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What a 20-year-old learns

190 topics open up at age 20, across 3 subjects. They are listed in the order they build on each other, not by importance: each one needs the ones above it.

age 19age 21
What should a 20-year-old be learning?

At age 20, 190 topics open up across Science, Mathematics, Computing. Each one has prerequisites that come first, so the order matters more than the list: Intensity in Single-Slit and Double-Slit Diffraction is a typical starting point.

How do I know what a 20-year-old already knows?

LightMySky starts by assuming everything below the learner’s level is already known, then checks the assumption as they practise. What holds up stays lit; what does not comes back for review.

Science

  1. Intensity in Single-Slit and Double-Slit DiffractionWaves, Light & Sound
  2. The Michelson InterferometerWaves, Light & Sound
  3. X-Ray Diffraction and the Bragg ConditionWaves, Light & Sound
  4. Rotational Kinematics and the Angular Velocity VectorForces & Motion
  5. Microstates, Multiplicity and Boltzmann's EntropyThermal & Statistical Physics
  6. The Boltzmann Factor and the Partition FunctionThermal & Statistical Physics
  7. Free Energy and the Direction of Spontaneous ChangeThermal & Statistical Physics
  8. Flux, Magnitudes and the Distance LadderAstronomy & Astrophysics
  9. Moment of Inertia by Integration and the Parallel-Axis TheoremForces & Motion
  10. Rotational Dynamics and Rolling Without SlippingForces & Motion
  11. Angular Momentum and Its ConservationForces & Motion
  12. The Two-Body Problem, Orbital Energy and Kepler's First Two LawsForces & Motion
  13. Viscosity, Poiseuille Flow and the Reynolds NumberForces & Motion
  14. Aromaticity: Huckel's Rule and Antiaromatic SystemsChemistry
  15. Directing Effects in Electrophilic Aromatic SubstitutionChemistry
  16. Hartree-Fock and What a Basis Set DecidesChemistry
  17. Density Functional Theory and Choosing a FunctionalChemistry
  18. Ligand field theory and pi bonding in complexesChemistry
  19. Electronic spectra of complexes and Tanabe-Sugano diagramsChemistry
  20. Nucleophilic Addition and the Tetrahedral IntermediateChemistry
  21. Acetals, Imines and Protecting GroupsChemistry
  22. Nucleophilic Acyl Substitution and the Reactivity LadderChemistry
  23. Enols, Enolates and Alpha SubstitutionChemistry
  24. Aldol and Claisen: Building Carbon SkeletonsChemistry
  25. Nucleophilic Aromatic Substitution, Diazonium Salts and HeterocyclesChemistry
  26. Organometallic Nucleophiles: Grignard and Organolithium ReagentsChemistry
  27. Pericyclic Reactions: The Diels-Alder and Orbital SymmetryChemistry
  28. The Molecular Partition FunctionChemistry
  29. Entropy from Counting: The Molecular Meaning of the Third LawChemistry
  30. Steady-State and Pre-Equilibrium ApproximationsChemistry
  31. Chain Reactions, Branching and Explosion LimitsChemistry
  32. Stability Constants and the Kinetics of Ligand SubstitutionChemistry
  33. Electron Counting and the Eighteen-Electron RuleChemistry
  34. Metals in Biology: Active Sites, Electron Transfer and ChelationChemistry
  35. Organometallic Ligands: Carbonyls, Phosphines and HapticityChemistry
  36. Elementary Steps of Catalysis: Oxidative Addition and InsertionChemistry
  37. Transition State Theory and the Eyring EquationChemistry
  38. Chemical Potential and Gibbs Energy That Depends on CompositionChemistry
  39. Computed Geometries, Frequencies and Reaction BarriersChemistry
  40. Real Gases, Fugacity and ActivityChemistry
  41. Phase Diagrams, the Clapeyron Equation and the Phase RuleChemistry
  42. Ideal and Real Solutions: Raoult, Henry and Excess FunctionsChemistry
  43. Electrolyte Solutions and the Debye-Huckel Limiting LawChemistry
  44. Ion Exchange and Electrophoretic SeparationsChemistry
  45. Mass Spectrometry: Ionisation Methods and Mass AnalysersChemistry
  46. Accurate Mass and Isotope Patterns as Formula EvidenceChemistry
  47. Hyphenated Techniques: GC-MS, LC-MS and Tandem FragmentationChemistry
  48. Reading a Proton NMR Spectrum: Shift, Integration and MultiplicityChemistry
  49. Carbon-13, DEPT and Coupling ConstantsChemistry
  50. Green Chemistry Metrics: Atom Economy, E-Factor and Solvent ChoiceChemistry
  51. Decay Modes, Decay Chains and Nuclear ReactionsQuantum & Modern Physics
  52. Fission Reactors and Fusion in the LaboratoryQuantum & Modern Physics
  53. Nuclear Models: the Liquid Drop and the Semi-Empirical Mass FormulaQuantum & Modern Physics
  54. Particle Accelerators, Detectors and Cross-SectionsQuantum & Modern Physics
  55. The Finite Well and Wavefunction PenetrationQuantum & Modern Physics
  56. Quantum Tunnelling and the Transmission ProbabilityQuantum & Modern Physics
  57. The Quantum Harmonic Oscillator and Zero-Point EnergyQuantum & Modern Physics
  58. Operators, Eigenvalues and MeasurementQuantum & Modern Physics
  59. Separating the Schrödinger Equation for HydrogenQuantum & Modern Physics
  60. The Standard Model and the Four InteractionsQuantum & Modern Physics
  61. Magnetisation: Diamagnetism, Paramagnetism and FerromagnetismElectricity & Magnetism
  62. Self-Inductance and Energy Stored in a Magnetic FieldElectricity & Magnetism
  63. Oscillations in LC and RLC CircuitsElectricity & Magnetism
  64. Impedance and Phasors in AC CircuitsElectricity & Magnetism
  65. Resonance and Power in AC CircuitsElectricity & Magnetism
  66. Displacement Current and Maxwell's EquationsElectricity & Magnetism
  67. Hydrostatic Equilibrium and the Structure of a StarAstronomy & Astrophysics
  68. Energy Transport and Nuclear Burning in StarsAstronomy & Astrophysics
  69. Stellar Spectra, Spectral Classes and Surface CompositionAstronomy & Astrophysics
  70. Sequencing reads and how a genome is assembledGenetics & Evolution
  71. Sequence alignment and reading a search resultGenetics & Evolution
  72. Distance and Likelihood Trees, and What Branch Support MeansGenetics & Evolution
  73. Using Trees: Ancestral States, Dates and Comparative TestsGenetics & Evolution
  74. Signal Integration: Cross-talk, Feedback and TerminationOrganisms & Life Processes
  75. Apoptosis: Caspase Cascades and the Decision to DieOrganisms & Life Processes
  76. Actin Filaments: Polarity, Treadmilling and Cell ShapeOrganisms & Life Processes
  77. Microtubules, Dynamic Instability and the SpindleOrganisms & Life Processes
  78. Motor Proteins and Directed Movement Along TracksOrganisms & Life Processes
  79. Junctions, Adhesion and the Extracellular MatrixOrganisms & Life Processes
  80. Building a Visual Scene: Pathways, Maps and Receptive FieldsThe Human Body
  81. Hormone Classes and Why Their Receptors Sit in Different PlacesThe Human Body
  82. Hypothalamic and Pituitary Axes and Layered FeedbackThe Human Body
  83. The Autonomic Nervous System and Its Two OutflowsThe Human Body
  84. Resistance, Flow and the Regulation of Blood PressureThe Human Body
  85. Lung Mechanics: Compliance, Surface Tension and the Work of BreathingThe Human Body
  86. Touch, Temperature and PainThe Human Body
  87. Hearing and Balance: Frequency, Place and Head MotionThe Human Body
  88. Ventilation and Perfusion Matching, and the Control of BreathingThe Human Body
  89. Acid-Base Balance: The Bicarbonate System and CompensationThe Human Body
  90. Morphogen Gradients and Positional InformationOrganisms & Life Processes
  91. Alternative splicing and one gene, many proteinsBiochemistry & Molecular Biology
  92. Small RNAs and silencing after transcriptionGenetics & Evolution
  93. Methylation, Histone Marks and Heritable Expression StatesGenetics & Evolution
  94. Gluconeogenesis and the Reciprocal Regulation of Opposing PathwaysBiochemistry & Molecular Biology
  95. The Citric Acid Cycle as a Hub, and Refilling ItBiochemistry & Molecular Biology
  96. Redox Potentials and the Proton-Motive ForceBiochemistry & Molecular Biology
  97. ATP Synthase as a Rotary MachineBiochemistry & Molecular Biology
  98. Fatty Acid Oxidation, Ketone Bodies and Choosing a FuelBiochemistry & Molecular Biology
  99. Organogenesis and the Model Organisms That Made It VisibleOrganisms & Life Processes
  100. Horizontal Gene Transfer: Uptake, Contact and DeliveryMicrobiology
  101. Pathogenesis: Adhesion, Invasion and Virulence FactorsMicrobiology
  102. Exotoxins, Endotoxin and How Damage Is Actually DoneMicrobiology
  103. Plasmids, Mobile Elements and How Resistance SpreadsMicrobiology
  104. CRISPR: A Bacterial Defence Turned into a ToolMicrobiology
  105. Optimal Foraging and the Currency of a Behavioural DecisionEcology & Conservation
  106. Sexual Selection, Mate Choice and Parental InvestmentEcology & Conservation
  107. Kin Selection, Relatedness and When Helping PaysEcology & Conservation
  108. Biogeography: Why a Species Is Here and Not ThereEcology & Conservation
  109. Demography: life tables, age structure and what they predictEcology & Conservation
  110. Life-History Trade-offs and How Reproduction Is AllocatedEcology & Conservation
  111. Competition, Niches and What Allows CoexistenceEcology & Conservation

Mathematics

  1. LU Factorisation, Pivoting and the Cost of a SolveLinear Algebra
  2. Double Integrals in Polar CoordinatesCalculus & Analysis
  3. The Completeness Axiom: Suprema and InfimaCalculus & Analysis
  4. Orthogonal Projection and Least SquaresLinear Algebra
  5. The Gram-Schmidt ProcessLinear Algebra
  6. Symmetric Matrices and the Spectral TheoremLinear Algebra
  7. The Singular Value DecompositionLinear Algebra
  8. Arc Length and Curvature of a Space CurveGeometry
  9. Convergence of Sequences, RigorouslyCalculus & Analysis
  10. Cauchy Sequences and the Bolzano-Weierstrass TheoremCalculus & Analysis
  11. Continuity and Uniform ContinuityCalculus & Analysis
  12. Differentiability and the Theorems Behind the RulesCalculus & Analysis
  13. Polynomial Interpolation and Its Error TermCalculus & Analysis
  14. Numerical Quadrature: Newton-Cotes and Adaptive RulesCalculus & Analysis
  15. The Riemann Integral and When a Function Is IntegrableCalculus & Analysis
  16. Pointwise and Uniform ConvergenceCalculus & Analysis
  17. Triple Integrals and Coordinates for SolidsCalculus & Analysis
  18. Change of Variables and the JacobianCalculus & Analysis
  19. Vector Fields and Line IntegralsCalculus & Analysis
  20. Conservative Fields and Path IndependenceCalculus & Analysis
  21. Green's Theorem in the PlaneCalculus & Analysis
  22. Divergence and CurlCalculus & Analysis
  23. Surface Integrals and FluxCalculus & Analysis
  24. Stokes' Theorem and the Divergence TheoremCalculus & Analysis
  25. Second-Order Linear Equations with Constant CoefficientsDifferential Equations
  26. Nonhomogeneous Equations and Particular SolutionsDifferential Equations
  27. Damped and Driven OscillationsDifferential Equations
  28. Fourier SeriesDifferential Equations
  29. Series Solutions About an Ordinary PointDifferential Equations
  30. The Laplace TransformDifferential Equations
  31. Harmonic Functions and What Analyticity ForcesComplex Analysis
  32. Contour Integrals Along Parametrised PathsComplex Analysis
  33. Cauchy's Theorem and Deforming a ContourComplex Analysis
  34. Cauchy's Integral Formula and Derivatives of Every OrderComplex Analysis
  35. Power Series and Analyticity in the Complex PlaneComplex Analysis
  36. Point Estimation: Bias, Variance and ConsistencyData & Statistics
  37. Maximum Likelihood EstimationData & Statistics
  38. Confidence Intervals for Means and ProportionsData & Statistics
  39. Hypothesis Tests for a Mean with the t-DistributionData & Statistics
  40. Errors, Power and the Design of a TestData & Statistics
  41. Generating FunctionsDiscrete Mathematics
  42. Graphs, Degrees and the Handshake LemmaDiscrete Mathematics
  43. Trees and Spanning TreesDiscrete Mathematics
  44. Fermat's Little Theorem and Euler's TheoremNumber Theory
  45. Primitive Roots and the Units Modulo nNumber Theory
  46. Metric Spaces: Distance as an AxiomTopology
  47. Open Sets, Closed Sets and Limit PointsTopology

Computing

  1. Test Doubles and Testing at a BoundarySoftware Engineering
  2. Test Strategy: Unit, Integration and End to EndSoftware Engineering
  3. Paging, Page Faults and Replacement PoliciesComputer Systems
  4. File Systems: Names, Blocks and MetadataComputer Systems
  5. Crash Consistency and the JournalComputer Systems
  6. Threads and Shared MemoryComputer Systems
  7. Race Conditions and Mutual ExclusionComputer Systems
  8. Deadlock and the Four ConditionsComputer Systems
  9. Clocks, Ordering and Happens-BeforeNetworks & Security
  10. Symmetric Encryption and What a Cipher PromisesNetworks & Security
  11. Dynamic Programming: Optimal Substructure and Overlapping SubproblemsAlgorithms & Data Structures
  12. Dynamic Programming on Two SequencesAlgorithms & Data Structures
  13. Dynamic Programming with a Capacity: KnapsackAlgorithms & Data Structures
  14. Shortest Paths with Negative Weights and Between All PairsAlgorithms & Data Structures
  15. Union-Find and Minimum Spanning TreesAlgorithms & Data Structures
  16. Isolation Levels and the Anomalies They AllowData & Databases
  17. Concurrency Control: Locks and SnapshotsData & Databases
  18. Crash Recovery and Write-Ahead LoggingData & Databases
  19. Failure Models and ReplicationNetworks & Security
  20. Consensus and Why Agreement Is HardNetworks & Security
  21. Consistency Models and the CAP Trade-offNetworks & Security
  22. Key-Value and Document StoresData & Databases
  23. Wide-Column and Graph StoresData & Databases
  24. Property-Based TestingSoftware Engineering
  25. Continuous Integration and the Build PipelineSoftware Engineering
  26. Regularisation: Charging for ComplexityMachine Learning
  27. Cross-Validation and Honest Model SelectionMachine Learning
  28. Beyond Accuracy: Precision, Recall and the Cost of an ErrorMachine Learning
  29. Decision Trees and EnsemblesMachine Learning
  30. Clustering: Structure Without LabelsMachine Learning
  31. Principal Components and Dimensionality ReductionMachine Learning
  32. Public-Key Cryptography and Key ExchangeNetworks & Security
What a 20-year-old learns · LightMySky