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)?
Algebraic K-theory of Monoid Algebras ISIRD, SRIC