Mathematics
What is true for certain, why, and how far the structures of number, space and change reach.
For: anyone starting the degree, anyone who liked maths until it became procedures, anyone curious what mathematicians actually do all day.
Shaped by MIT's Course 18 (real analysis, algebra, then electives across pure and applied) and the Cambridge Mathematical Tripos (Numbers and Sets, Groups, Analysis, Vectors and Matrices in the first year).
Year 1 · The language
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01ReadInteractiveQuiz10 min
Calculus I: Slopes, Totals and the Bridge Between Them
Limits, derivatives and integrals: the calculus every later course makes rigorous.
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02ReadInteractiveQuiz13 min
Linear Algebra: Machines That Move Space
Vectors and matrices: the most useful structure in mathematics.
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03ReadInteractiveQuiz12 min
Discrete Mathematics: Counting, Logic and Proof
Logic, counting and the first real proofs.
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04ReadInteractiveQuiz13 min
Foundations: What Mathematics Stands On
Sets, infinity and the axioms: what mathematics stands on, and its famous limits.
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05ReadInteractiveQuiz11 min
Programming Fundamentals: Instructions, State and Taking Things Apart
Mathematicians experiment with code before they prove.
Year 2 · Proof
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06ReadInteractiveQuiz10 min
Multivariable Calculus: Calculus on a Landscape
Calculus in many dimensions, the gateway to geometry and physics.
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07ReadInteractiveQuiz11 min
Differential Equations: Laws Written as Rates
Differential equations: where mathematics meets every science.
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08ReadInteractiveQuiz11 min
Real Analysis: Why Calculus Works
Why calculus works: limits, continuity and completeness, proved.
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09ReadInteractiveQuiz12 min
Abstract Algebra: Symmetry as a Structure of Its Own
Groups, rings and fields: symmetry as a structure of its own.
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10ReadInteractiveQuiz12 min
Probability & Statistics: Reasoning with Uncertainty
Probability: the mathematics of uncertainty.
Year 3 · The great subjects
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11ReadInteractiveQuiz14 min
Complex Analysis: Calculus on a Plane Where Everything Fits Together
Calculus with complex numbers, where everything is unexpectedly beautiful.
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12ReadInteractiveQuiz14 min
Number Theory: Primes, Clocks and the Secrets Built on Them
Primes, divisibility and the oldest open questions.
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13ReadInteractiveQuiz14 min
Topology: Shape Without Measurement
Shape without measurement: the doughnut and the coffee cup.
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14ReadInteractiveQuiz14 min
Partial Differential Equations: One Law for Every Point of a Field
The equations of heat, waves and quantum mechanics.
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15ReadInteractiveQuiz13 min
Numerical Analysis: Answers You Can Trust, and How Wrong They Are
Computing answers you can trust, and knowing how wrong they are.
Year 4 · Where it leads
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16ReadInteractiveQuiz13 min
Differential Geometry: Curvature You Can Feel From the Inside
Curvature, from soap films to Einstein's spacetime.
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17ReadInteractiveQuiz13 min
Theory of Computation: What Can and Cannot Be Computed
What can and cannot be computed: logic's other great limit.
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18ReadInteractiveQuiz12 min
Game Theory: When Your Best Move Depends on Mine
Strategy as mathematics.
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19ReadInteractiveQuiz11 min
Machine Learning: Programs That Write Their Own Rules
Learning from data is optimisation, linear algebra and probability.
MIT OpenCourseWare 18.100 (Real Analysis) and 18.701 (Algebra I), free; the textbooks named at the end of each subject. See also the library's 12 Mathematics for gentler first reads.
Physics
What the universe is made of, how it moves, and how we know: from the pendulum to the Big Bang.