Circle Method (March, 2011)
The circle method finds its genesis in the first letter
of Srinivasa Ramanujan to G.H. Hardy written in January 1913.
This method was developed in the celebrated paper of Hardy and
Ramanujan on the partition function. It ws further
developed in a series of papers by Hardy and Littlewood in their
work on Waring's problem and Goldbach's conjecture.
In the first lecture,
we give a short survey of the method. In the second lecture,
we show its application to Waring's problem. In the third
lecture, we give a short sketch of Vinogradov's solution to
Goldbach's problem that every odd number can be written
as a sum of three primes.
Special values of zeta and L-functions (Dec, 2011)
We will survey known results concerning special
values of zeta and L-functions and discuss several open questions
in the area.