William Beckner

William Beckner is a professor of mathematics and known for his significant contributions to harmonic analysis and geometric inequalities. He holds the Paul V. Montgomery Centennial Memorial Professorship in Mathematics at The University of Texas at Austin. Beckner earned his degree in physics from the University of Missouri in 1963. He pursued his Ph.D. in mathematics at Princeton University, completing it in 1975 under the mentorship of Elias M. Stein. Following his doctoral studies, he engaged in postdoctoral research at the University of Chicago under the guidance of A.P. Calderón.

Beckner's academic career includes faculty positions at the University of Chicago. He joined UT Austin in 1983. His research focuses on the interplay of symmetry, smoothness, and uncertainty in characterizing structures of geometric manifolds. He has developed new paradigms for fundamental inequalities and information encoding structures, with principal research themes encompassing symmetry, inequalities, and analysis on manifolds.

Beckner has received numerous accolades, including the Salem Prize in 1975 for his work in Fourier analysis. He has also been recognized as a Sloan Fellow and was an inaugural Fellow of the American Mathematical Society. Beckner has held visiting appointments at esteemed institutions such as Columbia University, Princeton University, UCLA, and in France at the Institut Henri Poincaré in Paris and Université de Cergy-Pontoise. At UT Austin, he has served as chair of the mathematics department (2007–2011) and chair of the Faculty Council (2014–2015). He has also contributed to various committees of the American Mathematical Society, including serving as managing editor of the Transactions of the AMS.

To examine how the interplay of symmetry, smoothness and uncertainty can characterize structure for geometric manifolds.

• Fourier Analysis
• Geometric Inequalities
• Lie Groups & Differential Geometry
• Mathematical Physics
• Partial Differential Equations
• Probability

Fourier Analysis - Sharp Inequalities and Geometric Manifolds

Geometric inequalities provide insight into the structure of manifolds. More directly, Sobolev embedding, the Fourier transform, convolution and fractional integrals are central tools for analysis on geometric manifolds. Questions concerning fractional smoothness, multilinear operators, product manifold structure and restriction phenomena on subvarieties are engaging directions for current research. The principal objective of my research is to develop a deeper understanding of the way that sharp constants for function-space inequalities over a manifold encode information about the geometric structure of the manifold. This direction seems fundamental to explore the interplay between geometry and analysis on locally compact non-unimodular Lie groups, including SL(2,R), hyperbolic space, and more generally, manifolds with nonpositive curvature (Cartan-Hadamard spaces). Asymptotic arguments identify geometric invariants that characterize large-scale structure. Weighted inequalities provide quantitative information to characterize integrability for differential and integral operators and reflect the dilation character of the manifold. Sharp estimates constitute a critical tool to determine existence and regularity for solutions to pde's, to demonstrate that operators and functionals are well-defined, to explain the fundamental structure of spaces and their varied geometric realizations, to calculate precise lower-order effects and to suggest new directions for the development of analysis on a geometric manifold. Model problems and exact calculations in differential geometry and mathematical physics are a source of insight and stimulus, particularly conformal deformation, fluid dynamics, many-body dynamics, quantum physics, statistical mechanics, stellar dynamics, string theory and turbulence. Functional forms that characterize smoothness lie at the heart of understanding and rigorously describing the many-body interactions that determine the behavior of dynamical phenomena.