2006/07 Clay Lectures on Mathematics
Lectures will be hosted by:
Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
Akshay Venkatesh (Courant Institute)
Harmonic analysis on the circle has been one of the central tools of analytic number theory. An early triumph was the 1918 paper Hardy-Ramanujan, giving an exact formula for the number of partitions of a large integer (e.g. 4=3+1=2+1+1=2+2=1+1+1+1 has five partitions). However, already in this paper, modular forms make an appearance, behind them lurks the nonabelian group SL2(R).
The theme of the lectures will be the role of nonableian symmetry groups and nonabelian harmonic analysis in analytic number theroy. With that in mind, Professor Venkatesh will give a discussion of some beautiful results from the last century, as well as a brief survey of current development.
Some theorems of Hardy, Littlewood and Ramanujan. Partitions
and sums of squares
Tuesday, November 28 at 4:30pm
Some theorems of Linnik, Duke and Iwaniec
Wednesday, November 29 at 4:30pm
Adding square numbers (Public lecture)
Thursday, November 30 at 8:00pm
The theme of the lecture will be adding together square numbers (1,4,9,16,25...) This simple operation gives rise to complex and beautiful patterns, which have motivated mathematicians from ancient times to the present.
A survey of
Friday, December 1, at 4:30 pm
Ben Green (University of Cambridge, CMI)
Additive Combinatorics is the name given to a branch of number theory concerned with additive properties of sets of integers. If a set A is somewhat closed under addition, what is the structure of A? What do we need to know about A in order to be able to locate very regular structures, such as arithmetic progressions inside A? How does the Fourier transform of A reflect the additive structure of A? Professor Green will talk about these questions and others, to elucidate this rapidly-developing area of mathematics.
The structure theory of set addition. Freiman's
Tuesday, November 28 at 5:45pm
Gowers norms and nilsequences
Wednesday, November 29 at 5:45pm
Adding prime numbers (Public lecture)
Thursday, November 30 at 6:30pm
It has on occasion been noted that it is more natural to multiply primes than to add them. However many famous open problems in number theory are concerned with adding primes, and the study of these problems has led to some fascinating mathematics. I hope to give a flavour of some of this in my talk.
The idempotent theorem: an
application of additive combinatorics to harmonic analysis
Friday, December 1 at 5:45 pm