Fuzzy Spaces
Research into fuzzy spaces was the focal point of my PhD (see PDF here ). Fuzzy spaces are finite-dimensional non-commutative geometries. The idea behind my research is that non-commutative geometry provide a framework where quantum theory and gravity (being viewed as the curvature of geometry) can co-exist without conceptual issues. More precisely, the claim is that non-commutative geometries describe our reality and the picture of General Relativity and even Quantum Field Theory are really just some type of low-energy description of a non-commutative geometry construction. This viewpoint of non-commutative geometry was really developed and propelled into popularity by the field-medalist Alain Connes (https://alainconnes.org/en/).
There are many exotic and complicated examples of non-commutative geometry, and it is easy to lose yourself in the study of these objects. But my tactic is to start simple and progressively get more complicated as the needs arise. So this brings us to fuzzy spaces - the simplest non-commutative geometries.
Some basics to whet your apetite
A fuzzy space is really just a matrix algebra with some extra structures defined on top. So you might be wondering how matrix algebras might be useful in the description of geometry?! And it’s a valid question.