|Office Hours||T-TH: 11:00-12:00, 3:00-4:00 pm, and by request|
|Phone||578-1608 and 225-337-2206 (cell)|
|Text||Noncommutative Harmonic Analysis, An Introduction by R. Fabec and G. Olafsson|
You can download the lecture note at Noncommutative Harmonic Analsysis using your departmental login and password. The aim is to cover variuous material from Capter 1-4. If there is time, then we will also introduce material from other chapters.There will be 2-3 homework due at least every second week and one longer list of homeworks around middterm. There will be a take home final due on Monday, Dec. 6, at 7:30 pm. You can replace the final by a 45 min. lecture on a selected topic.
The subject proper of harmonic analysis is to decompose functions into ''simpler'' functions, where the meaning of simple depends on the context we are working in. In the theory of differential equations that means to write an arbitrary functions as a superposition of eigenfunctions. If we have a symmetry group acting on the system, then we would like to write an arbitary function as a sum of functions that transforms in a simple and controllable way under the symmetry group. The simplest example is the use of polar coordinates and radial functions for rotation symmetric equations. Sometimes, we are working with general spaces than Rn. Here anlysis meets with Lie groups, geometry, representation theory and harmonic analysis to form abstract harmonic analysis.
The course begins with a short overview of classical Fourier analysis on the torus and Rn. This leads us to topics like:
We can view Rn as a set or as amanifold. But we can also view it as an abelian group. In that sense Rn is a part of abelian harmonic analysis. The simplest example of nonabelian harmonic analysis is the Heisenberg group Hn=R2n+1 (with a new group multiplication). The Heisenberg group is also a simple example of a Lie group and of a topological group . There are several other well known examples of topological groups like the group of rotations SO(n) , the group of all invertible matrices GL (n,R). Depending on the time and interest we will at the end discuss some advanced topics related to topological groups.
Representations of topological groups are central in several branches of mathematics: In number theory and the study of authomorphic functions and forms, in geometry as a tool to construct important vector bundles and differential operators, and in the study of Riemannian symmetric spaces. Finally, those are important tools in analysis, in particular analysis on some special homogeneous manifolds like the sphere, Grassmanians, the upper half plane and its generalizations. Representations even shows up in branches of applied mathematics as generalizations of the windowed Fourier transform and wavelets. Several examples of those applications in analysis and geometry will be discussed in the class.
Professor He will offer a more advanced course during the spring 2011.