Department of
Mathematics

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.

Research

Low-dimensional topology

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.

Selected Publications

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