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). From a number theoretical point of view a crucial fact is that the space of modular forms of a given weight on Γ is finite dimensional, thereby allowing to prove any given identity among them almost mechanically. 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.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/IV 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