# Program

## Schedule

The schedule is not yet avalaible. Its main features are as follows:

There are 6 courses (see below) each of them splitted into 6 lectures, by taking into account also tutorial sessions, and some specialized seminars (see below) will also be given. The duration of lectures and seminars is of one hour, with 10 minutes for questions or discussion included.

The working language is English.

- Wednesday, July 12: Arrival of participants at Seagull Hotel
- 12:00 – 14:00 Lunch
- 14:00 – 18:30 Registration
- 18:30 – 19:30 Welcome cocktail
- 19:30 – 21:00 Dinner

- Working day with 6 hours of activities. This applies to the following 5 days:

Thursday 13, Friday 14, Monday 17, Tuesday 18, Thursday 20, Friday 21.- 8:00 – 8:30 Shuttle from Hotel to ICISE
- 8:30 – 9:30 Lecture / tutorials or specialized seminar
- 9:30 – 10:00 Coffee Break
- 10:00 – 11:00 Lecture / tutorials or specialized seminar
- 11:00 – 12:00 Lecture / tutorials or specialized seminar
- 12:00 – 13:30 Lunch
- 13:30 – 14:30 Lecture / tutorials or specialized seminar
- 14:30 – 15:30 Lecture / tutorials or specialized seminar
- 15:30 – 16:00 Coffee Break
- 16:00 – 17:00 Lecture / tutorials or specialized seminar
- 17:00 – 18:30 Leisure time and informal discussions
- 18:30 – 19:00 Shuttle from ICISE to Hotel

- Working day with 3 hours of activities. This applies to the following 2 days:

Saturday, 15, Wednesday, 19.- 8:00 – 8:30 Shuttle from Hotel to ICISE
- 8:30 – 9:30 Lecture / tutorials or specialized seminar
- 9:30 – 10:00 Coffee Break
- 10:00 – 11:00 Lecture / tutorials or specialized seminar
- 11:00 – 12:00 Lecture / tutorials or specialized seminar
- 12:00 – 13:30 Lunch
- Free afternoon

- Sunday, 16: excursion.
- Saturday, July 22: Departure of participants

## 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.

### 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**,
Department of Physics, University of Naples Federico II, Italy

The applications of noncommutative geometry and noncommutative spaces to field theory will be introduced and some of the relevant examples will be discussed in detail. These will range from field theory on noncommutative and fuzzy spaces, to the applications to the standard model and particle physics, to the issues of quantization of gravity. The course will be self-contained and the relevant mathematical tools and relevant physics concepts will be introduced.

### 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:

- Alan Carey (Australian National University, Canberra, Australia)
- Wicharn Lewkeeratiyutkul (Chulalongkorn University, Thailand)
- Fedele Lizzi (Università di Napoli Federico II, Napoli, Italy)
- Yoshiaki Maeda (Tohoku University, Sendai, Japan)

## Posters

Participants are invited to submit a poster, see registration page. Poster discussions will be given at any occasion (Leisure time and informal discussions, coffee breaks) all during the period of the school.