Graduate topics course: Harmonic Analysis on Symmetric Spaces

Instructor: Simon Marshall

Instructor email: simon.marshall@unimelb.edu.au

Meeting time: Thursdays 1-3 pm Australian Eastern Time

Meeting dates: The first meeting will be held on July 30. The class will generally meet during the semester 2 teaching weeks at the University of Melbourne, which runs from July 27 to October 23. A full schedule can be found on the course canvas page (link below).

Location: Lectures will be in Peter Hall building, in room 107 for the first 2 weeks and room 162 thereafter. Room information is also listed on the course canvas page.

Zoom: Lectures will be streamed and recorded on zoom, at https://unimelb.zoom.us/j/87814660232?pwd=TInXrydbUSa1Acb2OTTGWShS7i7mfh.1

Mailing list: You can sign up to the mailing list for the class at https://lists.unimelb.edu.au/info/graduate-studies-a.

Canvas page: https://canvas.lms.unimelb.edu.au/courses/122542.

To get access to the canvas page, please do the following:

If you are not subscribed to the mailing list, please sign up to it.

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This will generate a list of email addresses for people who have recently been active on the mailing list, which will then be added to canvas. If you haven’t been added to canvas a week after doing this, please email the instructor.

Description

Globally symmetric spaces are a class of Riemannian manifolds with a transitive isometry group. The simplest example is hyperbolic space, but other examples include complex hyperbolic space, and the space of symmetric, positive definite, unimodular real matrices. There is an analogue of the Fourier transform, called the Harish-Chandra transform, on these spaces, and this class will provide an introduction to this transform. The main results covered will be the inversion and Plancherel formulas, and the Paley-Wiener theorem. We will discuss applications of these results to the study of eigenfunctions on locally symmetric spaces (which are the quotient of a globally symmetric space by an isometry group), including the trace formula and Weyl law, as well as other topics.

Globally and locally symmetric spaces are connected to several areas of mathematics, including:

• Analysis.
• Geometry: real and complex hyperbolic manifolds are examples of locally symmetric spaces.
• Number theory: Shimura varieties are also examples of locally symmetric spaces. Moreover, under some additional arithmetic assumptions, eigenfunctions on locally symmetric spaces form a class of automorphic forms called Maass forms.
• Representation theory: as the isometry group of a globally symmetric space is a semisimple Lie group, analysis on these spaces is linked to the representation theory of semisimple groups.
• Dynamics: the decay of Harish-Chandra spherical functions is connected to exponential mixing of the geodesic flow on the cotangent bundle of hyperbolic manifolds. Random matrix theory.

The material covered in this class may be of interest to people working in any of these areas.

Pre-requisites

Students should have the following background:

• Knowledge of Fourier analysis, including the inversion and Plancherel formulas, and Paley-Wiener theorem.
• A knowledge of basic results in distribution theory, including the definition of tempered distributions and their Fourier transform.
• Familiarity with the notion of a Riemannian manifold.

It is desirable, but not required that students know the definition of a Lie group and Lie algebra, including the exponential map and adjoint action. Please email the instructor if you want to take the class but do not have one of the pre-requisites, or if you have any other questions.