Take a look at pictures from the 2004 Istanbul Abstract Harmonic Analysis Conference held at Koc University.

### Abstract Harmonic Analysis

Abstract Harmonic Analysis refers to all aspects of analysis on locally compact groups. This class of groups is particularily suitable for analysis due to the existence of a left (or a right) invariant measure, called Haar measure, on any such group. Among other things, Haar measure is a basic tool in developing the representation theory of locally compact groups.

The class of locally compact groups is large enough to encompass many/most of the topological groups that arise in physics, geometry, number theory and other areas of mathematics and the natural sciences. Moreover, any abstract group is a topological group when endowed with the discrete toplogy. In all these examples of interesting groups, the representations of the groups have meaningful interpretations and applications.

I am particularly interested in the dual spaces of locally compact groups and the influence that the topological properties of the dual space have on the analysis of functions on the group.

If G is a locally compact group, the set of equivalence classes of irreducible unitary representations of G is denoted G^ and is called the dual space of G. There is a natural topology on G^ which usually fails to be Hausdorff. It is the non-Hausdorff character of most dual spaces that leads to much of the fun.
By clicking on the image below, you will (hopefully) see a big drawing of one of my favorite dual spaces. The group whose dual is pictured below is the fundamental group of the Klein bottle. It happens to also be one of the 17 two dimensional crystal groups and in that collection carries the name pg, so that is what I will call it here.
pg^ = Click on the small picture to get a full view.
Notes:
• The topological space pg^ has a dense open subset (the orange cylinder) that is homeomorphic to the cartesian product of S1 and (-1, 1).
• A copy of S1 is glued to each end of the cylinder, but in a double cover fashion like the boundary of a Moebius band.
• For 2 different points on one of these copies of S1, I have drawn a small basic neighbourhood. You can see that if these points were side-by-side, so to speak, they could not be separated.
• Each point in the orange cylinder corresponds to a two dimensional irreducible representation of pg, while each point on the boundary copies of S1 corresponds to a one dimensional irreducible representation.
• For a detailed treatment of pg and a description of its group C*-algebra, see Ken Davidson's excellent new book C*-Algebras by Example, Fields Institute Monographs 6, AMS Publishers.
Another two dimensional crystal group that has an interesting dual is pgg. A picture of pgg^ is shown below. I will leave it to you to try and sort out how the non-Hausdorff nature of the topology is being represented. The central square is an open square of 4-dimensional representations. The interesting pattern in the central square is just a jpeg artifact that means nothing. The points on the horizontal and vertical solid lines each represent 2-dimensional representations. The upper left and lower right big dots are each 2-dimensional representations and all the other dots are 1-dimensional.