## A course about the connections between Harmonic Analysis, Algebraic Geometry, Topology, and Representation Theory

University of Sydney 1996

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

Last updated: 27 November 2014

## Course description

Beginning on Monday 26 February 11am in Carslaw 350. In general the course will meet Monday 11am in Carslaw 350, Wednesday 10am in Carslaw 353, and Wednesday 2pm in Carslaw 830.

My intention is to begin by working through the notes of Macdonald on *Spherical functions on p-adic Lie groups*. By going through these notes I am hoping
that we will all learn how to feel comfortable working with
Spherical functions

Fourier inversion

${L}^{2}\left(G\right)$

${L}^{2}\left(G\text{.}K\right)$

Root systems

Affine root systems

Tits buildings

Chevalley groups

Hecke rings
and many other things that come up in many different fields of mathematics.

I will not assume much in the way of prerequisites, although I will assume a bit of mathematical maturity, i.e. an appreciation that mathematics is a beautiful thing
and some willingness to stare at a boardful of definitions. As far as mathematical prerequisites go, it will probably be helpful to be familiar with some basic group
theory, (groups, homomorphisms, cosets, normal subgroups) and some basic topology (open and closed sets, connectedness, compactness, discrete topology). I will review
any these basic things whenever necessary (for myself or for others).

The material in this course is designed to give an introduction to the techniques and concepts that have been crucial to some very exciting recent results of
Cherednik, Opdam, Lusztig and others which used the affine Hecke algebras to prove the famous Macdonald conjectures. I am also hoping to get a new understanding
of the Springer representations by these methods. The main idea is to have some fun!

Arun Ram

## Lecture Notes and class materials

## Notes and References

These are a typed copy of a scanned page entitled *A course about the connections between Harmonic Analysis, Algebraic Geometry, Topology, and Representation Theory*.

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