Undergraduate Colloquium – Tuesday October 27, 2020

Michael Hitrik (UCLA Mathematics)

Tuesday, October 27th

16:00 – 17:00, Undergraduate Colloquium

Zoom link: https://ucla.zoom.us/j/94137603833

Meeting ID: 941 3760 3833

Smooth and smoother: spaces of infinitely differentiable functions

Abstract. The elementary functions that we encounter in calculus are real analytic, in the sense that they can be expanded into convergent power series, and in analysis we then learn about the difference between real analytic and general smooth functions. The latter are flexible objects, and a classical theorem of Emile Borel states that for any sequence of numbers, we can construct a smooth function with that sequence as the coefficients of its Taylor series at the origin. In this talk, after sketching the proof of Borel’s theorem, we shall discuss classes of smooth functions defined by restricting the growth of their derivatives with the order of differentiation. Along the way, we shall prove the celebrated Denjoy-Carleman theorem characterizing quasi-analytic classes, which enjoy the unique continuation property, just like real analytic functions. We shall then talk about the important subclass of Gevrey functions, “interpolating” between analytic and general smooth functions, giving a bird’s eye view of their significance in PDE, from classical (the theory of the heat equation) to modern (the theories of diffraction and quantum scattering). No knowledge beyond undergraduate real and complex analysis will be assumed.

For more info, visit: https://secure.math.ucla.edu/seminars/display.php?&id=834200