Good day and welcome.
I am a final year PhD student supervised by Mark Pollicott—I started October 2017.
Research right now.
I am currently interested in dynamical systems arising naturally in the context of low dimensional geometry, more specifically, frame flows on negatively curved 3-manifolds.
If we consider the frame flow on a Riemannian manifold of pinched negative curvature, this system is a characteristic example of what is known as a partially hyperbolic dynamical system.
There are many interesting and useful results known for hyperbolic dynamical systems (these generalise geodesic flows on negatively curved manifolds: notice in both cases the strong geometric link!), which is a much stringer property. Of special mention are those results of Dolgopyat, which provide strong exponential mixing estimates and asymptotic formulae for the number of closed orbits less than a given length.
Although these results provably do not extend to the entire class of partially hyperbolic systems, it is our hope that some analogue exists for certain classes of partially hyperbolic systems, for instance frame flows.
At this stage I am currently getting up to speed with the literature, with a view to investigating this problem.
Triviality of Milnor Fibrations
At this moment I am also working with David Mond to write up results emerging from my MMath dissertation.
In brief; in the setting of prehomogeneous vector spaces (which are very natural objects of study for algebraic geometers and singularity theorists), we see naturally arising Milnor fibrations over the punctured plane.
James Damon proved that in the equidimensional setting, the monodromies associated to these fibrations are cohomologically trivial: inducing the identity on all cohomology classes of the fibre.
In light of this theorem it is natural to think that the monodromy is geometrically trivial (i.e. it is isotopic to the identity on the fibre), which is equivalent to the fibration being a straightforward projection (up to bundle equivalence).
Indeed in many settings, we found this to be true (an interesting collection of examples arising in the form of Quiver Spaces), but we also discovered examples in which the conjecture utterly fails.
Thus we are currently writing up these results. Further to this, we are curious to see if the (non)triviality of these fibrations is preserved under Castling Transformation.