I am Aidan Backus, a third-year Ph.D. candidate at Brown University; previously I was at UC Berkeley. My main research interest is PDE, especially its geometric and numerical-analytic aspects, but I generally enjoy interdisciplinary parts of mathematics.

I can be reached at aidan_backus@brown.edu. You might also read my CV, GitHub, and blog Some Compact Thoughts, or check out the community-written real analysis textbook, Clopen Analysis.

Recently I've been thinking about least-gradient and best-Lipschitz functions, advised by George Daskalopoulos; see my recent research statement for more on that. I've also recently been interested in the fractal uncertainty principle and the scattering theory of resonances.

Publications and Preprints

You can click on the titles of these papers to read their abstracts.

  1. The fractal uncertainty principle via Dolgopyat's method in higher dimensions (joint with James Leng and Zhongkai Tao, in preparation)
  2. Functions of least gradient in constant curvature (in preparation)
  3. The Breit-Wigner series for noncompactly supported potentials on the line (2020, arXiv version)
  4. An algorithm for computing root multiplicities in Kac-Moody algebras (joint with Peter Connick and Joshua Lin, 2019, arXiv, implementation in Sage)

Talks given

Fall 2021, Grad Student Seminar, Brown University
Nonlinear Laplace equations
Summer 2021, Fourier integral operators seminar, UC Berkeley
Properties of Fourier integral operators (Notes); Oscillatory integrals with linear phase (Notes)
Spring 2021, Grad Student Seminar, Brown University
An ultrapowerful proof technique, the Nullstellensatz, and Sendov's conjecture
Fall 2020, MUSA Math Mondays, UC Berkeley
Numbers Big and Small: Calculus, ultrapowers, and measurability
Summer 2020, Scattering theory seminar, UC Berkeley
Resonance-free regions I: The geometry of trapping, nontrapping estimates, and semiclassical defect measures; The scattering matrix in dimension 3

Teaching

The following is a list of my teaching posts. Prior to Fall 2021, all such teaching posts have been at UC Berkeley; after, all posts are at Brown University unless otherwise noted.

Spring 2022
Teaching Assistant, Math 100 — Calculus II
Fall 2021
Teaching Assistant, Math 90 — Calculus
Reader, Math 523 (Wesleyan University) — Topology
Summer 2021
Reader, Math 110 — Linear Algebra
Spring 2021
Reader, Math 140 — Metric Differential Geometry
Fall 2020
Reader, Math 202A — Topology and Analysis
Summer 2020
Reader, Math 110 — Linear Algebra
Spring 2020
Reader, Math 202B — Topology and Analysis
Organizer, MUSA 74 — Proof-Writing Skills
Fall 2019
Student Instructor, Math 185 — Complex Analysis
Organizer, MUSA 74 — Proof-Writing Skills
Summer 2019
Reader, Math 1A — Calculus
Spring 2019
Reader, Math 105 — Second Course in Analysis
Organizer, MUSA 74 — Proof-Writing Skills
Fall 2018
Reader, Math H104 — Honors Introduction to Mathematical Analysis
Spring 2018
Reader, Math 104 — Introduction to Mathematical Analysis
Fall 2017
Academic Intern, CS 61A — Structure and Interpretation of Computer Programs

Clopen Analysis

Clopen Analysis is a community-written graduate-level real analysis textbook that strives to operate at a high level of abstraction while still providing lots of concrete examples and applications. Come check it out!

Nonpublishable writing

Aidan Backus. The Breit-Wigner series and distribution of resonances of potentials
My undergraduate thesis.
Aidan Backus. Formalizations of analysis
My final project for nonclassical logic at UC Berkeley.
Naveen Vaidya, Angelica Bloomquist, et al. Mathematical Models for Linking Within-Host and Between-Host Viral Dynamics: The Effect of Antibodies on the Probability of Transmission
A report on work done at the 2018 San Diego State University REU.
Aidan Backus. My undergraduate analysis lecture notes
Banach-valued measure theory. Ergodic theory and the Hopf argument. The Toda lattice and its applications to the QR algorithm. Siegel's KAM theorem. The holomorphic functional calculus. C*-representation theory of locally compact groups. Noncommutative geometry. Hormander's \dbar estimates. Bergman kernels and their application to Chow's theorem. Pseudodifferential calculus and the FBI transform. The Cauchy problem in general relativity. Cosmic censorship in spherical symmetry.
Aidan Backus. My undergraduate logic lecture notes
The Goedel completeness and incompleteness theorems. Kleene's recursion theorem. Induction in weak subtheories of PA. Forcing and Cohen's theorem. Turing degrees and the axiom of determinacy. Consistency of GCH. Large cardinals under a measurable cardinal. Measurable cardinals and their inner models. The Kunen calamity. Supercompact cardinals and their inner models. The Ultimate L.