Stuff I’ve learned (provided by Ryan)

Math 109 (Mathematical Reasoning)
- Propositional logic; logical connectives
- Implications; direct proof; proof by cases; contradiction
- Induction
- Sets (union, intersection, complement), finite/infinite, countable
- Quantifiers (for all, there exists)
- Functions, bijections, domains/codomains
- Division; Euclid's Algorithm; congruence mod m; primes

Math 170A (Numerical Analysis: Linear Algebra)
- Flops; Big-O; Gaussian elimination; back substitution
- LU/PLU; permutation matrices; banded matrices
- Positive definiteness; symmetric matrices; Cholesky
- Norms; induced norms; condition numbers; perturbation theory
- Orthogonality; projectors; Householder reflectors; QR
- Least squares; SVD; pseudoinverse
- Eigenmethods; defective matrices; similarity; diagonalization
- Power method; Hessenberg form; QR iteration
- Jacobi; Gauss–Seidel

Math 170B (Numerical Analysis: Nonlinear)
- Taylor's theorem (Lagrange remainder); IVT; MVT
- Convergence and order; finite-precision arithmetic; roundoff
- Bisection; Newton; secant; fixed-point iteration; contraction mapping
- Polynomial interpolation (Newton, Lagrange); error; Horner's method
- Divided differences; Hermite; B-splines; cubic splines
- Numerical differentiation; Richardson extrapolation
- Numerical integration: Newton–Cotes, composite rules, Gaussian quadrature

Math 154 (Graph Theory & Discrete Mathematics)
- Degrees, neighborhoods; handshaking lemma; digraphs & networks
- Subgraphs; walks, tours; connected/Eulerian/Hamiltonian
- De Bruijn sequences; bridges; trees; BFS; bipartite graphs
- Prim's, Kruskal's, Dijkstra's
- Block decomposition; Menger's theorems; edge connectivity
- Matchings, factors; independent sets and covers; Hall's; König; Tutte
- Vizing's theorem; Brooks' theorem
- Max-flow min-cut; flows, capacities, cuts

Math 184 (Enumerative Combinatorics)
- Counting principles; inclusion–exclusion; bijective proofs
- Generating functions (ordinary/exponential); formal power series
- Compositions; partitions; binary strings; substring restrictions
- Recurrences; asymptotics; Catalan numbers
- Lagrange inversion; Cayley’s formula; counting graphs

Math 142A/B (Analysis I/II)
- N, Q, R; Archimedean property; sup/inf; completeness
- Sequences: limits, monotone conv., subsequences, limsup/liminf; series
- Continuity, uniform continuity; epsilon–delta; IVT, EVT; MVT
- Power series; uniform convergence; differentiation/integration of series
- Weierstrass approximation; M-test; Riemann/Darboux integrals; properties

Math 180A (Probability)
- Sample spaces, random variables; conditional probability; Bayes
- Independence; distributions; expectation; variance; MGFs
- Joint distributions; independence; CLT; LLN

Math 103A/B (Abstract Algebra I/II)
- Sets, cardinality; functions; partitions; equivalence relations
- Groups; isomorphisms; abelian groups; permutation/dihedral groups; subgroups
- Cosets; Lagrange; normal subgroups; homomorphisms; quotient groups
- First isomorphism theorem; inner automorphisms; commutators; simple groups
- Rings, integral domains, fields; ideals; principal ideals; prime/maximal
- Polynomial rings; evaluation; factor theorem; division algorithm
- Irreducibility tests; cyclotomic polynomials; Euclidean algorithm
- UFD/PID/ED hierarchy; prime vs irreducible
- Field extensions; degree; tower law; algebraic/transcendental; minimal polynomials
- Splitting fields; intro to Galois theory; insolubility of the quintic

Math 120A (Complex Analysis)
- Complex numbers; triangle inequality; conjugates; exponential form
- Roots; regions; mappings; limits (incl. infinity); continuity; derivatives
- Cauchy–Riemann; conditions for differentiability; harmonic functions; reflection
- Elementary functions; zeros & singularities; contours and integrals
- Bounds; Cauchy–Goursat; simply-connected domains; Cauchy integral formula

Math 130 (Dynamical Systems)
- Fixed points; (linear) stability; existence/uniqueness
- Bifurcations: saddle-node, transcritical, pitchfork
- Phase plane/portraits; linearization; conservative/gradient systems
- Index theory; closed orbits; Lyapunov functions; Dulac’s criterion

Math 187A (Cryptography)
- Classical ciphers (Caesar, Vigenère, Playfair, Hill, etc.)
- Modular arithmetic; GCD; frequency analysis; index of coincidence
- Known-plaintext attacks; G-test
- RSA; discrete log (ElGamal, Diffie–Hellman); elliptic curves (R/\(\mathbb{F}_p\))

Math 181A (Mathematical Statistics)
- Discrete/continuous distributions; gamma function; Monte Carlo
- MME, MLE; log-likelihood; parameter-dependent support
- Bias, variance, MSE; efficiency; Fisher information; CRLB; consistency
- CIs (mean, proportion, variance); tests; p-values; power
- Neyman–Pearson; UMP; GLR; chi-square, t, F
- Bayesian vs frequentist; priors/posteriors; conjugacy; Bayesian point estimates

Math 180B (Stochastic Processes)
- Conditional expectation; random sums; convolution
- Bivariate normal; covariance; linear combos
- Markov chains: transitions, Chapman–Kolmogorov, hitting times, absorption
- Classification, recurrence/transience, stationary distributions, periodicity
- Branching processes; Poisson processes; thinning/superposition; conditioning

Math 170C (Numerical ODEs)
- Finite differences; Richardson extrapolation; quadrature (Newton–Cotes, Gaussian)
- IVPs: Euler/Taylor/RK; embedded/adaptive; multistep (Adams–Bashforth/Moulton)
- Stability/consistency/convergence; root condition; Dahlquist; stiffness; A-stability
- Systems and higher order ODEs; autonomous systems
- BVPs: shooting, multiple shooting, finite differences, collocation

Math 194 (Mathematics of Finance)
- No-arbitrage, law of one price; compounding
- Binomial model (single/multi-period); risk-neutral measure; replication
- Portfolios; contingent claims; pricing by backward induction; hedging
- Martingales; stopping times; optional stopping; American options; Snell envelope
- EMM; fundamental theorems of asset pricing; martingale representation
- Continuous time: GBM; Black–Scholes