Research Projects

This project aims to explore major research areas in Affine Algebraic Geometry, focusing on the Zariski Cancellation Problem, necessary and sufficient conditions for finite generation of kernels of locally nilpotent derivations on the affine 4-space, automorphism groups of affine 3 and 4 spaces, and the generalized linearization problem for a k-dimensional torus action on the affine n-space. Techniques in the theory of locally nilpotent derivations, exponential maps, affine fibrations, torus actions on algebraic varieties, graded automorphisms, and related areas will be applied. The research will have major implications in the areas of Cryptography, as the Anshel-Anshel-Goldfeld key exchange protocol is based on the complexity of the conjugation element search problem in the group under consideration. The conjugation element search problem in the groups of automorphisms of the affine n-space, subgroup of tame automorphisms of the affine n-space, group of automorphisms of an affine variety, and subgroup of special automorphisms of an affine variety will be studied. One-way functions will be studied to address the factorization problem in automorphism groups of affine varieties, and the decomposition of elements of SAutX into exponents of locally nilpotent derivations.