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Analysis at Large

Dedicated to the Life and Work of Jean Bourgain

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  • © 2022

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Editors:
  1. Artur Avila

    1. Institute of Mathematics, University of Zurich, Zürich, Switzerland

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  2. Michael Th. Rassias

    1. Department of Mathematics and Engineering Sciences, Hellenic Military Academy, Athens, Greece

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  3. Yakov Sinai

    1. Department of Mathematics, Princeton University, Princeton, USA

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  • A wide spectrum of subjects influenced by Bourgain’s monumental contributions to mathematics
  • Pays homage to the life and work of a true pioneer of mathematics
  • Contributions by leading experts present the state of the art in the problems treated

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. On the Joint Spectral Radius

    • Emmanuel Breuillard
    Pages 1-16

  3. The Failure of the Fractal Uncertainty Principle for the Walsh–Fourier Transform

    • Ciprian Demeter
    Pages 17-28

  4. The Continuous Formulation of Shallow Neural Networks as Wasserstein-Type Gradient Flows

    • Xavier Fernández-Real, Alessio Figalli
    Pages 29-57

  5. On the Origins, Nature, and Impact of Bourgain’s Discretized Sum-Product Theorem

    • Alexander Gamburd
    Pages 59-100

  6. Cartan Covers and Doubling Bernstein-Type Inequalities on Analytic Subsets of \(\mathbb {C}^2\)

    • Michael Goldstein, Wilhelm Schlag, Mircea Voda
    Pages 101-124

  7. A Weighted Prékopa–Leindler Inequality and Sumsets with Quasicubes

    • Ben Green, Dávid Matolcsi, Imre Ruzsa, George Shakan, Dmitrii Zhelezov
    Pages 125-129

  8. Equidistribution of Affine Random Walks on Some Nilmanifolds

    • Weikun He, Tsviqa Lakrec, Elon Lindenstrauss
    Pages 131-171

  9. Logarithmic Quantum Dynamical Bounds for Arithmetically Defined Ergodic Schrödinger Operators with Smooth Potentials

    • Svetlana Jitomirskaya, Matthew Powell
    Pages 173-201

  10. The Slicing Problem by Bourgain

    • B. Klartag, V. Milman
    Pages 203-231

  11. On the Work of Jean Bourgain in Nonlinear Dispersive Equations

    • Carlos E. Kenig
    Pages 233-251

  12. On Trace Sets of Restricted Continued Fraction Semigroups

    • Alex Kontorovich
    Pages 253-272

  13. Polynomial Equations in Subgroups and Applications

    • Sergei V. Konyagin, Igor E. Shparlinski, Ilya V. Vyugin
    Pages 273-297

  14. Exponential Sums, Twisted Multiplicativity, and Moments

    • E. Kowalski, K. Soundararajan
    Pages 299-332

  15. The Ternary Goldbach Problem with a Missing Digit and Other Primes of Special Types

    • Helmut Maier, Michael Th. Rassias
    Pages 333-362

  16. A Note on Harmonious Sets

    • Yves François Meyer
    Pages 363-371

  17. On the Multiplicative Group Generated by Two Primes in Z∕Q Z

    • Péter P. Varjú
    Pages 373-388
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Keywords

About this book

​Analysis at Large is dedicated to Jean Bourgain whose research has deeply influenced the mathematics discipline, particularly in analysis and its interconnections with other fields. In this volume, the contributions made by renowned experts present both research and surveys on a wide spectrum of subjects, each of which pay tribute to a true mathematical pioneer. Examples of topics discussed in this book include Bourgain’s discretized sum-product theorem, his work in nonlinear dispersive equations, the slicing problem by Bourgain, harmonious sets, the joint spectral radius, equidistribution of affine random walks, Cartan covers and doubling Bernstein type inequalities, a weighted Prékopa-Leindler inequality and sumsets with quasicubes, the fractal uncertainty principle for the Walsh-Fourier transform, the continuous formulation of shallow neural networks as Wasserstein-type gradient flows, logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrödinger operators, polynomial equations in subgroups, trace sets of restricted continued fraction semigroups, exponential sums, twisted multiplicativity and moments, the ternary Goldbach problem, as well as the multiplicative group generated by two primes in Z/QZ.

It is hoped that this volume will inspire further research in the areas of analysis treated in this book and also provide direction and guidance for upcoming developments in this essential subject of mathematics.

Editors and Affiliations

  • Institute of Mathematics, University of Zurich, Zürich, Switzerland

    Artur Avila

  • Department of Mathematics and Engineering Sciences, Hellenic Military Academy, Athens, Greece

    Michael Th. Rassias

  • Department of Mathematics, Princeton University, Princeton, USA

    Yakov Sinai

About the editors

Artur Avila: Foreign Associate of the U.S. National Academy of Sciences (2019), Fields Medal (2014), TWAS Prize (2013), Michael Brin Prize in Dynamical Systems (2011), Plenary address at the ICM (2010), Grand Prix Jacques Herbrand of the French Academy of Sciences (2009), EMS Prize (2008), Salem Prize (2006), Bronze medal of the CNRS (2006), Gold Medal at the International Mathematical Olympiad (1995)
Michael Th. Rassias: Latsis Foundation Senior Fellowship (2018-2019), Certificate of Special Recognition for contributions to Mathematics and its Applications, ΣΣΕ, Athens (2016), Notara Prize, Academy of Athens (2014), C. Papakyriakopoulos Prize, NTUA (2006-2007), Silver Medal at the International Mathematical Olympiad (2003), Silver Medal at the Balkan Mathematical Olympiad (2002)


Yakov Sinai:  Abel Prize (2014), Leroy P. Steele Prize (2013), Foreign Member of the Royal Society (2009), Henri Poincaré Prize (2009),Lagrange Prize (2008), Nemmers Prize (2002), Wolf Prize (1997), Dirac Prize (1992), Dannie Heineman Prize (1990), Boltzmann Medal (1986).



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