Pure Mathematics Topics
Topic Title: Singular integrals beyond Calderon-Zygmund theory and applications
Supervisor: Dr The Anh Bui
One of the major themes of harmonic analysis is the study singular integrals on certain function spaces which rely on the theory of Calderon-Zygmund operators. We emphasize that the Calderon-Zygmund theory of singular integrals plays an important role in analysis and has a strong influence on other branches of mathematics including the study of partial differential equations. However, in practice there are a number of important settings which do not fall within the scope of the Calderon-Zygmund theory. The main aim of this project is to study the boundedness of singular integrals beyond the Calderon-Zygmund theory and applications in partial differential equations.
Topic Title: New mathematical theory for fluid motion on surfaces with holes
Supervisor: Dr Chris Green
The purpose of this project is to explore and develop new methods and theory for ideal fluid motion on surfaces with holes (compact Riemann surfaces of genus greater than zero). There are many possible directions the project could take depending on the particular interests of the student. The project will touch on a mix of mathematical topics, including complex analysis, special function theory, potential flows, and differential geometry.
Topic Title: Riesz transform for Grushin-type operators
Supervisor: A/Prof Adam Sikora