# Aidan Backus

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.

## Research

The proof that a Sierpiński carpet is not orthogonal to itself. This implies that the fractal uncertainty principle holds for the Sierpiński carpet in **R**^{4}.

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.

### Research preprints

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

19 June 2023: Regularity of sets of least perimeter in Riemannian manifolds

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

Convex duality between minimal laminations and tight calibrations

## Travel

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

## 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