My interests are broadly in topology and algebra.
Algebraic topology can be described as the study of topological spaces by means of algebraic objects associated to them. I have been working on structural aspects of compact groups acting freely on manifolds, equivariant maps, free rank, index theory, and recently topological complexity.
My interest in group theory has been in cohomology, automorphisms, (twisted)-conjugacy classes and other structural aspects of finite groups, generalized braid groups, Artin groups and Coxeter groups. Recently, I have also been thinking about mapping class groups of orientable surfaces.
Knot theory is the study of embedded circles in the 3-space. I have been interested in various ramifications and extensions of classical knot theory. Specifically, I have been working on cohomological, computational and structural aspects of quandles and their more general avatars. These objects give strong invariants for knots and appear naturally in diverse areas of mathematics including group theory, solutions of the Yang-Baxter equation, Hopf algebras and symmetric spaces, to name a few.
- (With Neha Nanda), Alexander and Markov theorems for virtual doodles, New York Journal of Mathematics 27 (2021), 272--295.
- (With Tushar Naik and Neha Nanda), Conjugacy classes and automorphisms of twin groups, Forum Mathematicum 32 (2020), 1095--1108.
- (With V.G. Bardakov and Manpreet Singh), Link quandles are residually finite, Monatshefte fuer Mathematik 191 (2020), 679--690.
- (With V.G. Bardakov and A.Y. Vesnin), Structural aspects of twin and pure twin groups, Geometriae Dedicata 203 (2019), 135--154.
- (With V.G. Bardakov and Manpreet Singh), Free quandles and knot quandles are residually finite, Proceedings of the American Mathematical Society 147 (2019), 3621--3633.
- (With I.B.S. Passi and M.K. Yadav), Automorphisms of Finite Groups, Springer Monographs in Mathematics, Springer, (2018), ISBN 978-981-13-2894-7, ISSN 1439-7382, xix + 217 pp.