Learning track · 30 concepts
Mathematics
Active math only: solve, derive, implement, simulate — never aesthetic consumption. Stack: probability & statistics → linear algebra → optimization → quant bridge. No artifact, no learning.
What mastery looks like
This track contains 30 connected concepts rather than an unordered reading list. Mastery means you can move from vocabulary to mechanisms, predict how the system behaves under pressure, and support a design decision with code, measurements, or a failure-recovery exercise. For Mathematics, use the track description as the boundary: learn enough detail to reason clearly about active math only: solve, derive, implement, simulate — never aesthetic consumption. stack: probability & statistics → linear algebra → optimization → quant bridge. no artifact, no learning.
A useful explanation names the state involved, the operation that changes it, the resource or safety constraint, and the observable signal that tells you whether the mechanism works. Avoid stopping at product names. Compare at least two approaches, state what each optimizes, and identify what breaks first as scale, concurrency, latency, or uncertainty increases.
Suggested study sequence
Start with Vectors & Vector Spaces, Matrices & Linear Transformations, Probability Fundamentals, Descriptive Statistics to establish the basic vocabulary. Continue through the core concepts by alternating explanation with an executable drill. Treat Eigenvalues & Matrix Decomposition, Multivariable Optimization, Information & Entropy, Bayesian Inference as integration work: they should combine earlier mechanisms rather than introduce disconnected facts.
At the end of each session, record one decision you can now make, one failure mode you can now predict, and one unanswered question. Revisit that question through the linked primary sources, then prove the answer in the Playground or a real repository. The track is complete when you can transfer the reasoning to an unfamiliar system, not when every page has been opened.
Roadmaps
Concepts in this track
intro
Vectors & Vector Spaces
Vectors as ordered lists, dot products, norms, orthogonality, and the geometric picture of n-dimensional space.
intro
Matrices & Linear Transformations
Matrix multiplication as composing linear maps: rotation, scaling, projection, and change of basis.
advanced
Eigenvalues & Matrix Decomposition
Eigenvectors as directions preserved by a transformation; eigenvalues as stretch factors; SVD as the universal factorization.
intro
Probability Fundamentals
Sample spaces, conditional probability, independence, and Bayes' rule for updating beliefs.
core
Random Variables & Distributions
Discrete and continuous distributions, expectation, variance, and the law of large numbers.
intro
Descriptive Statistics
Summarizing data: mean, median, variance, correlation, and when each summary lies.
core
Estimation & Confidence Intervals
Point estimates, standard error, confidence intervals, and what '95% confident' actually means.
core
Hypothesis Testing
Null and alternative hypotheses, p-values, significance, power, and Type I/II errors.
core
Linear Regression
Fitting a line (or hyperplane) by least squares; residuals, R², and the geometry of projection.
core
Derivatives & Gradients
Partial derivatives, the gradient vector, and reading a loss surface for descent direction.
advanced
Multivariable Optimization
Convexity, critical points, constrained optimization, and why SGD works on non-convex losses anyway.
advanced
Information & Entropy
Entropy as surprise, cross-entropy as a loss, KL divergence as a distributional distance.
core
Classical Distributions
Bernoulli, Binomial, Poisson, Normal, and Exponential — when each models the world and what to expect from them.
core
Sampling & the Central Limit Theorem
Sampling distributions, standard error of the mean, and why averages become Normal as n grows.
core
A/B Testing for Engineers
Sample size, statistical power, practical significance, SRM checks, and multiple-comparison traps in product experiments.
advanced
Bayesian Inference
Priors, posteriors, credible intervals, and when Bayesian updating beats frequentist tests.
advanced
Maximum Likelihood Estimation
Choosing parameters that make the observed data most probable; log-likelihood; connection to cross-entropy loss.
core
Covariance & Correlation
Covariance measures co-movement; correlation normalizes to [−1, 1]. Foundation for regression, PCA, and portfolio risk.
core
Bias, Variance & Overfitting
Underfitting vs memorizing noise; why in-sample greatness lies.
core
Rank, Basis & Subspaces
Column space, rank, independence — why matrices are often low-rank in practice.
advanced
PCA & Projection
Principal components as variance-maximizing orthogonal directions; projection as subspace approximation.
core
Returns & Volatility
Simple/log returns, realized volatility, annualization — basic quant units.
advanced
Stationarity & Autocorrelation
When series statistics are stable; memory in lags; noise vs signal.
core
Random Walks & Markov Chains
Markov property, random walks, transition matrices — generative story behind market efficiency intuition.
core
Sharpe, Drawdown & Portfolio Risk
Sharpe, max drawdown, correlation matrices — risk-adjusted comparison.
advanced
Momentum Backtest Discipline
Momentum rules, SPY/QQQ benchmarks, holdout honesty — why most backtests lie.
core
Queueing Theory
Little's law and the utilisation curve — why latency explodes before a system runs out of capacity.
core
Curse of Dimensionality
Why distances concentrate in high dimensions, and what that does to nearest-neighbour search.
core
Combinatorics
Counting without enumerating — permutations, combinations, inclusion-exclusion, pigeonhole.
core
Numerical Stability
Floating point, catastrophic cancellation, and the log-sum-exp trick.