K-theories are very important invariant associated to the algebras, varieties, schemes and spaces.
I work on K-theory of monoid algebras and toric varieties.

My another area of research is projective module over affine algebras, i.e.,
vector bundles over affine schemes. This is "essentially" a study of lineal algebra
over commutative rings with 1. Projective modules are "locally" free module.
Free modules over commutative rings are those modules which have a basis.
Then one would like to investigate 'how far" a projective module from being
a free module. Starting point of all these is a famous conjecture of J.-P. Serre.
The statement is "every projective modules over polynomial rings are free".
This is now a theorem due to Quillen and Suslin. The "equivalent" question in
linear algebra is: given a vector (a_1,...a_n) in R^n (here R is a commutative
ring with 1) such that a_1b_1+...+a_nb_n=1, then can we write down (or find
in an existential way) nxn invertible matrix such that first row is (a_1,...,a_n)?

Here is the link for JRF recruitment: http://www.iitkgp.ac.in/temporary-jobs (2nd adv. from the last)

All Publications

K-theory of monoid algebras and a question of Gubeladze by Sarwar H. P., Krishna A. Journal of the Institute of Mathematics Jussieu 18 1051-1085 (2019)
Existence of a unimodular element in a projective modules over symbolic Rees algebras by Sarwar H. P., Bhaumik C. Proceeding Mathematical Sciences, IAS 133 - (2023)
Negative K-theory and Chow group of monoid algebras by Sarwar H. P., Krishna A. K-Theory Conference (AMS, Contemporary Mathematics) 195-224 (2020)
Stability results for projective modules over Rees algebras by Sarwar H. P., Rao R. A. Journal of Pure and Applied Algebra 223 1-9 (2018)
Some results about projective modules over monoid algebras by Sarwar H. P. Communications in Algebra 44 2256-2263 (2016)
Serre dimension of monoid algebras by Sarwar H. P., Keshari M. K. Proc. of Mathematical Sciences (Indian acad Sci.) 127 269-280 (2017)
Ideal class groups of monoid algebras by Sarwar H. P. Journal of Commutative Algebra 9 303-312 (2017)
Serre dimension and Euler class group of overrings of polynomial rings by Sarwar H. P., Keshari M. K. Journal of Commutative Algebra 9 213-242 (2017)
K_0-stability over monoid algebras by Sarwar H. P. Annals of K-theory 6 629-649 (2021)
The third homology of symplectic groups and algebraic K-theory by Sarwar H. P., Schlichting M. American Journal of Mathematics - (Accepted/In-Press)

Completed Projects

Co-Principal Investigator

Alumni members

Ph. D. Students

Md Abu Raihan

Area of Research: Algebraic K-theory

Soumik Mukherjee

Area of Research: Algebraic Geometry

Sudip Manna

Area of Research: Algebraic Geometry