LightMySky
Explore

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

Mathematics

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

Computing

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