Research Projects

The subjects algebra and geometry are crucial to the development of not only mathematics but also other branches of basic and applied sciences, including those that have direct applications in day to day life. Affine spaces and polynomial rings are fundamental objects of interest which pervade almost the whole of modern algebra and algebraic geometry. Results on these objects can be needed by any mathematician in any field, at any unforeseen time. The project envisages investigations on central problems on these objects like embeddings of lines and quadratic planes in the affine three space, embeddings of hyperplanes in higher-dimensional spaces, characterisation and cancellation problems on affine spaces, characterisation of non-cancellative varieties, studies on automorphism groups, rings of invariants of G_a-actions, structures and properties of affine fibrations, affine forms, retracts and finite generation of subalgebras of polynomial algebras, and allied problems. Some of these investigations are likely to involve intricate questions on computational commutative algebra like the determination of explicit generators of a certain ideal or a subring. Results on even partial cases of embedding problems had enormous consequences in recent years. While I have solved the Zariski Cancellation Problem in positive characteristic, the problem remains open in characteristic zero and some approaches to it are connected to questions around embedding problems. Thus even partial contributions to embedding problems will have significant impact. The project also envisages investigations on the ring of invariants of certain algebraic group actions on affine varieties. The determination of the structure and properties of such rings, especially a question on these rings related to Hilbert's 14th problem, are among the central problems on affine spaces. The theory of locally nilpotent derivations and exponential maps have played a significant role in solving open problems in affine algebraic geometry. I plan to continue to make further advancements with fruitful applications to problems in affine algebraic geometry. The area of affine fibrations seeks to derive information about the structure and properties of algebras from information about their fibre rings. A deep work of Asanuma shows that any affine fibration over a regular local ring is a stable polynomial ring. Results and examples on affine fibrations have an impact on research in cancellation and epimorphism problems. For instance, Asanuma's candidate for a counterexample to Zariski Cancellation Problem emerged from his research on affine fibrations. Therefore it is important to study affine fibrations in further details and the project proposes to do so.