MATH5525 Noncommutative Harmonic Analysis

MATH5525 is a Special Topics course in Pure Mathematics course for Postgraduate or Honours students. See the course overview below.

Units of credit: 6

Prerequisites: Cycle of offering: Course is not offered every year.

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

More information:

 This recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains information about the course. (The timetable is only up-to-date if the course is being offered this year.)

If you are currently enrolled in MATH5525, you can log into UNSW Moodle for this course. 

Course Overview

What becomes of Fourier series and transforms when you abandon the familiar real line and circle group? It turns out that it is possible to define a notion of Fourier transform in the setting of a locally compact group of symmetries. There is a good Plancherel theory for some of these groups. Nearly every transform you have ever studied falls under the Fourier transfom on some Lie group: Laplace transforms, Bessel transforms, etc. and the familiar expansions in terms of exponential functions are replaced by expansions in other exotic special functions. The key to understanding all this is the study of representation theory for continuous groups. This also lies at the heart of the study of quantum mechanics, number theory and other arcane disciplines.

This is a huge subject, of which we can only see a fraction, but we will strive for some kind of overview. Techniques are used from algebra, topology, geometry, functional analysis and operator theory.