Many problems in the analysis of (partial) dierential equations are of a functional calculus nature. They involve giving a satisfying meaning to f(L), for an operator L acting on a Banach space X, and certain functions f. For example, the exponential of matrices can be used to solve systems of ordinary dierential equations, while the exponential of the Laplace operator gives solutions to the heat equation. Functional analysis includes several fundamental constructions of functional calculi for abstract operators, such as the spectral theorem which gives a functional calculus for Borel functions and self-adjoint operators on Hilbert spaces. However, these pure operator theoretic methods are usually not sucient to solve interesting problems. To exploit the dierential nature of the operators involved in applications, one uses harmonic analysis, and, in particular, Calderon-Zygmund theory.
A honours project on this topic focuses on a specic class of operators, arising in PDE, probability, and/or geometry. The specic case is chosen based on the student background
and interests. The fundamental prerequisite course is MATH3325.
The notes of my 2011 MATH3349 course (available at http://maths-people.anu.edu.au/~portal/page/teach.html) can give you a good feel for the topic.