### Noncommutative
Geometry

Quantum gravity motivates the idea that space or spacetime should be considered as quantum objects. One possible
realization of this idea is noncommutative geometry: essentially one generalizes the coordinates on a manifold to a
nonabelian algebra. Mathematicians have considered this possibility since the 80's. I would mention here two of the
physical motivations to take this possibility seriously. One is the argument by Doplicher

*et
al.* [

1],
which combines Heisenberg's microscope with gravitational effects, to deduce an uncertainty
principle between the coordinates of a particle (as opposed to coordinates and momenta). A different motivation is the
progress made by A. Connes with his `spectral triple' approach to noncommutative geometry, which has allowed him to
deduce some of the structure of the standard model from first principles [

2].
In my research, I studied a particular approach to noncommutative geometry based on a generalization of differential
geometry and generalizations of Lie groups called `Quantum Groups', or Hopf Algebras. I believe these mathematical
structures are worth studying for a physicist because they have a chance of being a description of some solutions of the
yet unknown quantum theory of gravity. Moreover, they predict interesting new phenomena when used as a background on
which particles and fields propagate (see, for example, the partial results like [

3],
in which the authors found a noncommutative spacetime structure as an effective description of particles propagating in
2+1-dimensional gravity). In the field of Noncommutative Geometry, I intend to develop the approach to field theory on a
noncommutative background spacetime for which I laid the foundations in [

4]. In that paper I developed
the basic differential-geometric tools (exterior and interior derivatives, Lie derivative, hodge-*, integrals) needed
to describe field theories in a way that respects the noncommutative structure of spacetime. The mid-term goal is to
study quantum field theories on noncommutative spacetimes in a way that is free of inconsistencies. This has the
potential to lead to testable predictions.