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Subfactors and Quantum Groups: A Combinatorial Approach to some Problems in Operator Algebras
Provider: Faculty of Science

Activity no.: 5001-19-07-32
Enrollment deadline: 20/03/2019
PlaceDepartment of Mathematical Sciences
Date and time29.04.2019, at: 09:00 - 03.05.2019, at: 16:00
Regular seats
ECTS credits2.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserRyszard Nest    E-mail address: rnest@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Course workload
Course workload categoryHours
Preparation / Self-Study40.00
Course hours25.00
Sum65.00
Content
This masterclass aims to provide a comprehensive introduction to the subfactor theory as well as to the combinatorics behind in order to highlight some new applications in the interplay of quantum information theory, quantum groups and operator algebras obtained by M. Brannan, A. Freslon and S. Vaes together with their coauthors. The lecturers are internationally leading experts in their respective areas, with publications in the most prestigious journals and they have participated and organized many international conferences. Moreover, they are known for their well-prepared, enthusiastic and pedagogical lectures, which will benefit undoubtedly the young students who start their research in this difficult subject.

Course 1 (M. Brannan). This course will start by an introduction to planar algebras and the Temperley-Lieb category. Next, the notions of entanglement and quantum channels, which are fundamental in quantum information theory, will be discussed showing the relation with operator algebras and precisely with compact quantum groups, which leads to the last part of the course where a recent result will be exposed concerning the construction of entangled subspaces and quantum channels coming from the representation theory of some compact quantum groups using non-random techniques, which differs from previous works.

Course 2 (A. Freslon). This course will start by an introduction to compact quantum groups (in the sense of S. L. Woronowicz), their representation theory and its duals, called discrete quantum groups. Next, a concrete class of such quantum groups, called easy quantum groups, will be introduced and their representation theory will be described precisely in some examples of interest (such as free versions of the orthogonal, permutation and reflection groups) using the combinatorial notion of non-crossing partitions. Finally, a recent result will be exposed concerning new examples of discrete quantum groups satisfying the Connes' embedding problem, based on the pioneer work of M. Brannan, B. Collins and R. Vergnioux.

Course 3 (S. Vaes). This course will start by a comprehensive introduction to the subfactor theory as well as to rigid C*-tensor categories in order to understand their connection. Next, some approximation properties for discrete quantum groups will be discussed with special attention to property (T) together with examples. The recent result that will complete this course will expose the construction of the first family of examples so far of genuine discrete quantum groups having property (T), based on a tensor category version of the Zuk's spectral gap criterion for property (T).

Learning outcome
After this course, students will be familiar with the main concepts and tools of subfactor theory and quantum groups obtaining a complete state-of-the-art together with the presentation of some of the ultimate results and applications in operator algebras as well as in quantum information theory.

Lecturers
Assistant Professor Michael Brannan, Texas A&M University, USA
Maître de Conférences Amaury Freslon, Université Paris-Sud 11, France
Professor Stefaan Vaes, KU Leuven, Brussels Belgium

Remarks

More details on the masterclass webpage.

On-line application only from the registration form at the masterclass page. The deadline for applications is 20 March 2019.

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