Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.

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Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes. 1.

Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.

Abstract. We survey recent developments on the Restriction conjecture.