**SUMMARY**:

This research proposal aims to develope excelence research, with a strong international component, in the field of Mathematical Physics. In particular, the main research subjects in the Project are quantum groups, Poisson-Lie groups and their applications in the fields of quantum gravity and classical and quantum integrable systems. The three main scientific objectives of the Project are the following:

- To foster the classification and explicit construction of noncommutative spacetimes arising from quantum deformations of Poincaré, de Sitter and anti de Sitter groups in (1+1), (2+1) and (3+1) dimensions, and to analyse their applicability as suitable models in quantum gravity.
- To apply techniques based on Poisson coalgebras and projective geometry in order to obtain new classical integrable systems on curved spaces, and to construct their quantum analogues through the correspondinng spectral problems of Schrödinger and Dirac type.
- To apply Poisson algebra techniques in order to obtain integrable systems of first-order ordinary differential equations, including coupled and non autonomous systems, to generalize such techniques for the case of bihamiltonian systems and to extend the theory of Lie-Hamilton systems through the use of defomed coalgebra structures.

**RESEARCH GROUP**:

Ángel Ballesteros, Alfonso Blasco, Francisco J. Herranz, I. Gutiérrez-Sagredo (U. de Burgos)

Decio Levi, Orlando Ragnisco (Roma Tre University)

Rafael Hernández Heredero (U. Politécnica de Madrid)

Catherine Meusburger (FAU Erlangen-Nürnberg)

Javier de Lucas (U. Warsaw)

Enrique Reyes (U. de Santiago de Chile)

**SUPPORTED BY**:

Agencia Española de Investigación (Spain), Project MTM2016-79639-P

The Project is running from 01/01/2017 until 31/12/2019.