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THE LIVING INDEX
A map of ideas
Mathematics, computer science, and theory of computation. A growing tree of definitions, results, and exercises.
Look up a definition, check a theorem’s assumptions, then test your understanding. Follow prerequisites when an idea is unfamiliar; follow connections when it sparks another question.
Foundations
The language of sets, maps, and proofs.
Algebra
Equations, polynomials, and arithmetic structures.
Linear algebra
Vector spaces, linear maps, and spectral structure.
Calculus & analysis
Limits, local change, accumulation, and approximation.
- Limits & continuityMake approximation statements precise by controlling errors.
- Derivatives & Taylor expansionThe derivative is a local linear model; Taylor expansion adds controlled higher-order terms.
- Integration & accumulationIntegrals combine local contributions; the fundamental theorem links accumulation and change.
- Gradients, Hessians & change of variablesLocal geometry is encoded by first and second derivatives.
Probability
Randomness, conditioning, and limiting behavior.
- Events, conditioning & BayesUpdate probabilities by restricting the sample space and renormalizing.
- Random variables & momentsA distribution describes a random quantity; moments summarize selected features.
- LLN, CLT & concentrationUnderstand what averaging guarantees—and how much data a guarantee needs.
- Modes of convergenceDistinguish pathwise behavior, small error probabilities, average error, and convergence of laws.
- Conditional expectationAverage using the information available; least squares and conditioning are versions of the same projection idea.
Statistics
Learning from data while accounting for uncertainty.
- Estimation & likelihoodChoose a model, then distinguish an unknown parameter from a statistic computed on data.
- Confidence intervals & testsExpress sampling uncertainty without confusing the data with the hypothesis.
- Linear regressionFit conditional means while keeping the assumptions behind inference visible.
Group theory
Symmetry, actions, and quotient structures.
Representation theory
Understand symmetry through linear transformations.
- Representations & invariant subspacesReplace abstract symmetries by matrices that preserve composition.
- Characters & decompositionTraces summarize finite-group representations without fixing a basis.
- Direct sums, tensor products & dualsBuild new representations and track how their dimensions and characters change.
Computer science
Data, logic, and the correctness of programs.
- Data structures & amortized costChoose a representation by the operations and guarantees the application needs.
- Boolean logic, circuits & satisfiabilityConnect logical specifications to finite computations on bits.
- Recursion, invariants & terminationA correct output and a guaranteed halt are separate obligations.
Algorithms
Design procedures, prove them correct, and bound their cost.
- Asymptotics & recurrencesState the input-size measure and computational model before counting operations.
- Sorting, binary search & lower boundsUse ordering to narrow possibilities, and distinguish algorithm cost from unavoidable cost.
- Graph traversal & shortest pathsChoose a traversal based on edge weights, direction, and the question being asked.
- Dynamic programming & greedy choiceIdentify what information a subproblem needs, or prove why a local choice is safe.
Theory of computation
Languages, machines, and the limits of computation.
- Finite automata & regular languagesFinite memory recognizes patterns whose relevant history has only finitely many cases.
- Context-free grammars & parsingA stack adds unbounded nesting to finite-state control.
- Turing machines, deciders & recognizersDistinguish computing an answer eventually from guaranteeing an answer on every input.
- Reductions & undecidabilityTransfer impossibility by showing how a hypothetical solver would solve a known impossible problem.
- P, NP & completenessAsk what can be computed efficiently after establishing that it can be computed at all.
Real analysis
Completeness, convergence, and rigorous approximation.
- Completeness & sequencesCompleteness guarantees that approximations whose internal error vanishes have a limit in the space.
- Metric compactness & fixed pointsCompactness controls arbitrary covers; contraction controls successive approximations.
- Series & uniform convergenceA pointwise limit answers one input at a time; a uniform limit controls all inputs at once.
Measure theory
Measure sets, integrate functions, and exchange limits safely.
- σ-algebras & measuresNot every subset needs to be measurable; specify the collection on which size is defined.
- Lebesgue integration & Lᵖ spacesBuild integrals from measurable simple functions, then identify functions that differ only on null sets.
- Convergence theorems & iterated integralsBefore exchanging a limit or order of integration, identify the theorem whose hypotheses apply.
Complex analysis
Holomorphic functions, contour integrals, and residues.
- Holomorphic functions & power seriesComplex differentiability in a neighborhood is far more rigid than real differentiability.
- Contour integration & Cauchy formulasContour integrals connect local values of a holomorphic function to its behavior around a boundary.
- Laurent series, poles & residuesThe coefficient of (z−a)⁻¹ records the local contribution to a contour integral.
Topology & fundamental groups
Continuity, deformation, and the algebra of loops.
Knots & braids
Embedded curves, crossings, and strand motions.
Category theory
Objects through maps, naturality, and universal constructions.
- Categories & morphismsDescribe structure through composable maps, without assuming the objects are sets with elements.
- Functors, naturality & YonedaTransport structure between categories and compare those transports coherently.
- Universal properties, products & adjunctionsSpecify an object by the maps it uniquely permits rather than by one construction.
Paths through the tree
From symmetry to matrices
Groups & subgroups Group actions & counting Representations & invariant subspaces Characters & decomposition
From randomness to inference
Events, conditioning & Bayes Random variables & moments LLN, CLT & concentration Confidence intervals & tests
From geometry to fitting data
Vector spaces & linear maps Orthogonality, least squares & SVD Linear regression
From programs to computational limits
Recursion, invariants & termination Asymptotics & recurrences Turing machines, deciders & recognizers Reductions & undecidability P, NP & completeness
From limits to integration
Completeness & sequences Series & uniform convergence Lebesgue integration & Lᵖ spaces Convergence theorems & iterated integrals
From loops to structure
Topological spaces & homotopy The fundamental group Braid groups & closures Functors, naturality & Yoneda
From integration to prediction
Lebesgue integration & Lᵖ spaces Conditional expectation Linear regression
From derivatives to residues
Holomorphic functions & power series Contour integration & Cauchy formulas Laurent series, poles & residues
A starting collection, not a complete encyclopedia. Further directions include differential equations, geometry, functional analysis, distributed systems, and programming languages.