# Jerusalem Mathematics Colloquium

Thursday, 29th November 2007, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Tamar Ziegler

(Technion)

"Polynomial progressions in primes"

** Abstract: **

In 1975 Szemeredi proved that any subset of the integers of positive density
contains arbitrarily long arithmetic progression. A couple of years later
Furstenberg gave an ergodic theoretic proof for Szemeredi's theorem. At
around the same time Furstenberg and Sarkozy independently proved that any
subset of the integers of positive density contains a perfect square
difference, namely elements x,y with x-y=n^2 for some positive integer n.
In 1995, Bergelson and Leibman proved, using ergodic theoretic methods, a
vast generalization of both Szemeredi's theorem and the Furstenberg-Sarkozy
theorem, establishing the existence of arbitrarily long polynomial
progression in subsets of the integers of positive density.

The ergodic theoretic methods are limited, to this day, to handling sets of
positive density. However, in 2004 Green and Tao proved that the question of
finding arithmetic progressions in some special subsets of the integers of
zero density - for example the prime numbers - can be reduced to that of
finding arithmetic progressions in subsets of positive density. In recent
work with T. Tao we show that one can make a similar reduction for
polynomial progressions, thus establishing, through the Bergelson-Leibman
theorem, the existence of arbitrarily long polynomial progressions in
the prime numbers.

Light refreshments will be served in the faculty lounge at 3:30.

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