Description:

Title: Fourier decay and spectral gaps in negative curvature

Additive combinatorics tells us that the Fourier transform of a probability measure on Euclidean space must decay polynomially at infinity except on a very sparse set of frequencies—unless much of the mass concentrates near linear subspaces at many scales. This dichotomy becomes especially powerful when the measure arises from dynamics, which imposes nontrivial relations among its Fourier coefficients, and constrains its behavior across scales. This principle was recently used to prove exponential mixing on geometrically finite manifolds, which is a problem in the realms of dynamics and geometry. The same principle has since led to further results in dynamics, including quantitative Fourier decay for self-conformal measures and progress toward the non-spherical spectral gap conjecture. The talk will aim to survey these connections, assuming no prior familiarity with these objects. Based in part on joint work with S. Baker and T. Sahlsten, and with D. Kelmer and P. Sarkar.​