Artur Avila & Michael Th. Rassias 
Analysis at Large 
Dedicated to the Life and Work of Jean Bourgain

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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.

Table of Content

On the joint spectral radius (E. Breuillard).- The failure of the fractal uncertainty principle for the Walsh-Fourier transform (C. Demeter).- The continuous formulation of shallow neural networks as Wasserstein-type gradient flows (X. Fernández-Real).- On the Origins, Nature and Impact of Bourgain’s Discretized Sum-Product Theorem (A. Gamburd).- Cartan Covers and Doubling Bernstein Type Inequalities on Analytic Subsets of C2 (M. Goldstein).- A Weighted Prekopa-Leindler inequality and sumsets with quasicubes (B. Green).- Equidistribution of affine random walks on some nilmanifolds (E. Lindenstrauss).- Logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrodinger operators with smooth potentials (S. Jitomirskaya).- The slicing problem by Bourgain (B. Klartag).- On the work of Jean Bourgain in nonlinear dispersive equations (E. Kenig).- On Trace sets of restricted continued fraction semigroups (A. Kontorovich).- Polynomial Equations in Subgroups and Applications (V. Konyagin).- Exponential sums, twisted multiplicativity and moments (E. Kowalski).- The ternary Goldbach problem with a missing digit and other primes of special types (Th. Rassias).- A note on harmonious sets (Y. Franc cois Meyer).- On the multiplicative group generated by two primes in Z/QZ (P. Varju).

About the author

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).