Research interests

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Research description

My research focuses on 3-dimensional manifolds; these are spaces which locally look like the standard three-dimensional Euclidean space (or half-space) but globally can be much more complicated. A common strategy is to try to equip them with various additional structures, such as metrics, decompositions into polyhedra (in particular triangulations), decompositions into lower-dimensional subsets (foliations), or group actions, which allow one to study the manifold from various different perspectives.

Aspects of 3-dimensional topology which are central in my work are fibrations over the circle with pseudo-Anosov monodromy and, more generally, pseudo-Anosov flows. I study them combinatorially using veering triangulations, certain ideal triangulations of the complement of finitely many closed orbits of the flow.

The main advantage of using veering triangulations to study pseudo-Anosov flows is that they are combinatorial objects with a very rigid structure. This often simplifies proofs of theorems relating pseudo-Anosov flows to other structures in 3-dimensional topology. See for instance the relation between the Teichmüller polynomials and certain twisted Alexander polynomials.

Moreover, the existing software to study triangulations (such as SnapPy or Regina etc.) can now be used to study pseudo-Anosov flows experimentally.