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.

A photo of me.

Research

A marked Sierpiński carpet.

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.


Research preprints


Expository / undergraduate preprints


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