Title: k-plane Transforms, related operators and their properties on Constant Curvature Spaces
Description: We will be looking at the X-Ray transform, its generalization, the Radon transform, and its further generalization, the k-plane transforms. A lot of research has been done in the field of reconstructing a function using the k-plane transforms on the Euclidean spaces. We will be interested in some properties of these transforms and their behaviour on (smooth) manifolds and spaces of constant curvature.
This project is a part of the PhD thesis, and collaboration is not possible for now.
Title: Boundedness of hypersingular Integrals along Hypersurfaces
Description: We are interested in knowing the L^p-boundedness of hypersingular integrals defined along hypersurfaces. Starting from the codimension 1 case, we would like to study the problem when the said hypersurface is of arbitrary codimension in a Euclidean space. The project also deals with understanding the curvature properties of the surfaces that allow L^p-boundedness. Further, we would also like to investigate whether we have boundedness of such integrals in other function spaces (such as L^p-L^q or Lorentz spaces).
Title: Boundedness of hypersingular integrals along lower-dimensional surfaces
Description: We are interested in studying hypersingular integral operators defined along (curves and) surfaces. Some work related to specific curves and hypersurfaces is available in the literature. We wish to generalise this to more general surfaces of any codimension.
Title: Topology of Non-Triangular Metric Spaces
Description: Recently, in 2020, the concept of non-triangular metric spaces was given. This new concept generalizes metric spaces by omitting the triangle inequality axiom. We will be interested in studying the topology generated by this new definition of "distance function". Eventually, we could answer questions about compactness, connectedness, continuity of functions and even the metrizability of non-triangular spaces.
Title: Topology of Non-Triangular Metric Spaces
Description: Recently, in 2020, the concept of non-triangular metric spaces was given. This new concept generalizes metric spaces by omitting the triangle inequality axiom. We will be interested in studying the topology generated by this new definition of "distance function". Eventually, we could answer questions about compactness, connectedness, continuity of functions and even the metrizability of non-triangular spaces.