MaPSS: Mathematical Postgraduate Seminar Series: Philip Bos (Sydney University) -- Modular Forms and Number Theory - an insight into the Ramanujan conjectures

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