Dynamical Approaches in Number Theory - TCC module
Starting with of in 1977, a rich story of cross-pollination of ideas between ergodic theory, multiplicative number theory and additive combinatorics developed. This course will cover some of the highlights in this story and the connections between them. Topics covered, depending on time and students' interest, may include
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The development of in ergodic theory paralleling in additive combinatorics.
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The that the primes contain arbitrarily long arithmetic progressions.
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The subsequent program to study , and how it required an understanding of the behaviour of certain special dynamical systems (nilsystems).
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Sarnak's influential conjecture on and the relation with an older .
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The special case of Chowla's conjecture , the relation with , and the relation with Host-Kra uniformity of multiplicative functions.
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of Green and Tao's work, showing that aperiodic multiplicative functions are Gowers uniform, and to the problem of partition regularity of Pythagorean triples.
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The of Bergelson and Richter, which led to of the classical prime number theorem.
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in sets of positive density and .