DescriptionFourier Theory is a powerful tool for studying functions, and it has found an array of applications both, theoretical (symmetric spaces, representation theory, automorphic forms) and practical (signal processing and other areas of engineering). On the other hand finite groups have attracted the interest of mathematicians not only becasue of their very interesting theoretical properties (classification, representation theory etc), but also for their many practical applications (error-correcting codes, cryptography etc).The main aim of this project is to apply the techniques of Fourier analysis to the study of finite groups and derive some very interesting applications. We will start with the book of Terras [4] and develop some basic concepts. From there the project can take various directions depending on interest. Below is a list of such possible directions or a combination of them is also possible. (For more see also the book of Terras). Applications to Error-Correcting Codes: One of the main results in the theory of error-correcting codes is the so-called MacWilliams identities which relate the weight distribution of a code with its dual. The main tool to derive such a relation is a Fourier transform (it is often called Hadamard transform in this setting). A vast generalisation of this can be achieved in the setting of Association Schemes (see also Wikipedia page) which have very interesting algebraic structure. Thanks to the work of Delsarte there is a very interesting connection with the theory of error-correcting codes. (Hamming Scheme, Johnson Scheme, etc) Gelfand Pairs and Representation Theory: Gelfand Pairs (see also Wikipedia page ) are met in representation theory of groups. In this project we will be mainly considering finite Gelfand pairs (that is representation theory of finite groups). Central to the study of Gelfand pairs are so-called spherical functions (Krawtchouk polynomials, Hahn polynomials, etc) and the spherical Fourier transform. These functions, and transforms play a prominent role in many applications. In some sense finite Gelfand Pairs can be thought as the discrete analogue of more advanced topics such as Symmetric Spaces. Random Walks on Distance-regular Graphs: Distance-regular graphs (see also Wikipedia page) are graphs with some nice properties. One can use Fourier analysis to study random walks on such graphs. These random walks model some very interesting diffusion processes such as the Bernoulli-Laplace, and the Ehrenfest diffusion process. The spherical functions mentioned above can be used to study the so-called cutoff phenomena of such models, namely to answer the question, how long does it take to mix things up?.
Resources
[2] Algebraic Combinatorics I, Association Schemes, E. Banai, T. Ito, The Benjamin/Cummings Publishing Company, 1984. [3] An algebraic approach to the association schemes of coding theory, P. Delsarte, Philips Research Reports, 1973. [4] Fourier analysis on finite groups and applications, A. Terras, CUP 1999. [5] Group Representations in Probability and Statistics , P. Diaconis, IMS Lecture Notes-Monograph Series, 1988. Pre-requisites
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