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: Harmonic Analysis in the setting of capacities
Description: Capacities occur in the study of the structure of rough and exceptional sets arising in partial differential equations. As a mathematical object, a capacity mimics an outer measure on the Euclidean space and therefore gives a notion of integration that is not linear. Recently, certain aspects of Harmonic Analysis, such as the study of maximal operators, Lebesgue and BMO spaces, was initiated by certain groups of Mathematicians. In this project, we wish to continue looking at the Harmonic Analysis aspects of capacities (particularly Hausdorff capacities, which play an important role in Geometric Measure Theory and the study of fractals). We wish to study weighted boundedness of capacitary maximal functions, singular integrals related to capacities and many other directions.
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.