“We know the past but cannot control it. We control the future but cannot know it.”

— Claude Shannon

About

I’m an engineer and researcher interested in subjects related to mathematics, engineering, computer science, economics, and complex systems. Professionally, I’ve worked on DARPA-funded AI research programs, built systems at large technology companies, and started a few ventures of my own.

Purpose

This is my research notebook. My current technical work uses symmetry, game theory, and control as tools to study organized systems. I also write essays about AI, political economy, art, and perception. Earlier tutorials and learning notes are collected below the current work.

Posts may be long and technical, and are written primarily for my own thinking rather than optimized for the reader. This is by design. I suggest reading the blog in conjunction with your preferred AI companion.

Contents

Essays

AI, Demand, and Institutions

Art, Value, and Culture

Mind and Perception

Do We See the Same Colors?

Symmetry and Theory Construction

Original Work

Foundations

Games and Agents

Original Work

Learning and Equilibria

Learning Equilibria by Gradient Descent

Geometric Control

Technical Notes and Tutorials

Machine Learning and Statistics

Inspection Bias

Inspection Bias

Learning Dynamical Systems

Automated Reasoning

E-Graphs

E-Graph Basics

Proof Assistant

Engineering Notes

Setting up Vector Search in AWS with Pinecone and SST

Disclaimer

All code and information on this blog is subject to error. Use at your own risk.

Errors

Please let me know if you see any errors.

Recent Posts

From Symmetry to Theory: A Computational Engine

In the dynamical similarity and allometry posts, we started with a group acting on a system and asked what survived the group action. For dynamical similarity, the survivors…

Algebra and Allometry

In the dynamical similarity post we looked at equivariant symmetries of the Lagrangian, which produce Noether-like quantities that are conserved not within a single…

Games, Invariants, and Alignment

In the post/paper draft here I wrote about using invariants to classify normal-form games. That post is a monster that I ultimately need to carve up and revise…

Invariant Coordinates for Normal-Form Games Modulo Strategy Relabeling

For an accessible motivation and overview of this framework, see Games, Invariants, and Alignment.

Thoughts on Demand

Robert Oppenheimer, when he had his security clearance questioned and then lifted when he was being punished for having resisted the development of the hydrogen bomb, was…

Financial Explanations of Art

In a previous post, we explored some functional theories of art. However, we left open the question of specific financializations. In this post I’ll try to understand…

Building a Minimal Computational Invariant Theory Library

Now that we’ve investigated the basics of invariant theory, we can look at how to compute invariants with code. In this post, I’ll investigate computational invariant…

Survey of Classical Invariant Theory

Invariant theory is the study of how symmetries constrain the structure of mathematical objects (similar to Noether’s theorem). In this post, I will give a brief…

How Will Humans Generate Value In a Post-AI Society?

The current dominant AI narrative asserts that “white-collar jobs are next”. This includes lawyers, software engineers, radiologists, writers, mathematicians, artists, and…

Does AI Make Totalitarianism More Likely?

Much of the contemporary AI risk discourse focuses on large-scale existential threats to the human species. However, there are more mundane risks that are also worth…

Thoughts on Selectorate Theory

Bruce Bueno de Mesquita and Alastair Smith’s The Dictator’s Handbook (2011) is one of the more successful pop-science books in political economics. Its central thesis is…

Do We See the Same Colors?

Neither would it carry any Imputation of Falshood to our simple Ideas, if by the different Structure of our Organs, it were so ordered, That the same Object should produce…

Functional Explanations of Art

What is art’s purpose in society? Why do we spend so much time and money consuming, producing, and criticizing it? What separates “good” art from “bad” art (is there even…

Reduction

In this post, I introduce gauge equivalence, and also investigate a few different types of reduction under symmetry (to build out a taxonomy).

Inspection Bias

Suppose we have a population of objects of different lifespans (starting at different times). Given a sample from a specific time point, we should expect “most” of the…

Dynamical Similarity and Equivariant Symmetry

We have continued our investigation of geometric controls by investigating conserved quantities derived from Noether’s (first) theorem. Here we look at a slight extension…

Are We Approaching Cultural Saturation?

In the Paradox of Taste, I looked at novels as information-theoretic objects. One question I asked was: what if all novels that could ever exist were enumerated and indexed…

Noether’s Theorem with Time

Let’s extend the discrete controls framework and Noether’s theorems to include time-invariance. We’ll start with extending the continuous version, then discretize.

Is a Picture Worth a Thousand Words?

In a previous essay I considered the problems associated with generating novels. With the introduction of the new Nano-Banana Pro, let’s revisit those same basic information…

Noether’s Theorem and Geometric Controls

What’s the point of geometric integrators? Why bother with the formalism around Lagrangians and Hamiltonians to manage basic control problems or simulations?

Controls from the Geometric Perspective

In my post on differential games and stag hunt I mentioned discrete controls. When looking into this topic, I ended up in a rabbit hole around geometric controls. To learn a…

Differential Games and Stag Hunt

In the last two posts in this series we looked at games as static functions with discrete strategies. That is, each player picked a strategy \(s_i \in S_i\) and at the end…

SINDy with Control

The SINDy method is useful for fitting governing equations to data drawn from a dynamical system. However, for engineering applications we often seek not just to analyze…

Learning Equilibria by Gradient Descent

Given a set of agents playing a game, how do we determine their optimal strategic behavior?

Differentiable Game Canonicalization

Given two games, how can we tell if they are “strategically equivalent”?

SINDy Method for Learning Dynamical Systems

In the last post I looked at DMD and EDMD, two methods for analyzing dynamical systems. Both methods look at pairs of snapshots \((x_k, x_{k+1})\) and fit a linear update…

Linear Methods for Learning Dynamical Systems

In this series of posts, I will record some of my notes and code snippets on estimating dynamical system from data¹.

Tactics Elaboration

The “reverse-chaining” style proofs (working backward from some desired theorem using out tactics engine) ought to be convertible to “forward-chaining” style proofs (working…

Retrospective: Proof Assistant

I set out to build a proof assistant for a few reasons:

Match Construct

In the Induction post, we were able to build some simple inductive types and write some (very simple proofs). However, the actual process of defining the inductive types and…

Setting up Vector Search in AWS with Pinecone and SST

Frequently, we want the ability to efficiently search and retrieve relevant information based on meaning rather than just keywords. This is a common pattern in LLM-based…

Tactics Engine

Commonly used proof assistants like Coq and Lean don’t prove theorems directly using the typing rules. Rather, they use tactics to reason backwards from some goal to the…

The Paradox of Taste: Borges, Preference Oracles, and the Platonic Theory of Art

In Collected Ficciones (1944), Borges tells the story of the fictional author Pierre Menard. The story is presented in the form of a literary criticism of Menard’s complete…

Sort Hierarchy

In the post on the calculus of constructions, we implemented several Sorts (SProp, Prop, Set, Type). A Sort can be thought of as a “type of types.” For example, booleans and…

Induction

To introduce more complex notions into our type system, we need induction.

Typing Rules

How can we tell if a term is well-typed? How can we tell if an environment is well-formed? We can be sure we have formed a proper term and environment (in the Calculus of…

Conversion Rules

Now that we can represent some basic theorems using a computer, we need to be able manipulate those representations. We also will want to be able to tell if two types are…

Calculus of Constructions

To verify (or prove) theorems using a computer, we need some way to represent a theorem and the steps used to prove it. The Calculus of Constructions (CoC) forms the basis…

E-Graph Basics

An e-graph (equivalence graph) is a type of data structure commonly used to reason about equalities and programs computationally.