This is a page about what I think.
I am broadly interested in low dimensional topology and geometry. My current research involves Riemann surfaces, hyperbolic geometry and Teichmuller theory. My main motivation for doing mathematics is pictures and visualizations so I love anything with pictures!
We study distance relations in various simplicial complexes associated with low-dimensional manifolds. In particular, complexes satisfying certain topological conditions with vertices as simple multi-curves. We obtain bounds on the distances in such complexes in terms of number of components in the vertices and distance in the curve complex. We then define new invariants for closed 3-manifolds and handlebody-knots. These are defined using the splitting distance which is calculated using the distance in a simplicial complex associated with the splitting surface arising from the Heegard decompositions of the 3-manifold. We prove that the splitting distances in each case is bounded from below under stabilizations and as a result the associated invariants converge to a non-trivial limit under stabilizations.
Let γ be a filling curve on a topological surface Σ of genus g ≥ 2. The inf invariant of γ, denoted mγ, is the infimum of the length function on the space of marked hyperbolic structures on Σ. This infimum is realized at a unique hyperbolic structure, Xγ, which we call the optimal metric associated to γ. In this paper, we investigate properties of the inf invariant and its associated optimal metric. Starting from a filling curve and a separating curve, we construct a two integer parameter family of curves for which we derive coarse length bounds and qualitative properties of their associated optimal metrics. In particular, we show that there are infinitely many pairs of filling curves, each pair having distinct inf invariants but the same self-intersection number. The inf invariants give rise to a natural spectrum, we call the inf spectrum, associated to the moduli space of the surface. We provide coarse bounds for this spectrum.
We look at filling closed curves on hyperbolic surfaces and consider its length infima over the Teichmuller space of the surface. In particular explore relations between this infimum and self-intersection number of curves. We construct curves with same intersection number but different length infimum.
This is joint work in progress with my advisor Ara Basmajian . Here are my slides from a talk I gave on my work. Also a postcard I made for a conference at ICERM.
Joint work in progress with Meenakshy Jyothis. Constructing complexes of curves that include non-simple curves in addition to simple ones.
For my masters thesis I studied relations between contact structures and co-dimension one foliations on 3-manifolds via open book decomposions. Here are slides from my thesis defense. Thesis: Codimension one foliations related to contact topology in low-dimensional manifolds via Open Books. Advisor: Joan Licata , Australian National University