DescriptionThis is a project in Pure mathematics whose main aim is to study the interplay between three different areas: Algebraic Combinatorics, Algebra (in particular representation theory of finite groups) and Random Walks on Graphs. This interplay has been proved quite fruitful and has led to some very interesting applications in Error-Correcting Codes (linear programming bounds) and (finite) Diffusion Processes (cutoff phenomena).Association Schemes (see also Wikipedia page) have their origin in statistics and were used to design experiments but it was soon realised that they have very interesting algebraic structure and now are considered as part of algebraic combinatorics. They attracted more interest after the work of Delsarte which established a very interesting connection with the theory of error-correcting codes. (Hamming Scheme, Johnson Scheme, etc) 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). Gelfand pairs provide a big pool of interesting examples of association schemes and they are very helpful in the study of the so-called spherical functions (Krawtchouk polynomials, Hahn polynomials, etc) and spherical Fourier transform, which play a prominent role in the theory. In some sense finite Gelfand Pairs can be thought as the discrete analogue of more advanced topics such as Symmetric Spaces. Distance-regular Graphs (see also Wikipedia page) are graphs with some nice properties, which can be understood as association schemes when one considers the graph distance to form the associations. As an example we mention the hypercube graph which is related to the Hamming scheme. Random walks on such graphs can be used to model some very interesting diffusion processes such as the Bernoulli-Laplace, and the Ehrenfest diffusion process. The first is related to the Johnson scheme mentioned above and the second to the Hamming. 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?. Depending on interest the project may take the following directions or a combination of them. 1) Association schemes and error-correcting codes. 2) Random walks on distance-regular graphs and cutoff phenomena. 3) Study of finite Gelfand Pairs and their spherical functions.
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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