Beckman Fellow 2007-08

Thomas A Nevins



The technique of harmonic analysis plays a central role in mathematics and its applications. In outline, harmonic analysis takes “complicated data” (e.g., a function that describes a sound wave) and describes the data in terms of simple building blocks (e.g., the contributions of “pure tones” to that sound wave). Difficult problems about the complicated data take on an easier form when expressed in terms of the simple building blocks and their contributions.

The principles of harmonic analysis have inspired one of the guiding paradigms for contemporary research in algebra, geometry, and number theory, the Langlands Program. This Program formulates a deep interplay between classical number-theoretic questions and analytic information arising from solutions of differential equations.

During his Center appointment, Professor Nevins will apply harmonic analysis, viewed through the lens of the “categorical Langlands Program” or Geometric Langlands Program, to the most important open problems concerning a new algebraic structure, the double affine Hecke algebra. This structure arose in the proof of a complicated equation in mathematical physics. Subsequent work has shown that the algebra underlies many important phenomena in mathematics and mathematical physics, and Professor Nevins’s research will point the way to features of the algebraic landscape not yet dreamed of.