## Final Activity Report Summary - EU-NCG (Noncommutative Geometry)

European scientists have played leading roles in many of the most exciting recent discoveries in the interaction between operator algebras, geometry, topology, algebraic geometry and quantum groups in pure mathematics, and algebraic and conformal field theory, string theory, statistical mechanics in theoretical physics. The primary achievement of this Research Training Network is to ensure that there is a new generation of young researchers in Europe with sufficient breadth of knowledge and training required for this interdisciplinary research. This was achieved through bringing them to the cutting edge of research across a broad front including the following Research Areas:

1. Operator-algebras - classification and structure.

2. Measure and isomorphism rigidity.

3. Free probability and random matrices.

4. KK-theory and applications.

5. Cyclic theory and index theory.

6. Interplay between harmonic analysis and topology.

7. Quantum groups.

8. Applications to number theory.

9. Algebraic quantum field theory.

10. Fuzzy physics.

11. Strings and non-commutative geometry.

12. Quantum dynamics, quantum spin systems, quantum information.

Major results include for example progress in the classification of non-simple C*-algebras in the Elliott programme, rigidity in the theory of von Neumann algebras, when group von Neumann algebras and group measure space von Neumann algebras remember the group and the action, a new field of study in quantum groups as well as deepening understanding of the role of the Dirac operator, the relationship between braided subfactors and twisted equivariant K-theory, exoticness of the Haagerup subfactor, algebraic quantum field theory using axiomatizations using operator product expansions, Buchholz-Verch scaling, conformal nets, spectral triples and JLO cocycles, applications of noncommutative geometry to number theory and relation with equilibrium states, free probability methods to study analytic number theory, fuzzy physics with a simple model exhibiting an exotic phase transition in which geometry emerges as the system is cooled, the role of noncommutative geometry in the standard model.

The unifying mathematical concept behind the programme was the use of noncommutative operator algebras and noncommutative geometry to understand singular spaces or quantum spaces, replacing classical topological or measure spaces and the associated commutative algebras of continuous or measurable functions with noncommutative algebras of operators. Bringing together groups in Europe having a common goal in pursuing the deep connections between various branches of mathematics and physics we addressed their conjectures and problems through a training network preparing young researchers equipped to work in operator algebras and noncommutative geometry. An innovative feature of the training was the use of focused semesters.

The scientific and research training aspects of the project were significantly aided in having the 4 years of the RTN being broken up into 8 consecutive Focused Semesters each running for 6 months, with individual nodes taking turn to be responsible for leading the training during those periods. These helped give the young researchers a structure for broadening their training far beyond what could be achieved in a single country let alone a single institution. These semesters and associated training events, workshops as well as the annual meetings contributed to the scientific output in keeping the network at the cutting edge of worldwide research, facilitating networking as well as contributing to the training and employability of the young researchers, as well as involving the recruited researchers in the planning and management of the project.

1. Operator-algebras - classification and structure.

2. Measure and isomorphism rigidity.

3. Free probability and random matrices.

4. KK-theory and applications.

5. Cyclic theory and index theory.

6. Interplay between harmonic analysis and topology.

7. Quantum groups.

8. Applications to number theory.

9. Algebraic quantum field theory.

10. Fuzzy physics.

11. Strings and non-commutative geometry.

12. Quantum dynamics, quantum spin systems, quantum information.

Major results include for example progress in the classification of non-simple C*-algebras in the Elliott programme, rigidity in the theory of von Neumann algebras, when group von Neumann algebras and group measure space von Neumann algebras remember the group and the action, a new field of study in quantum groups as well as deepening understanding of the role of the Dirac operator, the relationship between braided subfactors and twisted equivariant K-theory, exoticness of the Haagerup subfactor, algebraic quantum field theory using axiomatizations using operator product expansions, Buchholz-Verch scaling, conformal nets, spectral triples and JLO cocycles, applications of noncommutative geometry to number theory and relation with equilibrium states, free probability methods to study analytic number theory, fuzzy physics with a simple model exhibiting an exotic phase transition in which geometry emerges as the system is cooled, the role of noncommutative geometry in the standard model.

The unifying mathematical concept behind the programme was the use of noncommutative operator algebras and noncommutative geometry to understand singular spaces or quantum spaces, replacing classical topological or measure spaces and the associated commutative algebras of continuous or measurable functions with noncommutative algebras of operators. Bringing together groups in Europe having a common goal in pursuing the deep connections between various branches of mathematics and physics we addressed their conjectures and problems through a training network preparing young researchers equipped to work in operator algebras and noncommutative geometry. An innovative feature of the training was the use of focused semesters.

The scientific and research training aspects of the project were significantly aided in having the 4 years of the RTN being broken up into 8 consecutive Focused Semesters each running for 6 months, with individual nodes taking turn to be responsible for leading the training during those periods. These helped give the young researchers a structure for broadening their training far beyond what could be achieved in a single country let alone a single institution. These semesters and associated training events, workshops as well as the annual meetings contributed to the scientific output in keeping the network at the cutting edge of worldwide research, facilitating networking as well as contributing to the training and employability of the young researchers, as well as involving the recruited researchers in the planning and management of the project.