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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - QUANTUM DYNAMICS (New Geometry of Quantum Dynamics)

Teaser

Summary

The Project aims to make significant advances to the field of noncommutative geometry by developing new methods through substantial interaction within sub-fields of noncommutative geometry and two other areas of pure mathematics. The main focus will be to determine the topological non-triviality of new types of quantum fibrations. Also, we aim to construct quantum metric geometries of crossed products and graph algebras relating the Lipschitz-norm and Dirac-operator approaches. The Project will combine various areas, which although interacting on the fundamental level, have had their concrete and usable connections mostly unexplored.

The success of the Project depends on connecting centres of excellence in relevant topics for the exchange of ideas and and production of high-quality collaborative results. The network has been carefully chosen to include the world\'s leading experts as well as promising early career researchers. Not only does this guarantee the participants access to an enormous knowledge base, it will also ensure that new and innovative lines of research will continue to be developed long after the Project has finished.

In particular, the collaborative nature of the project will be of great benefit to the early career researchers involved. In pure mathematics, fields often become so specialised that only a small number of people around the world might be actively researching a particular topic. This can make career progression very difficult. The interdisciplinary nature of the Project will expose the participants to a host of new mathematics and new collaborations. Consequently, this diversification will result in significantly more career opportunities than would otherwise be available.

Work performed

\"1. EU-funded secondments: 21 person-months.

2. Other exchange visits between nodes: over 44 person-months.

3. Events: 3

- Conference at the Fields Institute entitled \"\"Geometry, Representation Theory and the Baum-Connes Conjecture\"\" (July 18 - 22, 2016)
- Simons Semester at Banach Center entitled \"\"Noncommutative geometry the next generation\"\" (September - November 2016)
- Conference at the Institute of Mathematics of the Polish Academy of Sciences entitled \"\"New geometry of quantum dynamics\"\" (January 15-19, 2018)

4. Talks: 189

5. Research papers: 37\"

Final results

Work Package 1
Title: Noncommutative Borsuk-Ulam-type conjectures
Number of papers: 6

Work Package 2
Title: Noncommutative bordism of quantum fibrations
Number of papers: 6

Work Package 3
Title: Trimming and growing of graphs for graph algebras
Number of papers: 9

Work Package 4
Title: Noncommutative metric geometry of C*-dynamical systems and graph C*-algebras
Number of papers: 9

Work Package 5
Title: Noncommutative spectral geometry for quantum group homogeneous spaces
Number of papers: 7

This gives the total of 37 research papers. By the end of the project, we expect more than double this output by having at least 80 research papers. Many of them should be published or submitted for publication in high impact research journals. About half of all papers ought to be published by the end of the project, and all of them within a couple of yeras since the end of the project.