Essays
These are our essays. They showcase applied mathematics across a wide span of subjects, from physics and engineering to finance, history and medicine: the models, the methods that solve them, the code that runs them and the algorithms behind them, often with a live model you can try for yourself. Each essay starts from a question and works it through to a number. Nothing here is a product or advice. The code we publish as Jupyter notebooks is under Notebooks, the live boards that each answer one practical question are under Tools, and the research papers are under Research Papers. Every essay belongs to one of our five research desks; the buttons below show one desk's essays at a time.
51. The Cube Root That Runs the Treasury
How much cash should a company keep? If its daily cash flow is a random walk and every transfer costs a fee, the best rule is a band: do nothing inside it, and jump back to a return point a third of the way up. The return point is a cube root, which makes the rule remarkably forgiving.
Read more →52. De Rerum Electione: The Choice That Is Not a Swerve
A sequel in verse to De Rerum Stochástica. Lucretius put free will in the random swerve; Cicero called that childish, Laplace took the swerve away, and Born gave chance back to physics. The mathematics of control puts freedom somewhere else: in the steering, not the noise.
Read more →53. The Win Rate That Means Nothing
Put the stop nine times further from the entry than the target and you win 90% of your trades on any stock, and make nothing. The optional stopping theorem shows why a win rate is a setting, not a skill, and what costs and exits do to a day trader's record.
Read more →54. The Lead That Never Changes Hands
A fund manager with no skill spends at least 80% of twenty years ahead of the index with probability 29.5%, and a record that looks fair, ahead 40 to 60% of the time, turns up only 12.8% of the time. Lévy's arcsine law explains why, and why the same curve decides when the lead last changed hands.
Read more →55. The Return That Takes Four Centuries
A share with 20% volatility and a 5% expected return gives away its volatility in a single day of one-minute prices, to within 0.72 of a point. Its expected return needs 400 years of data to be known to within one point. Quadratic variation and Girsanov's theorem explain why, and why sampling more often cannot help.
Read more →56. The Stagflation Trap: Optimal Monetary Policy as an HJB Problem
A linear-quadratic HJB model that makes the stagflation trap mathematically precise: why a persistent supply shock forces the optimal steady state away from zero inflation and zero output gap simultaneously.
Read more →57. De Rerum Stochástica: Lucretius’s Random Swerve and the Birth of SDEs
Epicurus added a random, uncaused swerve (Lucretius would call it the clinamen) to save atomism from determinism. Two thousand years later, it became the diffusion term in every SDE ever written.
Read more →58. From Plasma to Game Theory: The Unlikely Journey of an SDE
McKean-Vlasov SDEs emerged from plasma physics in the 1960s. Mean field games arrived from economics in 2006. They converged on the same equation from opposite directions.
Read more →59. Markov Chains in Supermarket Management
How Markov chains model checkout queues and customer flow — and what the stationary distribution tells a supermarket manager about staffing and store layout.
Read more →60. Kelly, the Growth-Optimal Portfolio, and the Benchmark Approach to Option Pricing
The Kelly criterion and Black-Scholes option pricing are the same mathematical object viewed from different angles. Both lead to the growth-optimal portfolio — Platen's benchmark — which prices derivatives under the real-world measure without requiring a risk-neutral one.
Read more →61. The Bookmaker’s Measure: Kelly, Martingale, and the Price of an Edge
The bookmaker is not running a gambling operation. They are constructing a risk-neutral measure. Understanding that changes everything about how you think about betting strategies — including why Martingale fails even when you have edge.
Read more →62. The Kelly Criterion — Why the Optimal Strategy Is Never Used
A 60/40 coin. A provably optimal betting strategy. And almost nobody uses it. Kelly's 1956 result is one of the few places in finance where the correct answer is known and rationally ignored.
Read more →63. Four Historical Decisions an ABM Would Have Changed
Smoot-Hawley was not unique. From Versailles to Vietnam to Stalin's grain quotas to the Maginot Line — the same pattern recurs. A model that forgets the other agents in the system.
Read more →64. The Trade War Nobody Chose: Smoot-Hawley and the Model That Would Have Stopped It
In 1930, a thousand economists begged Hoover not to sign the Smoot-Hawley tariff. He signed it anyway. An agent-based model would have shown exactly what happened next — before it happened.
Read more →65. CAD 530 a Month: What Your Car Insurance Premium Actually Pays For
A friend bought a Ford F-150 in Ontario. The insurance quote came in at 530 dollars a month. I built a mathematical model to find out how much of that is justified.
Read more →66. Langevin Dynamics and Why They Matter in Finance
The Langevin equation was written to describe a particle in a fluid. It turns out to be exactly the right language for interest rates, volatility, and forward curves.
Read more →67. Two Worlds, One Price: Entropy and the Risk-Neutral Measure
The change of measure at the heart of derivative pricing is an exponential tilt of the real-world probability measure — identical in structure to the Boltzmann distribution. The cost of that tilt is relative entropy.
Read more →68. Pricing a Gas Liquefaction Plant as a Real Option: An HJB Approach
We price the option to invest in a gas liquefaction plant using the Hamilton-Jacobi-Bellman PDE, with CIR gas price dynamics and a finite-horizon free boundary problem.
Read more →69. Heat Diffusion in an Iron Sphere — A FEniCSx Simulation
A 3D FEniCSx simulation of heat diffusing through an iron sphere with a hot core. Full PDE, boundary conditions, weak form, and DOLFINx code.
Read more →70. Solar Magnetohydrodynamics: Solving Magnetic Diffusion in a Sphere
Solving the MHD induction equation in spherical geometry with Dedalus — magnetic diffusion on the 2D sphere surface and inside a 3D conducting ball, with animated GIF simulations.
Read more →71. Solving the 2D Wave Equation with Spectral Methods
Using the Dedalus spectral PDE solver to simulate the 2D wave equation on a periodic domain, with animated GIF output and applications across physics, finance, and medical imaging.
Read more →72. The Square Root That Keeps Rain Positive: CIR Processes in Weather Modelling
The Cox-Ingersoll-Ross process was built for interest rates. Its real strength — staying positive as noise vanishes near zero — turns out to be exactly what rainfall modelling needs.
Read more →73. When Neurons Forget: The Ornstein-Uhlenbeck Process in Neuroscience
The leaky integrate-and-fire neuron — the workhorse model of computational neuroscience — is an Ornstein-Uhlenbeck process in disguise. Here is why that matters.
Read more →74. What Agent-Based Models See That Regressions Miss
Standard regression conflates micro configurations that produce identical macro observables. Four classical examples — wealth concentration from random exchange, Schelling segregation, phantom traffic jams, and bank runs — show that genuine causal structure can be invisible to aggregate statistics and only visible at the level of agent interactions.
Read more →75. The Mean Field Game of Mediocrity
Mean field game theory gives mediocracy a precise formulation: when conformity pressure exceeds rank incentive, the coupled HJB–Fokker–Planck system has a concentrated Nash equilibrium. A phase transition separates dispersed outcomes from universal indistinction.
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