I am Aidan Backus, a fourth-year Ph.D. candidate at Brown University, where I am a member of the geometric analysis group and a student of Georgios Daskalopolous. Previously, I was at UC Berkeley.I can be reached at [email protected]. See also my CV, GitHub, and blog Some Compact Thoughts, or check out the community-written real analysis textbook, Clopen Analysis.
The proof that a Sierpiński carpet is not orthogonal to itself. This implies that the fractal uncertainty principle holds for the Sierpiński carpet in R4.
I work in partial differential equations and geometric measure theory. My research thus far has focused on the geometry of p-elliptic equations and the fractal uncertainty principle.By p-elliptic equations I mean certain generalizations of the p-Laplace equation; here p is a parameter ranging between 1 and infinity. In the case that p is 2 we recover the classical Laplace equation, but for endpoint values of p, the ellipticity of the equation degenerates as solutions become strongly constrained by the geometry of their domain.The fractal uncertainty principle is best understood as part of the story of semiclassical physics, which studies problems in which the effects of quantum mechanics are present, but incredibly weak. The fractal uncertainty principle says that in the semiclassical limit, the position and momentum distributions of a particle field cannot both converge to fractals. See also my lecture notes on the fractal uncertainty principle.
2 November 2023: Minimal laminations and level sets of 1-harmonic functions
23 February 2023, joint with James Leng and Zhongkai Tao: The fractal uncertainty principle via Dolgopyat's method in higher dimensions
Expository / undergraduate preprints
10 December 2019, joint with Peter Connick and Joshua Lin: An algorithm for computing root multiplicities in Kac-Moody algebras
Convex duality between minimal laminations and tight calibrations
January 2024, San Francisco: Joint Mathematics Meetings
November 2023, Storrs: University of Connecticut PDE and Differential Geometry Seminar
July 2023, Madison: Summer School on the Fractal Uncertainty Principle
April 2023, Princeton: Geometry Festival
March 2023, Providence: Geometric Analysis Workshop
January 2023, Boston: Joint Mathematics Meetings
November 2022, Storrs: Northeast Workshop in Geometric Analysis
October 2022, Amherst: Fall Eastern AMS Sectional Meeting
September 2022, Princeton: A Celebration of Karen Uhlenbeck's 80th Birthday
My bachelor's thesis: The Breit-Wigner series and resonances of potentials
Another undergraduate project on analysis and logic: Formalizations of analysis