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What a 22-year-old learns

115 topics open up at age 22, 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 21age 23
What should a 22-year-old be learning?

At age 22, 115 topics open up across Science, Mathematics, Computing. Each one has prerequisites that come first, so the order matters more than the list: Hamilton's Principle and the Action is a typical starting point.

How do I know what a 22-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. Hamilton's Principle and the ActionForces & Motion
  2. The Hamiltonian, Phase Space and the Canonical EquationsForces & Motion
  3. Fluctuations, Linear Response and the Fluctuation-Dissipation RelationThermal & Statistical Physics
  4. The Grand Canonical Ensemble and the Chemical PotentialThermal & Statistical Physics
  5. Chain Growth and Step Growth: Two Routes to a PolymerChemistry
  6. Cross-Coupling in Practice: Choosing Ligand, Base and ConditionsChemistry
  7. Reading a Paper and Its Supporting InformationChemistry
  8. Searching the Chemical Literature by Structure and by ReactionChemistry
  9. Characterisation Standards: What Counts as Proof of a New CompoundChemistry
  10. Solid-State Synthesis: Controlling Phase, Defect and DopantChemistry
  11. Porous Frameworks: Pores, Guests and StabilityChemistry
  12. Strategy in Retrosynthesis: Strategic Bonds and Convergent RoutesChemistry
  13. Protecting Group Strategy: Orthogonality and the Order of RemovalChemistry
  14. Structure Elucidation from a Complete Data SetChemistry
  15. NMR After Assignment: Relaxation, Distance and ExchangeChemistry
  16. Reducible Representations and Predicting Infrared and Raman ActivityChemistry
  17. Choosing Between Vibrational Methods: Infrared, Raman and Resonance RamanChemistry
  18. Single-Crystal Diffraction: From Reflections to a Refined StructureChemistry
  19. Substrate Control of Stereochemistry: Models for Facial SelectivityChemistry
  20. Asymmetric Catalysis: Where Enantioselectivity Comes FromChemistry
  21. Organocatalysis: Enamine and Iminium ActivationChemistry
  22. States as Vectors: Dirac Notation and the State SpaceQuantum & Modern Physics
  23. Observables as Hermitian Operators and Their SpectraQuantum & Modern Physics
  24. Commutators, Compatible Observables and the General Uncertainty RelationQuantum & Modern Physics
  25. Poisson Brackets and Constants of the MotionForces & Motion
  26. Noether's Theorem: Symmetry and Conserved QuantitiesForces & Motion
  27. From Coordinates to Fields: the Lagrangian DensityForces & Motion
  28. Angular Momentum from Its Commutation RelationsQuantum & Modern Physics
  29. Unitary Time Evolution and the Heisenberg PictureQuantum & Modern Physics
  30. Covariant Electrodynamics and the Field TensorElectricity & Magnetism
  31. Gauge Freedom and the Potentials as the Working VariablesElectricity & Magnetism
  32. Ideal Quantum Gases and Bose-Einstein CondensationThermal & Statistical Physics
  33. Phase transitions, order parameters and broken symmetryThermal & Statistical Physics
  34. Dimensional Analysis, Scaling and the Order-of-Magnitude EstimateScientific Inquiry
  35. Fitting a Model to Data: Least Squares, Chi-Square and Goodness of FitScientific Inquiry
  36. Systematic Error, Controls and the Design of a MeasurementScientific Inquiry
  37. Alignment Scoring: Substitution Matrices, Gap Penalties and SignificanceGenetics & Evolution
  38. Cancer as Clonal Evolution: Drivers, Passengers and Selection in a TissueGenetics & Evolution
  39. Differential Expression from Counts: Dispersion, Fit and ShrinkageGenetics & Evolution
  40. Hidden Markov models for sequence: profiles, states and decodingGenetics & Evolution
  41. Models of Sequence Evolution and Maximum Likelihood TreesGenetics & Evolution
  42. Bayesian Phylogenetics: Priors, Sampling and Judging ConvergenceGenetics & Evolution
  43. Phylogenomics: Gene Trees, Species Trees and Why They DisagreeGenetics & Evolution
  44. Stochastic Gene Expression: Bursting, Noise and Cell-to-Cell VariationGenetics & Evolution
  45. Single-Cell RNA Sequencing: From Droplets to a Count MatrixGenetics & Evolution
  46. Mapping Regulatory Elements: Accessibility, Occupancy and ContactGenetics & Evolution
  47. What a Cell Has to Break to Become a TumourOrganisms & Life Processes
  48. Oncogenes and Tumour Suppressors: Why One Allele Is Sometimes EnoughGenetics & Evolution
  49. Designing a Biological Experiment: Units, Controls and RandomisationScientific Inquiry
  50. Pseudoreplication: When n Is Smaller Than It LooksScientific Inquiry
  51. Power and Effect Size, Decided Before the DataScientific Inquiry
  52. Researcher Degrees of Freedom and the Pre-registered Analysis PlanScientific Inquiry
  53. Reading a Primary Paper: The Claim, the Figure and the Missing ControlScientific Inquiry
  54. Gene Regulatory Networks: Motifs and What They ComputeGenetics & Evolution
  55. Checkpoint Blockade: Taking a Brake Off a T CellThe Human Body
  56. Genome Engineering: Cut Site, Repair Choice and Editing OutcomeGenetics & Evolution
  57. Base and Prime Editing: Changing DNA Without a Planned Double-Strand BreakGenetics & Evolution
  58. The Tumour Microenvironment: Stroma, Hypoxia and Immune ExclusionOrganisms & Life Processes
  59. Targeted Therapy and the Resistance That Follows ItOrganisms & Life Processes

Mathematics

  1. Reading a Mathematics PaperMathematical Thinking
  2. Writing a Proof Someone Else Can CheckMathematical Thinking
  3. Sigma-Algebras and Measurable SetsCalculus & Analysis
  4. Outer Measure and the Construction of Lebesgue MeasureCalculus & Analysis
  5. Measurable Functions and Approximation by Simple FunctionsCalculus & Analysis
  6. The Lebesgue Integral and What It RepairsCalculus & Analysis
  7. Monotone Convergence, Fatou and Dominated ConvergenceCalculus & Analysis
  8. Product Measure and Fubini's TheoremCalculus & Analysis
  9. Lp Spaces and the Inequalities They Rest OnCalculus & Analysis
  10. Normed Spaces, Completeness and Banach SpacesCalculus & Analysis
  11. The Fourier Transform on the LineCalculus & Analysis
  12. Classifying Second-Order PDEs and What the Type DecidesDifferential Equations
  13. Separation of Variables and the Heat EquationDifferential Equations
  14. The Wave Equation and Its CharacteristicsDifferential Equations
  15. Laplace's Equation and the Maximum PrincipleDifferential Equations
  16. Weak Solutions and Test FunctionsDifferential Equations
  17. Conditional Expectation Given a Sigma-AlgebraProbability
  18. Markov Chains and the Memoryless AssumptionProbability
  19. Stationary Distributions and Long-Run BehaviourProbability
  20. Ideals and Quotient RingsAbstract Algebra
  21. Normal Subgroups and Quotient GroupsAbstract Algebra
  22. Group Actions, Orbits and the Class EquationAbstract Algebra
  23. Polynomial Rings, Irreducibility and Unique FactorisationAbstract Algebra
  24. Field Extensions and Their DegreesAbstract Algebra
  25. Modules: Linear Algebra Over a RingAbstract Algebra
  26. Group Representations and the Group AlgebraAbstract Algebra
  27. Maschke's Theorem and Complete ReducibilityAbstract Algebra
  28. The Galois CorrespondenceAbstract Algebra
  29. The Sylow Theorems and Groups of Small OrderAbstract Algebra
  30. Definitions as Design DecisionsMathematical Thinking
  31. Counterexamples and Testing a ConjectureMathematical Thinking

Computing

  1. Complexity Beyond NP: Space, Randomness and the HierarchyAlgorithms & Data Structures
  2. Randomised Algorithms and the Probabilistic MethodAlgorithms & Data Structures
  3. SAT Solvers and What Makes Search PracticalAlgorithms & Data Structures
  4. Impossibility Under Asynchrony: What Consensus Cannot PromiseNetworks & Security
  5. Raft: Leader Election and Log ReplicationNetworks & Security
  6. Quorums, Leases and Changing the MembershipNetworks & Security
  7. Linearizability and Checking a HistoryNetworks & Security
  8. Distributed Transactions and the Blocking Case of Two-Phase CommitData & Databases
  9. Conflict-Free Replicated Data TypesData & Databases
  10. Log-Structured Storage and CompactionData & Databases
  11. Specifying Correctness: Safety, Liveness and InvariantsSoftware Engineering
  12. Concentration Bounds and a High-Probability GuaranteeAlgorithms & Data Structures
  13. Approximation Ratios and How One Is ProvedAlgorithms & Data Structures
  14. Linear Programming, Duality and RoundingAlgorithms & Data Structures
  15. Attention as a Learned LookupMachine Learning
  16. The Transformer Block: Heads, Residuals and NormalisationMachine Learning
  17. Tokenisation and What a Model Actually ReadsMachine Learning
  18. Pretraining by Predicting the Next TokenMachine Learning
  19. Scaling Laws and the Compute BudgetMachine Learning
  20. Adapting a Pretrained Model: Fine-Tuning and Low-Rank UpdatesMachine Learning
  21. Generative Models: Learning a Distribution You Can SampleMachine Learning
  22. Diffusion Models: Learning to Undo NoiseMachine Learning
  23. Reading a Systems or Machine-Learning PaperSoftware Engineering
  24. Benchmarks and the Traps in ThemMachine Learning
  25. Markov Decision Processes: States, Actions and ReturnMachine Learning
What a 22-year-old learns · LightMySky