DescriptionModular forms are functions in the complex upper half-plane with `infinitely many symmetries' in that they transform in a simple way under the action of a `discrete' subgroup Γ of SL(2, R) such as SL(2, Z). One of the most well known examples of a modular form is given by the famous Riemann zeta function. From a more general mathematical point of view one realises that modular forms occur quite naturally in an amazing variety of research areas in mathematics and physics. Hence one should rightfully expect a great impact and nice applications of modular forms in many other fields, including random matrix theory and mathematical physics. The project will begin with an
introduction to the hyperbolic space, upper half plane
model, modular subgroup, general Fuchsian subgroups,
fundamental domains, modular functions, Fourier series
of modular forms, theta multipliers. After this, it will
be possible to consider various further directions
including:
Prerequisites: Knowledge of Algebra 2 is necessary.
The knowledge of ENT 2 and NT III will help. Textbooks: H. Iwaniec, Topics in classical automorphic forms,
Graduate Studies in Mathematics, Volume 17. H. Iwaniec, Introduction to the spectral theory
of automorphic forms, Graduate studies in
Mathematics, Volume 53.
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email: Pankaj Vishe