Dolgopyat’s method and the fractal uncertainty principle

Notes on fractal uncertainty principle version 0.5 (October 4, 2017)

The Uncertainty Principle.

If a function ip(x) is mostly concentrated in a box Q, while its Fourier transform $>(£) is concentrated mostly in Q', then we say ijj is microlocalized in Q X Q' in (x, £)-space. The uncertainty principle says that Q X Q' must have volume at least 1. We will explain what it means for ip to be microlocalized to more complicated regions S of volume ~ 1 in (x, £)space. To a differential operator ...

The stem cell uncertainty principle

Manuscript Number: Full Title: The stem cell uncertainty principle Article Type: PSA 2012 Contributed Paper

The Hardy Uncertainty Principle Revisited

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic L2-estimates and the invertibility of the Fourier transform on L2(Rn) and S′(Rn).

Gerbes and Heisenberg’s Uncertainty Principle

We prove that a gerbe with a connection can be defined on classical phase space, taking the U(1)–valued phase of Feynman path integrals as Čech 2–cocycles. A quantisation condition on the corresponding 3–form field strength is derived and proved to be equivalent to Heisenberg’s uncertainty principle.