Courses

Compute with random variables, justify statistical inference from data, and model systems that evolve at random.

1. Calculus one page Limits, derivatives, integrals, sequences, and series in one variable. Open
2. Probability Theory 8 chapters · 175 exercises Random variables, conditioning, expectation, the laws of large numbers, and the central limit theorem. Start Chapter 1, Probability Spaces and Counting
3. Mathematical Statistics 13 chapters · 262 exercises Estimation, likelihood, testing, regression, experimental design, and Bayesian inference. Start Chapter 1, Statistical Models, Parameters, and Identifiability
4. Stochastic Processes one page Markov chains, Poisson processes, renewal theory, queues, and martingales. Open

Set up linear, convex, dynamic, and combinatorial decision problems, solve them, and read what the solution says.

1. Linear Algebra one page Vectors, matrices, rank, orthogonality, eigenvalues, and matrix decompositions. Open
2. Multivariable Calculus 9 chapters · 184 exercises Gradients, Jacobians, Hessians, multiple integrals, and vector calculus. Start Chapter 1, Vectors, Curves, and Geometry in Space
3. Continuous Optimization 10 chapters · 202 exercises Linear and convex optimization, duality, KKT conditions, and the algorithms that solve them. Start Chapter 1, Optimization Models and Feasible Geometry
4. Dynamic Programming and Optimal Control 10 chapters · 196 exercises Bellman recursions, Markov decision processes, and optimal control. Start Chapter 1, Sequential Models and Sufficient State
5. Discrete Optimization 10 chapters · 196 exercises Networks, matching, integer programming, branch-and-bound, and cutting planes. Start Chapter 1, Discrete Models, Encodings, and Complexity

Model strategic interaction, compute equilibria, design auctions and mechanisms, and shape what others learn.

1. Discrete Mathematics 8 chapters · 195 exercises Logic, proof, counting, recurrences, graphs, and relations. Start Chapter 1, Logic and the Language of Proof
2. Probability Theory 8 chapters · 175 exercises Random variables, conditioning, expectation, the laws of large numbers, and the central limit theorem. Start Chapter 1, Probability Spaces and Counting
3. Continuous Optimization 10 chapters · 202 exercises Linear and convex optimization, duality, KKT conditions, and the algorithms that solve them. Start Chapter 1, Optimization Models and Feasible Geometry
4. Game Theory 10 chapters · 194 exercises Nash equilibrium, dynamic and repeated games, incomplete information, auctions, and cooperative games. Start Chapter 1, Strategic Form, Best Responses, and Dominance
5. Information Design 3 chapters · 65 exercises Signaling, cheap talk, the value of information, Bayesian persuasion, and information design. Start Chapter 1, Beliefs as the Object of Choice

Handle Brownian motion, Itô's formula, and stochastic differential equations on a measure-theoretic footing.

1. Probability Theory 8 chapters · 175 exercises Random variables, conditioning, expectation, the laws of large numbers, and the central limit theorem. Start Chapter 1, Probability Spaces and Counting
2. Stochastic Processes one page Markov chains, Poisson processes, renewal theory, queues, and martingales. Open
3. Real Analysis 8 chapters · 187 exercises Rigorous limits, continuity, differentiation, Riemann integration, and metric spaces. Start Chapter 1, Completeness of the Real Numbers
4. Measure Theory 10 chapters · 185 exercises Sigma-algebras, measures, Lebesgue integration, and the convergence theorems. Start Chapter 1, Sigma-Algebras and Measures
5. Measure-Theoretic Probability 10 chapters · 197 exercises Probability on measure spaces, conditional expectation, and martingale convergence. Start Chapter 1, Random Elements, Laws, and Expectation
6. Stochastic Calculus one page Brownian motion, quadratic variation, the Itô integral and formula, and stochastic differential equations. Open

Carry rigorous limits from the real line through Lebesgue integration to Banach and Hilbert spaces.

1. Calculus one page Limits, derivatives, integrals, sequences, and series in one variable. Open
2. Real Analysis 8 chapters · 187 exercises Rigorous limits, continuity, differentiation, Riemann integration, and metric spaces. Start Chapter 1, Completeness of the Real Numbers
3. Measure Theory 10 chapters · 185 exercises Sigma-algebras, measures, Lebesgue integration, and the convergence theorems. Start Chapter 1, Sigma-Algebras and Measures
4. Functional Analysis 10 chapters · 197 exercises Banach and Hilbert spaces, bounded operators, duality, and compactness. Start Chapter 1, Normed Spaces and Completeness

Solve and analyze ordinary and partial differential equations, and compute with them reliably.

1. Calculus one page Limits, derivatives, integrals, sequences, and series in one variable. Open
2. Linear Algebra one page Vectors, matrices, rank, orthogonality, eigenvalues, and matrix decompositions. Open
3. Differential Equations 8 chapters · 160 exercises Ordinary differential equations, linear systems, stability, and phase analysis. Start Chapter 1, Differential Equations as Models
4. Multivariable Calculus 9 chapters · 184 exercises Gradients, Jacobians, Hessians, multiple integrals, and vector calculus. Start Chapter 1, Vectors, Curves, and Geometry in Space
5. Partial Differential Equations 10 chapters · 185 exercises First-order equations, the heat, wave, and Laplace equations, characteristics, and energy methods. Start Chapter 1, PDE Models and Classification
6. Numerical Analysis 10 chapters · 216 exercises Error, floating point, root finding, interpolation, quadrature, and numerical linear algebra. Start Chapter 1, Floating-Point Arithmetic and Conditioning

Read and write proofs about groups, rings, fields, and quotient structures.

1. Discrete Mathematics 8 chapters · 195 exercises Logic, proof, counting, recurrences, graphs, and relations. Start Chapter 1, Logic and the Language of Proof
2. Abstract Algebra one page Groups, rings, ideals, quotients, fields, and algebraic proof. Open

Size a two-stage launcher for a stated payload and orbit, and defend every number.

1. Calculus one page Limits, derivatives, integrals, sequences, and series in one variable. Open
2. Rocket Science 12 chapters · 260 exercises Size a two-stage launcher from the rocket equation to a converged design. Start Chapter 1, Momentum and the Rocket Equation

Handle everyday exchanges in Italian at level A2.

1. Italian A2 one page Everyday Italian to level A2, from greetings and routines to travel, appointments, and short writing. Open