Research Interests
I work in low-dimensional topology, geometry, and dynamics; and in particular on surfaces, 3-manifolds, and the connection between low-dimensional topology and computational complexity. More specifically, I have worked on taut foliations, 3-manifolds, mapping class groups of surfaces, the computational complexity of knot genus, as well as special algebraic integers arising in dynamics.
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Link to the Geometry Group at King's College London: Geometry (kcl.ac.uk)
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Foliations of 3-manifolds
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A (two-dimensional) foliation of a 3-manifold is a decomposition of the 3-manifold into injectively immersed surfaces (called leaves) such that locally the surfaces fit together like a stack of papers. A particularly important class of foliations are taut foliations. A foliation is taut if for every leaf L there is a closed curve c(L) intersecting L and transverse to the foliation at every point. It is known that the existence of a taut foliation gives a good deal of information about the ambient manifold, and they can be used to study the complexity of embedded surfaces inside the 3-manifold. I am interested in questions regarding the existence and properties of these foliations and its connection to other theories such as Thurston norm, universal circles, and left-orderable groups.
Mapping class groups of surfaces
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The mapping class group of a surface is the group of orientation-preserving homeomorphisms of the surface up to isotopy. This group has importance both for the study of surfaces and 3-manifolds. By Nielsen-Thurston classification, every surface homeomorphism can be isotoped to be in one of the three standard forms: periodic, reducible, and pseudo-Anosov. Pseudo-Anosovs are considered as 'generic' elements of the mapping class groups. Therefore understanding pseudo-Anosov maps is crucial for studying the mapping class group. A pseudo-Anosov map comes equipped with an algebraic integer, called the stretch factor. This number (or its logarithm) arises in different contexts such as the topological entropy of the homeomorphism or the length of closed geodesics on the Moduli space of surfaces equipped with the Teichmuller metric. Questions such as What algebraic integers arise in this way? or What is the smallest possible such algebraic integer for a fixed surface? can be quite interesting and challenging.
Knot theory and computational complexity
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A knot is an embedding of a circle inside the three-space or a 3-manifold. Usually, we visualize knots through their projections onto a plane, where there are over/under crossings in this diagram. The problem of deciding whether two knots are equivalent or whether a given diagram is equivalent to the unknot is a challenging question. The difficulty comes from the fact that sometimes in order to untangle a diagram, one has to increase the crossing number and so make the diagram more complicated (using only Reidemeister moves). It has been known since Haken that the question of deciding whether a diagram is equivalent to the unknot is solvable. Similarly one can decide if two knots are equivalent but the methods do not give any reasonably simple way of deciding. Questions such as How much an unknot diagram has to become complicated before it gets untangled? and Are there alternative ways to simplify a diagram monotonically? are among my interests.