Research Projects

The research focuses on projective modules and complete intersections, specifically the theory of Euler class groups, existence of unimodular elements, cancellation problem of projective modules, efficient generation of ideals, lifting problems, and orbit spaces of unimodular rows. They have worked on the top-rank projective modules over smooth affine algebras, such as $X=Spec(R)$ and $P$, and are interested in exploring these areas. The author's research question is whether $P\simeq A\oplus Q$ over smooth real affine algebras, where $C_n(P)=0$ in $CH_0(X)$. They also want to investigate the structure of $Um_{d+1}(R)/SL_{d+1}(R)$ and $Um_{d+1}(R)/E_{d+1}(R)$ when $X(\mathbb{R})$ is orientable. They also want to investigate the group $Um_{d+1}(R)/E_{d+1}(R)$ when $X(\mathbb{R})$ is orientable and every complex maximal ideal of $R$ is a complete intersection. The author also aims to understand the relations between the Euler class groups of $A$, $E^n(A)$, $E^n(A[T])$, and $E^n(A(T))$. They also have a few more questions related to efficient generation of ideals. In summary, the author's research aims to explore various intriguing problems in projective modules, complete intersections, Euler class groups, and efficient generation of ideals.