´ RECURRENCE AND NUMBER THEORY: POINCARE THIRTY YEARS LATER BRYNA KRA Hillel Furstenberg’s 1981 article in the Bulletin gives an elegant introduction to the interplay between dynamics and number theory, summarizing t

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COMPLEXITY AND DIRECTIONAL ENTROPY IN TWO DIMENSIONS RYAN BRODERICK, VAN CYR, AND BRYNA KRA Abstract. We study the directional entropy of a dynamical system associated to a Z2 configuration in a finite alphabet. We show

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CONVERGENCE OF MULTIPLE ERGODIC AVERAGES FOR SOME COMMUTING TRANSFORMATIONS NIKOS FRANTZIKINAKIS AND BRYNA KRA Abstract. We prove the L2 convergence for the linear multiple ergodic averages of commuting transformations T

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´ THE POLYNOMIAL MULTIDIMENSIONAL SZEMEREDI THEOREM ALONG SHIFTED PRIMES NIKOS FRANTZIKINAKIS, BERNARD HOST, AND BRYNA KRA Abstract. If ~q1 , . . . , ~qm : Z → Z` are polynomials with zero constant terms and E ⊂

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NONEXPANSIVE Z2 -SUBDYNAMICS AND NIVAT’S CONJECTURE VAN CYR AND BRYNA KRA Abstract. For a finite alphabet A and η : Z → A, the Morse-Hedlund Theorem states that η is periodic if and only if there exists n ∈ N suc

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UNIFORMITY SEMINORMS ON `∞ AND APPLICATIONS BERNARD HOST AND BRYNA KRA Abstract. A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or

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1 1 Multiple recurrence and nilsequences Vitaly Bergelson1 , Bernard Host2 , Bryna Kra3 with an Appendix by Imre Ruzsa4

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From combinatorics to ergodic theory and back again Bryna Kra∗ Abstract. Multiple ergodic averages, such as the average of expressions like f1 (T n x) f2 (T 2n xfk (T kn x), were first studied in the ergodic th

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VARIATIONS ON TOPOLOGICAL RECURRENCE BERNARD HOST, BRYNA KRA, AND ALEJANDRO MAASS Abstract. Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topologica

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GIVING A TALK BRYNA KRA No one likes to sit through a bad talk, but unfortunately everyone does it much too frequently. And no one sets out to give a bad talk, but probably all of us have done so. Paul Halmos [1] wrote a

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