## Periodic Reporting for period 1 - NCDIFFGEO (Models of noncommutative differential geometries)

Reporting period: 2016-02-01 to 2018-01-31

The project builds on a specific body of knowledge accumulated over the last 25 years related in part to the notion of `quantum groups' or Hopf algebras. These fully emerged in the 1980s out of quantum integrable systems as a new more general notion of symmetry but they can also feature as `quantum symmetries' of quantum spacetimes. The mathematical theory of `noncommutative Riemannian geometry' was also developed recently. This was originally motivated by the wish to include the NCG of quantum groups themselves (just as classical geometry was driven on part by the geometry of Lie groups) but, as a general scheme, it potentially applies much more widely. The aim of the project was to obtain and study a new generation of quantum spacetime models and their noncommutative differential geometries in this context.

At the same time a significant amount of training was undertaken in the techniques of `noncommutative Riemannian geometry', centering around the notion of a `bimodule connection'. A novel feature of the algebraic framework is that we do not need to work over real or complex numbers but can do geometry over any field. We took this nearly to the extreme by working over the field F_2 of 2 elements {0,1}, which we called `digital NCG'. Even the classification of digital NCGs in the affine case where the `co-ordinate algebra' is just polynomials in n<4 variables turned out to be very interesting. The differential calculi on such algebras turned out to be constructed by n-dimensional commutative algebras. We also began the classification of digital NCGs in the finite case where the `coordinate algebra' is a finite-dimensional algebra over F_2.

The work was disseminated in the form of 4 papers on the arXiv preprint repository with three of these already published in regular research journals and the fourth under review by a journal. The work was also disseminated extensively by the MSC fellow in the form of 9 talks at conferences and seminars. These included a very large `new frontiers in physics' conference among others. The MSC fellow also helped organise a conference in Poland on a theme related to the project and helped to run the regular `Quantum Algebras' seminar at the host.

In addition the MSC fellow undertook outreach activities such as speaking at the event `Women count' at the host during International Women's week, and at a Career Advice event for PhD students also at the host. On training, the MSC fellow trained for and acquired a certificate of Higher Education Academy (ADEPT). As a result of her activities the MSC fellow was offered and accepted a UK academic, ending the grant 4 months early. Therefore the grant can also be considered highly successful in its career development and training elements.

Neither experimental QG nor actual digital or quantum computing were in the scope of this theoretical project but these are two potential impacts in the longer term. Similarly we did not envisage immediate societal or socio-economic impact but in the longer term both topics have the potential to open the door to dramatic new technologies. Both interact with other EU supported scientific projects such as the COST network on quantum spacetime and EU investments in quantum technology.