Norbert Wiener Assistant Professor
Department of Mathematics
503 Boston Avenue
Medford, MA 02155
Finsler geometry, Anosov flows, Negatively-curved spaces, Laplace-like operators
I am working in geometry and dynamical systems, and, mostly, I'm
interested in the interaction between the two. For instance, what
can be deduced about the geometry of a space, knowing its dynamics?
And vice-versa. Among the objects we like to study in relation with
this question are the geodesic flow and objects constructed from it
(like the entropy, or some invariant measures like the Bowen-Margulis
measure) of course, but also global analytical tools like the Laplacian.
I study these type of questions in Finsler geometry, which is a generalization
of Riemannian geometry where the metric is given by a family of norms not
necessarily coming from a scalar product. As Finsler geometry is much wider
than Riemannian geometry, the links between geometry and dynamics are generally
either less strong, or even harder to prove. But this wider generality leads
to my two favorite phenomena:
Finsler geometry also lacks some of the global analysis tools that
Riemannian geometry enjoys. As part of my Ph.D., I introduced a Laplace
operator in Finsler geometry. This definition is very nice (I think) and
simple enough to allow the study of this operator.
- Some surprising rigidity results
- Some new behaviors that initially didn't seem to be possible
(did you know that there exists Finsler metrics on the sphere with
only two periodic geodesics?)
There is a lot of things that can be studied about a Laplacian. Among the
things I started studying are the construction of harmonic measures on (Finsler)
negatively curved manifolds and what happens when the harmonic measures coincide
with the Bowen-Margulis measure (for Riemannian metric, it implies that the space
is locally symmetric). I am also interested in more specifically Laplacian questions,
like finding bounds on the eigenvalues.
Outside of geometry, I also study Anosov flows (i.e., flows that locally
looks like the geodesic flow of a negatively curved manifold) for themselves.
With questions related to either their regularity (and what can be deduced from
regularity assumptions) or topological questions for Anosov flows in 3-manifolds.