Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents

Abstract. A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory. 1. Background For hundreds of years, mathematicians have made conjectures about patterns in the primes: one of the simplest to state is that the primes contain arbitrarily long arithmetic progressions. It is not clear exactly when this conjecture was first formalized, but as early as 1770 Lagrange and Waring studied the problem of how large the common difference of an arithmetic progression of k primes must be. A

Abstract. We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {p, 2p,..., kp}. We then derive several combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all ε> 0 and every subset of the integers Λ the set n ∈ N: d ∗ ( Λ ∩ (Λ + p1(n)) ∩ (Λ + p2(n)) ∩ (Λ + p3(n)) )> (d ∗ (Λ)) 4 − ε} has bounded gaps for “most ” choices of integer polynomials p1, p2, p3. Contents

Let P = {p1,..., pr} ⊂ Q[n1,..., nm] be a family of polynomials such that pi(Zm) ⊆ Z, i = 1,..., r. We say that the family P has the PSZ property if for any set E ⊆ Z with d ∗ |E∩[M,N−1]| (E) = lim supN−M→ ∞ N−M> 0 there exist infinitely many n ∈ Zm such that E contains a polynomial progression of the form {a, a + p1(n),..., a + pr(n)}. We prove that a polynomial family P = {p1,..., pr} has the PSZ property if and only if the polynomials p1,..., pr are jointly intersective, meaning that for any k ∈ N there exists n ∈ Zm such that the integers p1(n),..., pr(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If p1,..., pr ∈ Q[n] are jointly intersective integral polynomials, then for any finite partition Z = ⋃k i=1 Ei of Z, there exist i ∈ {1,..., k} and a, n ∈ Ei such that {a, a+p1(n),..., a+pr(n)} ⊂ Ei.

Abstract. For any measure preserving system (X, X, µ, T) and A ∈ X with µ(A)> 0, we show that there exist infinitely many primes p such that µ ` A ∩ T −(p−1) A ∩ T −2(p−1) A ´> 0 (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 (µ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p. 1.

Abstract. Szemerédi’s Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s Theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.

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Abstract. We prove L p boundedness of certain non-translation-invariant discrete maximal Radon transforms and discrete singular Radon transforms. We also prove maximal, pointwise, and L p ergodic theorems for certain families of non-commuting operators.

Abstract. Szemerédi’s Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized it, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s Theorem corresponds to the linear case of this polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We give a multiset extension of the polynomial theorem and derive several combinatorial consequences, including lower bounds for the size of the intersection and an “anti-Ramsey ” rainbow result. We also prove a structure theorem for the polynomial multicorrelation sequences ∫ f0 · T p1(n) f1 ·... · T pk(n) fk dµ. 1.