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St. CharlesCity in Germany 1 · school ＋ Arithmetic Numbers, fractions, percentages and the order of operations. St. CharlesCity in Germany 2 · school 𝑥 Algebra Equations, inequalities, factoring, functions and systems. St. CharlesCity in Germany 3 · school △ Geometry Triangles, circles, polygons, solids and coordinates. St. CharlesCity in Germany 4 · school ∿ Trigonometry Angles, the unit circle, identities and trig equations. St. CharlesCity in Germany 5 · school ƒ Precalculus Functions in depth, complex numbers, sequences, conics and vectors. St. CharlesCity in Germany 6 · school σ Statistics & Probability Averages, spread, counting and probability. St. CharlesCity in Germany 7 · school ⊢ Discrete Math & Logic Counting, logic, truth tables, sets and proof. St. CharlesCity in Germany 8 · college ∫ Calculus Limits, derivatives, integrals, series and optimisation. St. CharlesCity in Germany 9 · college ⊞ Linear Algebra Matrices, determinants, inverses, row reduction and eigenvalues. St. CharlesCity in Germany 10 · college ℕ Number Theory Primes, divisibility, gcd, modular arithmetic and sequences. St. CharlesCity in Germany 11 · college ẏ Differential Equations Equations whose unknown is a function: growth, oscillation, and how physics is written. St. CharlesCity in Germany 12 · college ∇ Multivariable Calculus Partial derivatives, gradients, multiple integrals, vector fields. St. CharlesCity in Germany 13 · college ε Real Analysis Limits made rigorous: sequences, series, convergence, improper integrals, complex numbers as functions. St. CharlesCity in Germany 14 · college ⋔ Combinatorics & Graph Theory Counting, generating functions, graphs, trees, colourings and networks. St. CharlesCity in Germany 15 · college ∈ Set Theory & Logic Sets, relations, functions, cardinality, proof and the foundations of mathematics. St. CharlesCity in Germany 16 · college 𝔾 Abstract Algebra Groups, rings, fields: symmetry made into structure. St. CharlesCity in Germany 17 · college 𝑃 Probability Theory Random variables, distributions, expectation, limit theorems. St. CharlesCity in Germany 18 · college ≈ Numerical Methods Root finding, approximation, numerical integration and error. St. CharlesCity in Germany 19 · college ℂ Complex Analysis Holomorphic functions, contour integrals, residues, the rigidity of the complex plane. St. CharlesCity in Germany 20 · graduate ◎ Topology Open sets, continuity, compactness, connectedness: geometry without distance. St. CharlesCity in Germany 21 · graduate μ Measure Theory Lebesgue measure and integral, σ-algebras, convergence theorems. St. CharlesCity in Germany 22 · graduate ‖·‖ Functional Analysis Banach and Hilbert spaces, operators, spectral theory. St. CharlesCity in Germany 23 · graduate κ Differential Geometry Curves, surfaces, curvature, manifolds: the language of general relativity. St. CharlesCity in Germany 24 · graduate π₁ Algebraic Topology Fundamental groups, homology: counting holes with algebra. St. CharlesCity in Germany 25 · graduate → Category Theory Objects, arrows, functors, universal properties: mathematics about mathematics. St. CharlesCity in Germany 26 · graduate ∂ Partial Differential Equations Heat, waves, potentials and transport: Fourier methods, energy estimates, Sobolev spaces and scaling. St. CharlesCity in Germany 27 · graduate ≋ Fluid Dynamics Continuity, vorticity, the Euler and Navier-Stokes equations, and what is proved about them. St. CharlesCity in Germany 30 · research ∞ Frontiers The open problems (Riemann, Navier-Stokes, P vs NP) broken into what you need to even read them.