# Aidan Backus

I am Aidan Backus, a third-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.

## Research

I am interested in partial differential equations (PDE). Traditionally thought of as purely the domain of analysis, modern research in PDE uses techniques not just from analysis, but also geometry (especially Riemannian and Lorentzian geometry), computer science (via numerical analysis), and physics (where relativity and quantum mechanics serve as a major source of intuition). The interdisciplinary nature of PDE is a big part of what drew me to, and what keeps coming back, to the field.

The proof that a Sierpiński carpet is not orthogonal to itself. This implies the fractal uncertainty principle holds for the Sierpiński carpet.

Roughly speaking, I have two topics that I spend most of my effort in graduate school towards, those being 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. I'm interested in the application of p-elliptic equations to solve problems in hyperbolic and complex algebraic geometry. Since p-elliptic equations can be solved numerically using convex optimization, I also am interested in applications of p-elliptic equations to practical problems in computational geometry.The fractal uncertainty principle is best understood as part of the story of semiclassical analysis: this is a theory which has its origins in semiclassical physics, the branch of 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.

### Preprints

23 February 2023, joint with James Leng and Zhongkai Tao: The fractal uncertainty principle via Dolgopyat's method in higher dimensions

28 April 2020: The Breit-Wigner series for noncompactly supported potentials on the line

10 December 2019, joint with Peter Connick and Joshua Lin: An algorithm for computing root multiplicities in Kac-Moody algebras

### Coming soon

Regularity of sets of least perimeter in Riemannian manifolds

Minimal laminations and level sets of 1-harmonic functions

## Miscellany

My bachelor's thesis: The Breit-Wigner series and resonances of potentials

Another undergraduate project on analysis and logic: Formalizations of analysis

Some notes on logic and analysis from when I was an undergraduate