Aidan Backus
I am Aidan Backus, a-fifth 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 am on the job market for a postdoc.I can be reached at [email protected]. See also my CV and research statement.
Research
I work in geometric measure theory, the study of geometric structures which are too singular to study using differential calculus. I'm particularly interested in calculus of variations in L∞ and BV, and its applications to geometry. Classically the calculus of variations has studied partial differential equations which arise as solutions of optimization problems in L2, or more generally a reflexive Banach space. The analysis becomes much trickier in L∞ and BV, but in these spaces the problems have much clearer geometric interpretations (for example, the prototypical BV problem is concerned with finding area-minimizing hypersurfaces) and my research attempts to use geometric and convex duality-based techniques to make progress.
Research publications
To appear, Analysis and Partial Differential Equations, joint with James Leng and Zhongkai Tao: The fractal uncertainty principle via Dolgopyat's method in higher dimensions (arXiv).
8 August 2024, Journal of Geometric Analysis: Minimal laminations and level sets of 1-harmonic functions (arXiv).
Research preprints
1 January 2025: The max flow/min cut theorem and the topological least gradient problem
29 November 2024: The canonical lamination calibrated by a cohomology class
12 March 2024, joint with Ng Ze-An: The Lipschitz extension problem with prescribed local Lipschitz constants and eikonal mappings
Preprints, not intended for publication
2 April 2024: An ∞-Laplacian for differential forms, and calibrated laminations
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
Informal Notes
Some interesting open problems: A list of problems that I've thought about at least a little bit. If you know how to make progress on them, let's talk about it!
Functions of least gradient and area-minimizing laminations: A guide to my work on functions of least gradient and the laminations produced from their level sets, plus some motivation from the Daskalopoulos--Uhlenbeck theory of geodesic laminations.
The Fenchel-Rockafellar theorem: A short self-contained proof of the Fenchel-Rockafellar convex duality theorem.
Travel
June 2025, Providence: Geometric Analysis Workshop
April 2025, Hartford: Spring Eastern AMS Sectional Meeting
February 2025, Chicago: University of Chicago Geometric Analysis Seminar
May 2024, Des Moines: Algorithmic Fractal Dimensions
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