Trjitzinsky Memorial Lectures
September 19-21, 2005
Jean
Bourgain
Institute for Advanced Study
will present
A combinatorial approach to exponential
sum estimates and applications
All lectures will be held at 4:00 p.m. in 245 Altgeld Hall
A reception will be held from 5:10 - 6:00 p.m. in Room 321 Altgeld Hall on Monday, September 19.
Over recent years, there has been a certain amount of research around
'sum-product phenomena', that roughly expresses the property that the
sumset A+A and product set A*A can't be both small. This is in particular
the case in finite fields of prime order. This combinatorial device offers
a new way to establish non-trivial bounds on exponential sums (in particular
Gauss sums, Mordell type sums and sums involving exponential functions)
in situations that seem beyond reach of classical methods such as Stepanov's
approach. There is an extended list of applications to number fields,
cryptography and problems involving pseudo-randomness that will be reviewed.
For instance, new results on the validity of the Diffie-Hellman indistinguishability
assumption are obtained and uniform distribution results for RSA generators.
Also the relevance to a recent line of research on the construction of deterministic extractors will be discussed. |
