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

Mathematics

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

Computing

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