PCMEA: Pointwise Convergence of Multiple Ergodic Averages

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The study of multiple ergodic averages has a long and rich history, dating back at least to H. Furstenberg's ergodic theoretic proof of Szemerédi's Theorem. The first breakthrough in multiple ergodic averages along non-linear polynomial orbits was due to H. Furstenberg and B. Weiss in the 1990s, which lead to a flurry of research, ultimately leading to M. Walsh's celebrated convergence result, an optimal convergence result in the category of norm convergence of multiple ergodic averages.Understanding convergence phenomena in the pointwise setting is far more subtle. Indeed, the issue of pointwise convergence of single function along a polynomial orbit remained open until breakthrough work of Bourgain in the late 1980s and early 1990s; the issue of norm convergence follows from elementary Hilbert space orthogonality arguments.In recent work with M. Mirek (Rutgers University) and T. Tao (UCLA), I established the first pointwise convergence result in the category of multiple (non-linear) polynomial ergodic averages, namely a pointwise bilinear polynomial ergodic theorem.The theme of this proposal is to develop the theory of pointwise convergence of multiple polynomial ergodic averages in as close an analogy to Walsh's result as possible. Although the questions posed are dynamical, results of this line of inquiry will lead to major advances in additive combinatorics, combinatorial number theory, and harmonic analysis.