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