Adam Henderson

“One of the principle objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.” - Josiah Willard Gibbs

This is the core thread of my research / professional career across Theoretical Physics (Quantum Gravity & General Relativity), applied math, machine learning, and software engineering.

List of Publications from my quantum gravity era. Without the academic pressure to write papers I am less and less inclined to share research.

Function Notation : Evaluation

Math
Notation

A survey of notations for function evaluation — \(f(x)\), subscripts, juxtaposition, Lisp-style — and the argument that the variety is valuable, not a flaw. Each notation emphasizes a different isomorphic representation of the same function space.

Jun 1, 2024
5 min

Convolutions

Math
Analysis

Convolution is just a binary operation and a measure. Tracing that observation from CNNs to Dirichlet convolution to graph neural networks, and asking when the Fourier convolution theorem survives.

Mar 13, 2022
6 min

Coordinates

Math

“Coordinates” is underspecified in every standard definition. A better formulation: coordinates are a bijection that makes the relevant structures on a space efficiently computable. This reformulation connects directly to algorithmic complexity.

Jul 11, 2021
6 min

Identity Elements for Binary Operations

Math
Algebra

The uniqueness proof for identity elements is clean — but left and right identities can be non-unique, and software occasionally produces near-identities that break the axioms. The right fix is to quotient them away.

Mar 3, 2020
6 min

Symmetries in Machine Learning I

Invariance and covariance under a group of transformations, made precise. The idealized example — translation-invariant functions on cyclic bit strings — reduces to counting necklaces, and the quotient space lacks obvious coordinates.

Feb 22, 2020
6 min

Editing Binary Data : Gran Turismo 2

Fun
When sitting down to play through Gran Turismo 2 with a young child I faced a conflict. I wanted to field frequent requests to purchase and race new cars while not…
Mar 28, 2019
6 min

Quotients of Algebraic Structures

Math
Algebra

Quotient constructions across algebra — groups, rings, modules — all require an equivalence relation compatible with the algebraic operations. The key condition is invariance under diagonal action, which unifies normal subgroups, ideals, and their generalizations.

Feb 4, 2019
7 min

Spectrum of Bianchi I in Loop Quantum Cosmology

Physics
Quantum-Gravity
Quantum-Cosmology
A nearly complete fragment of a paper on Loop Quantum Cosmology identifying a simple form of Hamiltonian constraint operator for Bianchi I.
Jan 24, 2019
12 min

Bash Arg Parsing

Software
Eng
As much as I prefer nearly any other language to Bash (or other Shell scripting) time and again I find myself scripting in Bash.
Oct 20, 2018
5 min

Asymptotic Notation as Equivalence Relation

\(O\), \(\Omega\), and \(\Theta\) as binary relations — a pre-order and an equivalence relation on functions. The quotient \(\mathbb{R}^{\mathbb{N}}/\Theta\) carries a Max-Times algebra, and the sub-exponential functions of L-notation fill the gaps between familiar complexity classes.

Jul 28, 2018
5 min

Statistical Inference on the Circle

Statistics

What is the natural analog of the Gaussian on the circle? The diffusion equation picks out the theta function. Maximum likelihood then reveals a posterior that is a mixture peaked at each possible mean value — a direct consequence of the circle having no unique notion of average.

Aug 10, 2017
12 min

Notation in Statistics

Statistics
Notation

The distribution is the primary object in probability theory. Expectation values and moments are tools for summarizing it — tools that can fail. A case for thinking in measures.

Mar 24, 2017
7 min

State Space Modeling - Kalman Filter

ML
Statistics
State Space

State space models, MAP smoothing, the Kalman filter, and sparse extensions — all derived from a single joint density over observations and states.

Feb 21, 2016
13 min
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