# Mathematics > Active math only: solve, derive, implement, simulate — never aesthetic consumption. Stack: probability & statistics → linear algebra → optimization → quant bridge. No artifact, no learning. Canonical: https://learn.significanthobbies.com/curriculum/tracks/mathematics Active math only: solve, derive, implement, simulate — never aesthetic consumption. Stack: probability & statistics → linear algebra → optimization → quant bridge. No artifact, no learning. This track contains 30 connected concepts. Mastery means explaining each mechanism, predicting its failure modes, and supporting decisions with code, measurements, or a reviewable design artifact. ## Roadmaps - [30-Day Math Rating Climb](https://learn.significanthobbies.com/curriculum/roadmaps/math-rating-climb-30d) — Raise your per-roadmap ELO by climbing from linear algebra through statistics to calculus — each week unlocks harder drills at your edge. - [30-Day Probability & Statistics](https://learn.significanthobbies.com/curriculum/roadmaps/prob-stats-30d) — Active probability & statistics — solve, derive, implement. No passive video watching as a substitute for problems. - [12-Week Active Math Stack](https://learn.significanthobbies.com/curriculum/roadmaps/math-stack-12w) — Math → AI systems → distributed/data → quant tools. Active only: solve, derive, implement, simulate — never aesthetic consumption. ## Concepts - [Vectors & Vector Spaces](https://learn.significanthobbies.com/curriculum/concepts/vectors-and-spaces) (intro) — Vectors as ordered lists, dot products, norms, orthogonality, and the geometric picture of n-dimensional space. - [Matrices & Linear Transformations](https://learn.significanthobbies.com/curriculum/concepts/matrices-and-transformations) (intro) — Matrix multiplication as composing linear maps: rotation, scaling, projection, and change of basis. - [Eigenvalues & Matrix Decomposition](https://learn.significanthobbies.com/curriculum/concepts/eigenvalues-decomposition) (advanced) — Eigenvectors as directions preserved by a transformation; eigenvalues as stretch factors; SVD as the universal factorization. - [Probability Fundamentals](https://learn.significanthobbies.com/curriculum/concepts/probability-fundamentals) (intro) — Sample spaces, conditional probability, independence, and Bayes' rule for updating beliefs. - [Random Variables & Distributions](https://learn.significanthobbies.com/curriculum/concepts/random-variables) (core) — Discrete and continuous distributions, expectation, variance, and the law of large numbers. - [Descriptive Statistics](https://learn.significanthobbies.com/curriculum/concepts/descriptive-statistics) (intro) — Summarizing data: mean, median, variance, correlation, and when each summary lies. - [Estimation & Confidence Intervals](https://learn.significanthobbies.com/curriculum/concepts/estimation-confidence) (core) — Point estimates, standard error, confidence intervals, and what '95% confident' actually means. - [Hypothesis Testing](https://learn.significanthobbies.com/curriculum/concepts/hypothesis-testing) (core) — Null and alternative hypotheses, p-values, significance, power, and Type I/II errors. - [Linear Regression](https://learn.significanthobbies.com/curriculum/concepts/regression-basics) (core) — Fitting a line (or hyperplane) by least squares; residuals, R², and the geometry of projection. - [Derivatives & Gradients](https://learn.significanthobbies.com/curriculum/concepts/derivatives-and-gradients) (core) — Partial derivatives, the gradient vector, and reading a loss surface for descent direction. - [Multivariable Optimization](https://learn.significanthobbies.com/curriculum/concepts/multivariable-optimization) (advanced) — Convexity, critical points, constrained optimization, and why SGD works on non-convex losses anyway. - [Information & Entropy](https://learn.significanthobbies.com/curriculum/concepts/information-entropy) (advanced) — Entropy as surprise, cross-entropy as a loss, KL divergence as a distributional distance. - [Classical Distributions](https://learn.significanthobbies.com/curriculum/concepts/classical-distributions) (core) — Bernoulli, Binomial, Poisson, Normal, and Exponential — when each models the world and what to expect from them. - [Sampling & the Central Limit Theorem](https://learn.significanthobbies.com/curriculum/concepts/sampling-and-clt) (core) — Sampling distributions, standard error of the mean, and why averages become Normal as n grows. - [A/B Testing for Engineers](https://learn.significanthobbies.com/curriculum/concepts/ab-testing-engineering) (core) — Sample size, statistical power, practical significance, SRM checks, and multiple-comparison traps in product experiments. - [Bayesian Inference](https://learn.significanthobbies.com/curriculum/concepts/bayesian-inference) (advanced) — Priors, posteriors, credible intervals, and when Bayesian updating beats frequentist tests. - [Maximum Likelihood Estimation](https://learn.significanthobbies.com/curriculum/concepts/maximum-likelihood) (advanced) — Choosing parameters that make the observed data most probable; log-likelihood; connection to cross-entropy loss. - [Covariance & Correlation](https://learn.significanthobbies.com/curriculum/concepts/covariance-correlation) (core) — Covariance measures co-movement; correlation normalizes to [−1, 1]. Foundation for regression, PCA, and portfolio risk. - [Bias, Variance & Overfitting](https://learn.significanthobbies.com/curriculum/concepts/bias-variance-overfitting) (core) — Underfitting vs memorizing noise; why in-sample greatness lies. - [Rank, Basis & Subspaces](https://learn.significanthobbies.com/curriculum/concepts/matrix-rank-basis) (core) — Column space, rank, independence — why matrices are often low-rank in practice. - [PCA & Projection](https://learn.significanthobbies.com/curriculum/concepts/pca-projection) (advanced) — Principal components as variance-maximizing orthogonal directions; projection as subspace approximation. - [Returns & Volatility](https://learn.significanthobbies.com/curriculum/concepts/returns-volatility) (core) — Simple/log returns, realized volatility, annualization — basic quant units. - [Stationarity & Autocorrelation](https://learn.significanthobbies.com/curriculum/concepts/stationarity-autocorrelation) (advanced) — When series statistics are stable; memory in lags; noise vs signal. - [Random Walks & Markov Chains](https://learn.significanthobbies.com/curriculum/concepts/random-walks-markov) (core) — Markov property, random walks, transition matrices — generative story behind market efficiency intuition. - [Sharpe, Drawdown & Portfolio Risk](https://learn.significanthobbies.com/curriculum/concepts/portfolio-risk-metrics) (core) — Sharpe, max drawdown, correlation matrices — risk-adjusted comparison. - [Momentum Backtest Discipline](https://learn.significanthobbies.com/curriculum/concepts/momentum-backtest) (advanced) — Momentum rules, SPY/QQQ benchmarks, holdout honesty — why most backtests lie. - [Queueing Theory](https://learn.significanthobbies.com/curriculum/concepts/queueing-theory) (core) — Little's law and the utilisation curve — why latency explodes before a system runs out of capacity. - [Curse of Dimensionality](https://learn.significanthobbies.com/curriculum/concepts/curse-of-dimensionality) (core) — Why distances concentrate in high dimensions, and what that does to nearest-neighbour search. - [Combinatorics](https://learn.significanthobbies.com/curriculum/concepts/combinatorics) (core) — Counting without enumerating — permutations, combinations, inclusion-exclusion, pigeonhole. - [Numerical Stability](https://learn.significanthobbies.com/curriculum/concepts/numerical-stability) (core) — Floating point, catastrophic cancellation, and the log-sum-exp trick.