Type: Seminar

Distribution: World

Expiry: 1 Aug 2017

CalTitle1: MaPSS: Mathematical Postgraduate Seminar Series

Auth: casella@10.17.26.133 (acas5565) in SMS-WASM

Dear All, We are delighted to present the MaPSS Seminar topic of Monday 08/05; please see the abstract below. **This Semester the Seminar will always run on Monday, at 5:00pm in 535A** Following the talk, there will be pizza on offer. Speaker: Philip Bos (Sydney University) Title: Modular Forms and Number Theory - an insight into the Ramanujan conjectures Abstract: The vector space of modular forms will be explained as complex-valued functions on the upper half plane with periodic-like properties. They are in some sense the hyperbolic geometrical equivalent of periodic functions of Euclidean space. We will explain that sense. As periodic functions on the one hand, we can develop a Fourier series expansion for modular forms. As a vector space on the other, they permit linear operators and in the 1930 the German mathematician, developed the so-called Hecke operators. With such a development, we discover that the vector space of Hecke operators form a unitary commutative algebra called the Hecke Algebra. When we apply these two results together, we can solve significantly difficult analytic number theory questions. The gifted Ramanujan had great insights into such relationships but could not prove all his conjectures. We will give an example of the Ramanujan tau function and outline the method of Hecke that shows the proof as a "natural" consequence of the above ideas. This is a far as we will go in our talk, though Hecke’s student Petersson went further showing Hecke operators are Hermitian with respect to the Petersson inner product, allowing us to derive bases for the vector spaces of modular forms. Continuing in this direction and far further allowed Andrew Wiles to solve Fermat’s Last Theorem. Supervisors, please encourage your students to attend. Thanks, MaPSS Organizers

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