Mathematical Foundations

1. M0 Calculus Toolkit 0 / 4 Gradient and direction, divergence and flux, Taylor expansion and "assume a straight line", differentiating under the integral and the total derivative along a path.

   1. M0.0 Descending Through Fog: Gradients and Direction
   2. M0.1 Counting Gate Crossings: Divergence and Flux
   3. M0.2 Thirty Seconds Without GPS: Taylor Expansion and Straight-Line Prediction
   4. M0.3 A Drifting Thermometer: Total Derivatives, Integral Differentiation, and JVPs
2. M1 Probability and Conditional Expectation 0 / 5 Multivariate Gaussians, conditional expectation, the minimizer of MSE, Bayes and the posterior, and denoising as the posterior mean (Tweedie).

   1. M1.0 Two Scales: Linear Combinations of Gaussians
   2. M1.1 Guessing Height from Shoes: Conditional Expectation
   3. M1.2 Which Temperature Should We Forecast? MSE Selects the Conditional Mean
   4. M1.3 A Positive Test: Bayes' Rule and the Posterior
   5. M1.4 Denoising as a Posterior Mean: Tweedie's Formula
3. M2 Deterministic Dynamics 0 / 5 Vector fields and ODEs, pushforward and flow maps, the continuity equation, Euler error and higher-order methods.

   1. M2.0 Arrows on a River: Vector Fields and ODEs
   2. M2.1 From One Leaf to a Cloud: Flow Maps and Pushforwards
   3. M2.2 Change in the Crowd = Inflow − Outflow: The Continuity Equation
   4. M2.3 Navigation Updates Every 30 Seconds: Euler's Method and Its Error
   5. M2.4 One More Look Halfway Through: Higher-Order Methods
4. M3 Stochastic Dynamics 0 / 5 From random walks to Brownian motion, SDEs as noisy ODEs, Fokker–Planck, Langevin and stationary distributions, weak vs. strong convergence.

   1. M3.0 Where Is the Walker After an Hour? From Random Walks to Brownian Motion
   2. M3.1 Flow Plus Fluctuations: Stochastic Differential Equations
   3. M3.2 A Thousand Leaves in a Noisy River: The Fokker–Planck Equation
   4. M3.3 Random Walkers in a Valley: Langevin Dynamics and Stationarity
   5. M3.4 Matching Paths or Matching Statistics: Strong and Weak Convergence
5. M4 Markov Chains 0 / 4 Markov chains and transition matrices, absorbing states, continuous-time Markov chains and rate matrices, time reversal and ratios.

   1. M4.0 Tomorrow Depends on Today: Markov Chains and Transition Matrices
   2. M4.1 No Way Back: Absorbing States and Absorption Times
   3. M4.2 The Phone Can Ring Anytime: Continuous-Time Markov Chains
   4. M4.3 Can You Detect a Reversed Movie? Time Reversal and Probability Ratios
6. M5 Optimization, Convexity, and Distances 0 / 4 Jensen's inequality, KL divergence and cross-entropy, moving targets and EMA, differentiating through a generator (the KL gradient is a difference of scores).

   1. M5.0 Why Insurance Can Make Sense: Convexity and Jensen's Inequality
   2. M5.1 The Cost of a Wrong Probability Table: KL Divergence and Cross-Entropy
   3. M5.2 Tracking a Moving Dog: EMA, Moving Targets, and Stop-Gradient
   4. M5.3 Adjusting a Sample-Producing Machine: Differentiating a Generator
7. M6 Optimal Transport 0 / 4 The warehouse problem (Monge / Kantorovich), the W₂ distance, one-dimensional monotone matching, and large-scale approximations (Sinkhorn, minibatch).

   1. M6.0 Moving Goods Between Warehouses: Monge, Kantorovich, and Couplings
   2. M6.1 How Far Apart Are Two Population Maps? The W₂ Distance
   3. M6.2 Uncross the Routes: One-Dimensional Monotonicity and Brenier's Theorem
   4. M6.3 Fifty Thousand Warehouses: Sinkhorn and Minibatch Approximations

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