Notes on mathematics, computer science, and AI.https://blog.mohijitsingh.com/https://blog.mohijitsingh.com/blog/imo-2026-problems/https://blog.mohijitsingh.com/blog/imo-2026-problems/The six problems from the 2026 International Mathematical Olympiad, held in Shanghai, China — with space left to fill in my own solutions.Sun, 26 Jul 2026 00:00:00 GMTmathcompetitionhttps://blog.mohijitsingh.com/blog/voltrade-multi-tenant-exchange/https://blog.mohijitsingh.com/blog/voltrade-multi-tenant-exchange/How Voltrade is structured — the registry of independent exchanges, the auth and persistence model, the bot fleet, and the process of getting there.Sun, 26 Jul 2026 00:00:00 GMTengineeringvoltradehttps://blog.mohijitsingh.com/blog/quantum-mechanics-from-fundamentals/https://blog.mohijitsingh.com/blog/quantum-mechanics-from-fundamentals/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.Sat, 11 Jul 2026 00:00:00 GMTphysicshttps://blog.mohijitsingh.com/blog/general-relativity/https://blog.mohijitsingh.com/blog/general-relativity/How the equivalence of gravitational and inertial mass leads to spacetime curvature, the Friedmann equations, and gravity's uneasy relationship with quantum mechanics.Sun, 05 Jul 2026 00:00:00 GMTphysicshttps://blog.mohijitsingh.com/blog/linear-algebra-from-fundamentals/https://blog.mohijitsingh.com/blog/linear-algebra-from-fundamentals/Building up vector spaces, linear maps, and eigenstructure from first principles, ending at the singular value decomposition.Sun, 28 Jun 2026 00:00:00 GMTmathalgebrahttps://blog.mohijitsingh.com/blog/ramsey-theory/https://blog.mohijitsingh.com/blog/ramsey-theory/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.Sat, 27 Jun 2026 00:00:00 GMTmathcombinatoricshttps://blog.mohijitsingh.com/blog/continuous-nowhere-differentiable/https://blog.mohijitsingh.com/blog/continuous-nowhere-differentiable/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.Sun, 21 Jun 2026 00:00:00 GMTmathanalysishttps://blog.mohijitsingh.com/blog/constructible-polygons-gauss-wantzel/https://blog.mohijitsingh.com/blog/constructible-polygons-gauss-wantzel/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.Sat, 13 Jun 2026 00:00:00 GMTmathalgebrahttps://blog.mohijitsingh.com/blog/local-global-principle/https://blog.mohijitsingh.com/blog/local-global-principle/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.Sun, 07 Jun 2026 00:00:00 GMTmathnumber-theoryhttps://blog.mohijitsingh.com/blog/quadratic-forms-and-sums-of-squares/https://blog.mohijitsingh.com/blog/quadratic-forms-and-sums-of-squares/From an elementary congruence check to Gaussian integers, class groups, and quadratic reciprocity.Sun, 31 May 2026 00:00:00 GMTmathnumber-theoryhttps://blog.mohijitsingh.com/blog/backtests-convincing-and-untrustworthy/https://blog.mohijitsingh.com/blog/backtests-convincing-and-untrustworthy/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.Sat, 30 May 2026 00:00:00 GMTtradingquanthttps://blog.mohijitsingh.com/blog/finding-alpha-trading-strategies/https://blog.mohijitsingh.com/blog/finding-alpha-trading-strategies/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.Sun, 24 May 2026 00:00:00 GMTtradingquanthttps://blog.mohijitsingh.com/blog/group-theory-foundations/https://blog.mohijitsingh.com/blog/group-theory-foundations/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.Sat, 16 May 2026 00:00:00 GMTmathalgebrahttps://blog.mohijitsingh.com/blog/high-dimensional-geometry/https://blog.mohijitsingh.com/blog/high-dimensional-geometry/Four numerical experiments showing how volume, angles, and distances behave completely differently once you leave two or three dimensions.Sun, 10 May 2026 00:00:00 GMTmathprobabilityhttps://blog.mohijitsingh.com/blog/quantum-computing-2026/https://blog.mohijitsingh.com/blog/quantum-computing-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.Sun, 03 May 2026 00:00:00 GMTphysicscshttps://blog.mohijitsingh.com/blog/p-vs-np/https://blog.mohijitsingh.com/blog/p-vs-np/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.Sat, 02 May 2026 00:00:00 GMTcscomplexity-theoryhttps://blog.mohijitsingh.com/blog/riemann-zeta-function/https://blog.mohijitsingh.com/blog/riemann-zeta-function/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.Sun, 26 Apr 2026 00:00:00 GMTmathnumber-theoryhttps://blog.mohijitsingh.com/blog/fermats-last-theorem/https://blog.mohijitsingh.com/blog/fermats-last-theorem/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.Sun, 12 Apr 2026 00:00:00 GMTmathnumber-theoryhttps://blog.mohijitsingh.com/blog/neural-networks-and-backpropagation/https://blog.mohijitsingh.com/blog/neural-networks-and-backpropagation/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.Sun, 05 Apr 2026 00:00:00 GMTmathai