples, \(K\)-cycles, index pairing
  • Lecture 3 – \(KK\)-theory, relation to \(K\)-theory and \(K\)-homology, existence of the Kasparov product
  • Lecture 4 – Unbounded Kasparov product
  • Lecture 5 – Construction of spectral tripes on noncommutative \(C^*\)-algebras, gauge transformations
  • Lecture 6 – Examples: theta deformations, crossed products from conformal group actions, Cuntz algebras

Equilibrium states on operator-algebraic dynamical systems

Nathan Brownlowe, School of Mathematics and Statistics, University of Sydney, Australia.

Equilibrium states, or KMS states, are mathematical objects used to describe quantum mechanical systems when in equilibrium. We will model quantum mechanical systems with operator-algebraic dynamical systems, which consist of an action (representing time evolution) of the real line on a \(C^*\)-algebra (representing the observable of the system). We will start with some background on \(C^*\)-algebras and states on \(C^*\)-algebras, before looking at the physical motivation behind the KMS condition. We will spend the second half of the course looking at recent work on the KMS-state structure of various \(C^*\)-algebras, including directed graph \(C^*\)-algebras and Noncommutative solenoids.

Noncommutative Geometry and Field Theory

Patrizia Vitale and Fedele Lizzi, Department of Physics, University of Naples Federico II, Italy.

Some of the applications of noncommutative geometry and noncommutative spaces to physics will be introduced and some of the relevant examples will be discussed in detail. In particular gauge theory on noncommutative spaces, and spectral action and the applications to the standard model of particles physics. The course will be self-contained and the relevant mathematical tools and relevant physics concepts will be introduced.

  • Lecture 1 – Noncommutative spaces, physical origins, mathematical foundations (F. Lizzi).
  • Lecture 2 – Short introduction to gauge theory. Derivation based differential calculus for noncommutative associative algebras (P. Vitale).
  • Lecture 3 – Noncommutative gauge theory I (Moyal space-time) (P. Vitale).
  • Lecture 4 – Noncommutative gauge theory II (Lie algebra type non-commutativity) (P. Vitale).
  • Lecture 5 – The standard model of fundamental interactions as an almost commutative geometry (F. Lizzi).
  • Lecture 6 – The spectral action confronts “real physics” (F. Lizzi).

Introduction to Noncommutative Analysis and Integration

Fedor Sukochev, School of Mathematics and Statistics, University of New South Wales, Sydney, Australia.

In NCG, the singular (Dixmier) traces have become an indispensable tool. These traces are defined via dilation invariant extended limits on the space of bounded measurable functions. We will discuss in details the formulae relating Dixmier traces and zeta-function residues and underlying concepts of NC integration theory with respect to such traces. These formulae are established under various additional conditions on these extended limits.

Formal and non-formal Quantization and Index Theorems

Ryszard Nest, Department of Mathematical Sciences, Copenhagen, Denmark.

Deformation quantization produces a lot of interesting examples of “noncommutative spaces”. After an introduction to deformation quantization and some formality theorems, this lecture will focus on some algebraic index theorems, on the algebras of pseudodifferential and Fourier integral operators, and finally on geometric quantization and its relation to physics and representation theory, especially in a NCG-approach to Loop Quantum Gravity.

Noncommutative Topology and Topological Quantization

Johannes Kellendonk, Institut Camille Jordan, Université Claude Bernard, Lyon, France.

In quantum physics, noncommutative topology is one of the most successful tools to describe topological quantization in presence of disorder. In particular, it allows to identify some topological invariants. This lecture is an introduction to noncommutative topology with applications to topological quantization. It will address operator K-theory, cyclic cohomology, K-homology, Quantum Hall Effect, classification of topological insulators…

Seminars

Seminars will be given by:

Posters

Participants are invited to submit a poster, see registration page. A posters session will take place during the school on Monday, July 17, 2017 (See Schedule). The posters will be displayed during all the school.

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