I am a graduate student supervised by Latham Boyle. My main research focus is on extending the framework of (Connes) Non-commutative geometry to accept Non-associative input data. There are two main motivations for this work:
First, usually when constructing physical theories physicists take as a primary piece of input data a symmetry group. Many of the most interesting group structures are non-commutative, and so restricting to commutative groups seems somewhat unnatural. When constructing physical models within the framework of Non-commutative spectral geometry one instead takes an algebra as a main piece of input data. The symmetries of the theory then arise as the automorphisms of that algebra. Many of the most interesting algebraic structures are non-associative (eg. Lie, Jordan, Cayley, Brown, Malcev). Imposing the restriction of associativity on the input data is similarly unnatural, and amounts to blinding ourselves to something essential that the formalism is trying to tell us
Second, there are many interesting questions left unanswered by Non-commutative spectral geometry. The most interesting GUTs, which address these questions, are out of the range of the spectral formalism in its current incarnation. For example, GUTs based on E6 are out of range of the theory as all of the exceptional groups arise as the automorphisms of non-associative algebras. Spectral geometry might have something interesting to tell us about grand unification - but first we need to extend the formalism to accept non-associative input data.
For more information on non-associative geometry see: http://arxiv.org/abs/1303.1782
`Building a Non-associative Geometry' - Caltech.
`Unification of all Fundamental Forces: A Spectral Approach' - Monash University.