Blog
Notes on mathematics, computer science, and AI.
- IMO 2026 ProblemsJuly 26, 2026
The six problems from the 2026 International Mathematical Olympiad, held in Shanghai, China — with space left to fill in my own solutions.
- Building Voltrade: architecture of a hosted, multi-tenant trading-exchange simulatorJuly 26, 2026
How Voltrade is structured — the registry of independent exchanges, the auth and persistence model, the bot fleet, and the process of getting there.
- Quantum mechanics from fundamentals: what the equations actually sayJuly 11, 2026
The empirical puzzle that forced physics to quantize energy, the postulates of quantum mechanics stated precisely, and two real systems — the infinite square well and the harmonic oscillator — solved completely, with plotted wavefunctions.
- General relativity, from two puzzles to an expanding universeJuly 5, 2026
How the equivalence of gravitational and inertial mass leads to spacetime curvature, the Friedmann equations, and gravity's uneasy relationship with quantum mechanics.
- Linear algebra, from vectors to singular valuesJune 28, 2026
Building up vector spaces, linear maps, and eigenstructure from first principles, ending at the singular value decomposition.
- The Party Problem: Order Hidden in Every CrowdJune 27, 2026
A complete elementary proof that among any six people, three either all know each other or are all mutual strangers — and how that one puzzle opens onto Ramsey theory, a field built on proving order is unavoidable while barely being able to compute it.
- Continuous everywhere, differentiable nowhere: the Weierstrass functionJune 21, 2026
Weierstrass's monster shattered the 19th-century belief that a continuous curve must be smooth almost everywhere — a full proof of continuity via the M-test, a real (if not fully sharp) estimate for nowhere-differentiability, and interactive plots showing the partial sums get jagged at every scale.
- Constructible Polygons: Why Gauss Could Draw a 17-gon but Not a 7-gonJune 13, 2026
From bisecting an angle to Gauss's 17-gon: how field extensions of degree a power of two decide exactly which regular polygons compass and straightedge can build.
- Solvable Everywhere, Solvable NowhereJune 7, 2026
An equation can have a solution modulo every integer and over the real numbers, yet still have no rational solution. Hasse–Minkowski, Selmer's cubic, and the Brauer–Manin obstruction.
- Which primes are a sum of two squares?May 31, 2026
From an elementary congruence check to Gaussian integers, class groups, and quadratic reciprocity.
- Backtests: convincing and untrustworthyMay 30, 2026
The more a strategy backtest is refined, the better it looks and the less it should be trusted — the statistical mechanism behind that, and what actually helps.
- Finding alpha is a search problem, not a prediction problemMay 24, 2026
Why a real trading edge is hard in a way that ordinary predictive modeling isn't — an adversarial, non-stationary opponent, the stationarity trap, overfitting with correlated rows, and what small versus large datasets actually demand of you.
- The Shape of Symmetry: A Ground-Up Tour of Group TheoryMay 16, 2026
Starting from the symmetries of a square, we build group theory from scratch with real proofs: Lagrange's theorem, the First Isomorphism Theorem, quotient groups, and a full proof of Cauchy's theorem.
- High-Dimensional Geometry Is Not Like You ThinkMay 10, 2026
Four numerical experiments showing how volume, angles, and distances behave completely differently once you leave two or three dimensions.
- Quantum computing: interference, not parallelismMay 3, 2026
What a qubit actually is, why superposition alone buys you nothing, and why interference is the real computational resource — with the linear algebra, Grover and Shor, and an honest look at where the hardware stands in mid-2026.
- P vs NP: Checking Is Easy. Is Finding?May 2, 2026
A tour of the biggest open question in computer science — what it actually asks, why it's hard to answer, and why the answer would reshape cryptography, optimization, and maybe creativity itself.
- The Riemann zeta function: from primes to the critical lineApril 26, 2026
Where ζ(s) comes from, why it extends past Re(s) > 1, what ζ(-1) = -1/12 actually means (and doesn't), and why its zeros control the primes.
- Fermat's Last Theorem: The Two Cases You Can Actually ProveApril 12, 2026
A complete, rigorous proof of Fermat's Last Theorem for exponents 3 and 4, plus an honest map — not a derivation — of the Frey curve / modularity argument needed for the general case.
- How neural networks learn: deriving backpropagation from scratchApril 5, 2026
A from-scratch derivation of backpropagation — from a single neuron to the chain-rule recursion that makes training networks with millions of parameters computationally tractable.