Low dimensional Topology
Low dimensional Topology
After completing his MSc from Indian Institute of Technology Bombay, Tejas Kalelkar received his PhD from Indian Statistical Institute Bangalore. His topic of research was low-dimensional topology. He was then a postdoctoral fellow at Institute of Mathematical Sciences, Chennai and a Chauvenet Postdoctoral Fellow at the Washington University in St Louis for three years. He joined IISER Pune in 2013.
Dr. Tejas Kalelkar's area of research is low-dimensional topology. Topology can be thought of as 'rubber-sheet geometry' and is the study of properties that remain invariant under controlled deformations. Low-dimensional topology deals with the study of 3-manifolds, which are objects that locally look like our 3-dimensional space. His focus is primarily on foliations, triangulations, Heegaard splittings and hyperbolic geometry of 3-manifolds.
A closed book looks like a 3-dimensional object but is in fact a union of 2-dimensional pages stacked together. Similarly every 3-manifold can be built by stacking 2-dimensional surfaces together into what is called a foliation. Dr. Kalelkar studied a special class of foliations called taut foliations, which imply useful topological properties for the 3-manifold.
On cutting open a 3-manifold along a special embedded surface, called the Heegaard-splitting surface, we end up with two simple pieces called handlebodies. Every 3-manifold has such splitting surfaces. Dr. Kalelkar obtained a structural form for such surfaces.
Every 3-manifold can be built by suitably sticking tetrahedra together along faces. Normal surfaces are surfaces embedded particularly 'nicely' with respect to such a triangulation. Dr. Kalelkar's work has proved a few results about this useful class of surfaces.
Triangulations of a manifold allow us to use combinatorial algorithms to resolve problems in geometric topology. A basic problem in this area is to distinguish between manifolds using their triangulation data. Dr. Kalelkar's recent work involves obtaining such algorithms for geometrically triangulated constant-curvature manifolds.
Kalelkar, T., Euler characteristic and quadrilaterals of normal surfaces, Proceedings Mathematical Sciences, Indian Academy of Science, Volume 118, Number 2, 2008, 227-233
Kalelkar T., Incompressibility and normal minimal surfaces, Geometriae Dedicata, Volume 142, 2009, 61-70
Gadgil S. and Kalelkar T., Chain complex and Quadrilaterals for normal surfaces, Rocky Mountain Journal of Mathematics, Volume 43, Number 2, 2013, 479-487
Kalelkar T. and Roberts R., Taut foliations in surface bundles with multiple boundary components, Pacific Journal of Mathematics, Volume 273, Number 2, 2015, 257-275
Kalelkar T., Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds, Proceedings of American Mathematical Society, Volume 148, Number 10, 2020, 4527-4529
Kalelkar T. and Phanse A., Geometric bistellar moves relate geometric triangulations, Topology and its Applications, Volume 285, 2020, 107390-107397
Kalelkar T. and Phanse A., An upper bound on Pachner moves relating geometric triangulations, Discrete and Computational Geometry, Volume 66, Number 3, 2021, 809-830
Kalelkar T. and Raghunath S., Bounds on Pachner moves and systoles of cusped 3-manifolds, Accepted in Journal of Algebraic & Geometric Topology