Paper II — Mathematics: References
93 SOURCES · 30 QUESTIONS · ALL FREELY AVAILABLE
Each source carries a technical description and a plain one; authors’ own pages and encyclopaedia entries are linked where they exist. Every address on this page was verified live at publication. The paper itself: Paper II.
1. State the Riemann Hypothesis precisely and explain why its truth constrains the distribution of primes.
The Riemann hypothesis (official problem description)
The prize formulation: zeros of ζ on the critical line, with the equivalence to sharp prime-counting error terms.
In plain terms: The official statement, free from the Clay Institute, of the most famous unsolved problem — and of what it would pin down about primes.
Survey of the evidence — zero statistics, function-field analogues, random matrix laws — and the principal lines of attack.
In plain terms: A readable free account of why nearly everyone believes it, what has been checked, and why belief is not proof.
An essay on the Riemann hypothesis
A Fields medallist's map of strategies — trace formulae, noncommutative geometry, the function-field proof as template.
In plain terms: What it might actually take: one of the deepest living mathematicians thinks aloud, freely, about roads to a proof.
The Riemann hypothesis (problem page)
The problem's standing page, with rules and the official description attached.
In plain terms: The front door: the million-dollar problem stated for any reader, with the formal documents linked.
Background: Riemann hypothesis · Prime-counting function
2. Prove that there is no continuous surjection from a compact interval onto a square, or explain why the naïve intuition here fails.
The refuting object: continuous surjections from the interval onto the square exist — the question's premise is false.
In plain terms: The curve that fills a square completely — the counterexample showing the 'obviously impossible' map exists after all.
The true theorem in the vicinity: no continuous injection lowers dimension; Brouwer's invariance is what intuition was reaching for.
In plain terms: What is genuinely impossible here — a one-to-one continuous correspondence — and the deep theorem that says so.
The early development of set theory
The history the question replays: Cantor's dimension-breaking bijection, Netto, and the road to invariance of domain.
In plain terms: The original scandal, freely told — when Cantor matched the line with the plane and wrote 'I see it, but I don't believe it.'
Background: Space-filling curve · Invariance of domain
3. Give an account of what independence results (e.g., the Continuum Hypothesis) show about mathematical truth.
The independence results stated exactly, with the realist and pluralist readings of what they show about truth.
In plain terms: The standing scholarly account of a question mathematics proved it cannot settle from its own axioms — and what that means.
The method behind independence, made genuinely accessible: generic extensions and why CH fails in some models.
In plain terms: Cohen's magic trick explained for outsiders — how mathematicians build universes in which the disputed statement goes each way.
Independence and large cardinals
The programme of new axioms: what stronger principles decide, and where CH stubbornly remains untouched.
In plain terms: The attempted repair — adopting bolder axioms — and the honest report of how far it reaches.
Background: Continuum hypothesis · Forcing (mathematics)
4. Sketch the structure of a proof that the Navier–Stokes existence-and-smoothness problem remains open in three dimensions; identify the precise obstruction.
Existence and smoothness of the Navier–Stokes equation (official problem description)
The precise open statement: global regularity or blow-up in three dimensions, with the known partial results.
In plain terms: The exact question, officially posed: do the standard equations of fluid flow ever tear themselves apart?
Why global regularity for Navier–Stokes is hard
The obstruction identified: supercriticality — every known conserved quantity weakens exactly where blow-up would occur.
In plain terms: The clearest free explanation of the barrier: our tools lose their grip precisely at the small scales where trouble would form.
Finite time blowup for an averaged three-dimensional Navier–Stokes equation
Sharpens the obstruction: a model respecting the energy identity does blow up, so any proof must exploit finer structure.
In plain terms: A landmark free paper building a machine out of near-Navier–Stokes fluid that self-destructs — fencing off whole strategies.
Navier–Stokes equation (problem page)
The standing problem page with the official description attached.
In plain terms: The front door to the prize problem, for orientation and the formal documents.
Background: Navier–Stokes existence and smoothness · Millennium Prize Problems
5. What does the P versus NP question ask, and why is a proof so elusive?
The P versus NP problem (official problem description)
The founder's formulation: verification versus search, NP-completeness, and the consequences either way.
In plain terms: The official free statement, by the man who framed it, of whether finding answers can be as easy as checking them.
The definitive modern survey of why proof is elusive: relativisation, natural proofs, algebrisation — the barrier theorems.
In plain terms: A free tour of the astonishing fact that we can prove theorems about why this theorem is hard to prove.
Mathematics and computation (open book)
The field's panorama: P vs NP situated within complexity theory's structure, connections, and philosophy.
In plain terms: An entire free book by an Abel laureate placing the question at the centre of the theory of computation.
The standing problem page with rules and the official description attached.
In plain terms: The front door to the prize problem.
Background: P versus NP problem · NP-completeness
6. Explain the role of the axiom of choice in a theorem of your choosing, and what is lost without it.
The axiom's equivalents, its indispensable uses, and the pathologies it purchases — the full ledger.
In plain terms: The complete free account of mathematics' most debated assumption: what it buys, and what it costs.
The working mathematician's summary: statements equivalent to AC and the standard theorems that need it.
In plain terms: A concise professional reference for exactly which everyday theorems lean on the axiom.
A celebrated exercise in choicelessness: cardinal division by three proved effectively, showing what care replaces choice.
In plain terms: A famously entertaining free paper doing by honest toil what the axiom would do by magic — the loss made visible.
Background: Axiom of choice · Zorn's lemma
7. Prove that the algebraic numbers are countable.
Definition and structure of the algebraic numbers — the objects whose enumeration the question demands.
In plain terms: The professional reference for the numbers in question: all roots of whole-number polynomial equations.
Countability and its closure properties — countable unions of countable sets — the proof's engine.
In plain terms: The key concept: infinite sets that can still be listed, and the rules for combining them that make the proof go.
The early development of set theory
Cantor's 1874 paper — where the countability of the algebraic numbers first appears, against the uncountable reals.
In plain terms: The result's birthplace, freely recounted: the very first theorem contrasting listable and unlistable infinities.
Background: Algebraic number · Countable set
8. What is a motive, and why did Grothendieck want one?
Motives — Grothendieck's dream
What a motive is meant to be: the universal cohomology through which all Weil cohomologies factor, and the standard conjectures.
In plain terms: A master expositor's free essay on Grothendieck's dream — one common source from which every way of measuring a shape flows.
The two-page distillation: motives as the postulated common refinement of cohomology theories.
In plain terms: The famous short answer, free from the AMS — the whole idea in the time it takes to drink a coffee.
The living technical entry: pure and mixed motives, categories of motives, and the modern realisations.
In plain terms: A continuously maintained free reference tracking how the dream has been partially built.
Background: Motive (algebraic geometry) · Alexander Grothendieck
9. Give a rigorous statement of Gödel's second incompleteness theorem and explain its hypotheses.
Gödel's incompleteness theorems
The rigorous statement demanded: effective axiomatisation, sufficient arithmetic, the derivability conditions behind the second theorem.
In plain terms: The careful free account of what Gödel actually proved about consistency, and of the fine print people skip.
The theorems in the context of Gödel's programme, with his own reading of their philosophical import.
In plain terms: The man and the theorem together — what the author of the result thought it did and did not show.
The base theory the hypotheses quantify over — what 'sufficient arithmetic' concretely contains.
In plain terms: The standard axioms of arithmetic, stated plainly: the stage on which the incompleteness theorems perform.
Background: Gödel's incompleteness theorems · Hilbert's second problem
10. Construct a non-measurable set, or explain what forbids it constructively.
The construction and its dependence on choice; Solovay's model where every set is measurable.
In plain terms: Both halves of the question in one free entry: how the monstrous set is built, and the universe in which it cannot be.
The classical construction: choice over cosets of the rationals defeating translation-invariant measure.
In plain terms: The original recipe — pick one point from each of infinitely many families, and measurement breaks.
The measure whose totality is at stake — the properties the Vitali set is engineered to contradict.
In plain terms: What 'measurable' means in the first place, so the construction's target is clear.
Background: Vitali set · Solovay model
11. Why is the classification of finite simple groups considered complete, and what would undermine that claim?
The status of the classification of the finite simple groups
The insider's audit at the moment of completion: the quasithin gap closed, and what 'complete' means for a 10,000-page proof.
In plain terms: A leader of the effort explains, freely, why the mammoth proof is considered finished — and what standard that claim answers to.
A brief history of the classification of the finite simple groups
The programme's architecture across generations, and the second-generation project consolidating the proof.
In plain terms: The whole campaign told free of charge: dozens of mathematicians, decades, and the ongoing rewriting meant to secure it.
The objects classified — the atoms of finite group theory — and the statement of the classification.
In plain terms: What was actually catalogued: the indivisible building blocks from which every finite symmetry is assembled.
Background: Classification of finite simple groups · Monster group
12. Explain the sense in which the exponential of a matrix solves a linear system.
Differential Equations (18.03SC, open course)
Full course covering linear systems x′ = Ax and the exponential solution e^{tA}x₀, with the diagonalisable and defective cases.
In plain terms: A complete free MIT course in which the matrix exponential earns its keep solving systems of equations.
An elementary introduction to groups and representations
The exponential map treated rigorously — convergence, one-parameter groups, and exp as the bridge from algebra to flow.
In plain terms: Free lecture notes explaining the deeper sense of the formula: the exponential turns instantaneous rules into finished motion.
Introduction to Real Analysis (open textbook)
The analytic underpinning: series convergence and uniform limits that make e^{tA} well-defined and differentiable.
In plain terms: A complete free analysis textbook supplying the fine print — why the infinite sum defining the exponential behaves.
Background: Matrix exponential · Linear differential equation
13. What is the Langlands correspondence trying to unify?
Lectures on the Langlands program and conformal field theory
The unification stated: Galois representations, automorphic forms, and the geometric analogue, with the physics bridge.
In plain terms: The standard free introduction to mathematics' grand unification project — number theory and harmonic analysis as one subject.
An elementary introduction to the Langlands program
The classic entry point: reciprocity from Gauss to Artin to Langlands' functoriality conjectures.
In plain terms: The much-loved free survey tracing the dream from schoolbook reciprocity to its modern form.
Proof of the geometric Langlands conjecture I
The opening paper of the series proving the geometric correspondence over function fields — the programme's first completed continent.
In plain terms: History in the open: the first instalment of the 2024 proof that settled the geometric half of the dream.
Background: Langlands program · Robert Langlands
14. Prove the fundamental theorem of Galois theory, or state it and give its content precisely.
Fields and Galois theory (course notes)
A complete rigorous course: the Galois correspondence proved, with solvability and the classical applications.
In plain terms: A full free textbook by a renowned expositor, containing the theorem, its proof, and what it conquers.
A modern annotated course, unusually explicit about the correspondence's content and standard misreadings.
In plain terms: Recent free lecture notes prized for saying carefully what the celebrated theorem does and does not assert.
The theorem's statement and setting in reference form: extensions, groups, and the inclusion-reversing bijection.
In plain terms: The concise professional summary of the dictionary between equations and symmetries.
Background: Galois theory · Fundamental theorem of Galois theory
15. In what sense is the Banach–Tarski paradox not a paradox?
Locates Banach–Tarski among choice's consequences: a theorem of ZFC, paradoxical only to measure-theoretic intuition.
In plain terms: The context that dissolves the shock — the doubling is a proved theorem, and the free account shows which assumption powers it.
The structural reading: non-amenability of the free subgroup of rotations, with non-measurable pieces as the price.
In plain terms: The free technical entry explaining why the pieces cannot be physical — they have no volume at all, even zero.
What the construction respects and what it must abandon: no finitely additive isometry-invariant extension to all sets in ℝ³.
In plain terms: The bookkeeping the 'paradox' never violates — because the strange pieces stand outside the books entirely.
Background: Banach–Tarski paradox · Amenable group
16. Define a scheme and motivate the definition against classical varieties.
The Rising Sea: foundations of algebraic geometry (open notes)
The standard modern free text: schemes built from rings with full motivation — nilpotents, generic points, functor of points.
In plain terms: A famous free book that patiently explains why geometry was rebuilt on stranger foundations, and what the strangeness buys.
The definition in reference form — locally ringed spaces locally of the form Spec R — with its categorical setting.
In plain terms: The concise technical entry for the object itself, kept current by the community.
Algebraic geometry (course notes)
The classical theory of varieties — exactly the backdrop against which the scheme definition is motivated.
In plain terms: The 'before' picture, free and complete: the classical geometry whose limitations schemes were invented to transcend.
Background: Scheme (mathematics) · Algebraic geometry
17. State and interpret the spectral theorem for self-adjoint operators.
Introduction to Functional Analysis (18.102, open course)
Full course with notes proving the spectral theorem for bounded self-adjoint operators and developing the functional calculus.
In plain terms: A complete free MIT course in which the theorem is built honestly, from Hilbert spaces to diagonalisation in the limit.
The hypothesis class exactly — including the unbounded case where self-adjointness outruns symmetry.
In plain terms: The professional reference for the operators the theorem is about, including the subtlety physicists learn the hard way.
The theorem's forms — multiplication-operator, projection-valued measure — and their interpretation.
In plain terms: The free entry stating the result's modern shapes: every such operator is, in disguise, multiplication by a real function.
Background: Spectral theorem · Self-adjoint operator
18. What does it mean for a PDE to be well-posed in the sense of Hadamard?
Hadamard's triple — existence, uniqueness, continuous dependence — stated as the definition of a properly formulated problem.
In plain terms: The three demands a physical problem must meet before its mathematics can be trusted, in reference form.
The complementary theory: Hadamard's failures as a field of study, with regularisation as the response.
In plain terms: What happens when the demands fail — and the free account of the modern art of taming such problems anyway.
Introduction to Partial Differential Equations (18.152, open course)
Well-posedness in action across the canonical equations — heat forward and backward, wave, Laplace with wrong data.
In plain terms: A free MIT course where the definition earns its keep: the same equation, posed two ways, sane and insane.
Background: Well-posed problem · Cauchy problem
19. Explain why the halting problem is undecidable, and connect this to Gödel.
The model, the universal machine, and the diagonal proof of the halting problem's undecidability.
In plain terms: The free scholarly account of Turing's machine and of the short, devastating argument that no program can foresee all programs.
Why the halting result binds all effective computation — the thesis that converts a theorem about one model into a limit on method.
In plain terms: The bridge premise: every mechanical procedure is capturable by Turing's machine, so its limits are everyone's.
Gödel's incompleteness theorems
The connection made exact: undecidability yields incompleteness — a complete effective theory would decide halting.
In plain terms: The promised link, free and careful: Gödel's theorem falls out of the halting problem in a few lines.
Background: Halting problem · Entscheidungsproblem
20. What is the significance of the Atiyah–Singer index theorem?
The Atiyah–Singer index theorem
The modern survey: analytic index equals topological index, the heat-kernel and K-theory proofs, and the physics afterlife.
In plain terms: A free authoritative tour of the theorem joining counting solutions of equations to the shape of the space they live on.
The statement and its many descendants in reference form — families, equivariant, and local index theorems.
In plain terms: The living entry cataloguing the theorem's forms and the fields it seeded.
Abel Prize 2004: Atiyah and Singer
The citation and accompanying materials assessing the theorem's significance across mathematics and physics.
In plain terms: The prize record — free documentation of why this result is ranked among the twentieth century's greatest.
Background: Atiyah–Singer index theorem · Michael Atiyah
21. Give an example of a statement true in the standard model of arithmetic but unprovable in Peano Arithmetic.
Gödel's incompleteness theorems
Documents the natural examples the question requests — Goodstein and Paris–Harrington — true in ℕ, unprovable in PA.
In plain terms: The free source for the astonishing specimens: ordinary-looking arithmetic truths that arithmetic's own axioms cannot reach.
Independence and large cardinals
The mechanism: such statements are proved in stronger systems whose consistency PA cannot certify.
In plain terms: Why the examples are provable at all — from a higher vantage arithmetic itself cannot officially trust.
The exact theory PA relative to which the unprovability claims are made.
In plain terms: The fixed rulebook in question, stated — so 'unprovable in PA' has a definite meaning.
Background: Goodstein's theorem · Paris–Harrington theorem
22. Describe the role of compactness in first-order logic and give one striking consequence.
Compactness as first-order logic's signature: finite satisfiability yields satisfiability, with its model-building consequences.
In plain terms: The free scholarly account of the theorem that lets logicians conjure infinite structures from finitely many promises.
Reference statement of compactness and Löwenheim–Skolem — the twin pillars and their limitative flavour.
In plain terms: The professional summary of the theorem and its equally strange sibling about sizes of models.
Elementary Calculus: An Infinitesimal Approach (open textbook)
The striking consequence, industrialised: compactness underwrites the hyperreals, and calculus runs on honest infinitesimals.
In plain terms: A whole free textbook teaching calculus with genuine infinitely small numbers — legitimate offspring of the compactness theorem.
Background: Compactness theorem · Nonstandard analysis
23. What is homotopy type theory, and what does it propose to found?
Homotopy Type Theory: Univalent Foundations of Mathematics (open book)
The founding text: types as spaces, equalities as paths, univalence, and mathematics natively machine-checkable.
In plain terms: The free book proposing new foundations in which 'equal' means 'connectable' and every proof is computer-ready by birth.
The lineage — Russell to Martin-Löf — into which the homotopy interpretation arrives, and what founding on types means.
In plain terms: The background story: the century-old rival to set theory that HoTT upgrades into a foundation for everything.
The living technical entry: models, univalence, higher inductive types, and the state of the programme.
In plain terms: The continuously updated free reference on how far the new foundations have actually been built.
Background: Homotopy type theory · Univalent foundations
24. Prove that π is irrational, or outline the strategy honestly.
A simple proof that π is irrational
The one-page classic: a polynomial-integral that would be a positive integer below one if π were rational.
In plain terms: The complete proof, free at the source, and famously short — a single page that closes a two-thousand-year question.
Niven's argument unpacked line by line, with the strategy — bounded positive integers below one — made explicit.
In plain terms: The honest walkthrough the question invites: every step of the famous page explained, freely.
The stronger fact and its method: Lindemann's theorem places π beyond all algebraic equations.
In plain terms: The sequel: π is not merely irrational but transcendental — and this free entry records the theorem that says so.
Background: Proof that π is irrational · Transcendental number
25. Explain the difference between pointwise and uniform convergence and why it matters analytically.
The stronger mode and its dividends: preservation of continuity, interchange of limit with integral and derivative.
In plain terms: The professional statement of the good kind of convergence — the one that lets you swap limits without lying.
The weaker mode, with the standard counterexamples where continuity and integrals are lost in the limit.
In plain terms: The deceptive kind: every point behaves, yet the whole misbehaves — with the classic cautionary examples.
Real Analysis (18.100A, open course)
Full course with lecture notes developing both modes and the theorems that separate them.
In plain terms: A complete free MIT course in which the distinction is built properly and its analytic consequences proved.
Background: Uniform convergence · Pointwise convergence
26. State the Hodge Conjecture and explain what kind of object it concerns.
The Hodge conjecture (official problem description)
The prize statement: rational (p,p)-classes on smooth projective varieties as classes of algebraic cycles.
In plain terms: The official free formulation, by a Fields medallist, of the question whether certain shapes-within-shapes always come from equations.
Hodge conjecture (problem page)
The standing page with orientation and the official description attached.
In plain terms: The problem's front door, stating for any reader what kind of object is at issue.
The statement in its cohomological habitat — Hodge structures, cycle class maps, known cases.
In plain terms: The free technical entry situating the conjecture among the tools invented to measure complex shapes.
Background: Hodge conjecture · Algebraic cycle
27. Why is the Birch–Swinnerton-Dyer conjecture a statement about arithmetic, despite its analytic form?
The Birch and Swinnerton-Dyer conjecture (official problem description)
The prize statement: analytic rank equals arithmetic rank — the L-function's vanishing order counting rational points.
In plain terms: The official free formulation, by the conqueror of Fermat, of how a smooth analytic function is believed to count whole-number solutions.
The arithmetic side surveyed: Mordell–Weil ranks, what is known, and BSD as the organising principle.
In plain terms: A free expert tour of the whole-number side of the story — the quantity the analytic formula is supposed to predict.
Birch and Swinnerton-Dyer conjecture (problem page)
The standing page with the official description attached.
In plain terms: The problem's front door, for orientation and the formal documents.
Background: Birch and Swinnerton-Dyer conjecture · Elliptic curve
28. What is a topos, and in what sense is it a generalised space?
The gentlest serious introduction: from sheaves on a space to elementary topoi as universes of variable sets.
In plain terms: A celebrated free primer on the idea of a 'generalised space' — a place where mathematics itself can be done differently.
The definition and its double life — geometric (sheaves, sites) and logical (internal language) — in reference form.
In plain terms: The living entry on the object's two faces: a kind of space, and a kind of universe.
An informal introduction to topos theory
The standard free survey: subobject classifiers, generalised spaces, and why Set is the trivial example.
In plain terms: The much-recommended plain-spoken paper explaining, without machinery worship, what a topos is for.
Background: Topos · Category theory
29. Give a probabilistic argument for a purely combinatorial fact.
Some remarks on the theory of graphs
The founding instance: random two-colourings give exponential Ramsey lower bounds — existence with no example.
In plain terms: The free three-page original where a coin flip first proved a combinatorial fact no construction could reach.
Probabilistic Methods in Combinatorics (open notes)
A full modern course: first moment, alterations, Lovász local lemma, concentration — the method as a discipline.
In plain terms: A complete free textbook on the art of proving that something must exist because randomness would find it.
Probabilistic Methods in Combinatorics (18.226, open course)
The companion course with problem sets — the method practised, not merely admired.
In plain terms: The same subject as a free MIT course, for readers who want to wield the trick themselves.
Background: Probabilistic method · Ramsey's theorem
30. Is there a mathematical statement you regard as true but expect never to be proved? Defend the attitude.
The 3x+1 problem: an annotated bibliography
The problem's curator documents a century of partial results — and the absence of any plausible route to proof.
In plain terms: The complete free dossier on the notorious 3n+1 puzzle: everything tried, everything known, nothing sufficient.
Almost all orbits of the Collatz map attain almost bounded values
The strongest modern result — and its author's admission that 'almost all' may be the method's ceiling.
In plain terms: The best that the best can do, free to read: nearly every number behaves, while the full claim stays out of reach.
The epistemology the attitude requires: truth outrunning provability, and what warrants belief without proof.
In plain terms: The framework for the confession itself — whether a mathematician may rationally believe what no one will ever prove.
Background: Collatz conjecture · Mathematical proof