Paper II — Mathematics
35 QUESTIONS
Paper II covers mathematics as a whole subject rather than as a service to the sciences. Its questions run from the classical open problems — the Riemann Hypothesis, P versus NP, Navier–Stokes existence and smoothness, Hodge, Birch–Swinnerton-Dyer — through the foundations, where independence results, Gödel’s second theorem, compactness, the axiom of choice and its pathologies, homotopy type theory and the topos are examined; through algebra and geometry, where Galois theory, schemes, motives, the Langlands correspondence and the Atiyah–Singer index theorem are set; and through analysis and its rigour, where measurability, convergence, the spectral theorem and well-posedness are put to work. Four questions reach into music and geography: the impossibility of a just fifth in any equal division of the octave, the ambiguity of rhythms sharing an interval content, the impossibility of an undistorted plane map of the sphere, and the length of a coastline.
The paper examines mathematical maturity rather than recall. Several questions demand a proof and will accept nothing less; several ask instead for an honest sketch of a strategy, and reward the candidate who marks precisely where the argument would become hard. Throughout, the requirement is to state hypotheses exactly, to know which of them is load-bearing, and to say what a theorem does and does not assert — the difference between a result being open, undecidable, and false being treated as a matter of technical fact rather than of attitude. The final question invites a candidate to name a statement believed true but expected never to be proved, and to defend the belief; the paper is willing to examine mathematical taste, provided it is disciplined.
Conduct of the response
Answer in the idiom of the college: provocations invite a disciplined essay; technical problems require correct, quantitative reasoning; interdisciplinary questions require two fields to be brought together. A response is an argument, not a survey; where the question admits a position, one is to be taken at the outset and defended. It must be exactly as technical as the question requires, and no more, and its course must remain followable by the educated layman. It may not exceed three thousand words, exclusive of equations, figures, tables, and references; the limit is a ceiling, not a target, and of two answers of equal force the shorter is judged the better. The full statement of the response required is given on the Responses page.
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1.
State the Riemann Hypothesis precisely and explain why its truth constrains the distribution of primes.
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2.
Prove that there is no continuous surjection from a compact interval onto a square, or explain why the naïve intuition here fails.
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3.
Give an account of what independence results (e.g., the Continuum Hypothesis) show about mathematical truth.
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4.
Sketch the structure of a proof that the Navier–Stokes existence-and-smoothness problem remains open in three dimensions; identify the precise obstruction.
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5.
What does the P versus NP question ask, and why is a proof so elusive?
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6.
Explain the role of the axiom of choice in a theorem of your choosing, and what is lost without it.
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7.
Prove that the algebraic numbers are countable.
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8.
What is a motive, and why did Grothendieck want one?
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9.
Give a rigorous statement of Gödel's second incompleteness theorem and explain its hypotheses.
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10.
Construct a non-measurable set, or explain what forbids it constructively.
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11.
Why is the classification of finite simple groups considered complete, and what would undermine that claim?
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12.
Explain the sense in which the exponential of a matrix solves a linear system.
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13.
What is the Langlands correspondence trying to unify?
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14.
Prove the fundamental theorem of Galois theory, or state it and give its content precisely.
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15.
In what sense is the Banach–Tarski paradox not a paradox?
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16.
Define a scheme and motivate the definition against classical varieties.
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17.
State and interpret the spectral theorem for self-adjoint operators.
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18.
What does it mean for a PDE to be well-posed in the sense of Hadamard?
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19.
Explain why the halting problem is undecidable, and connect this to Gödel.
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20.
What is the significance of the Atiyah–Singer index theorem?
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21.
Give an example of a statement true in the standard model of arithmetic but unprovable in Peano Arithmetic.
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22.
Describe the role of compactness in first-order logic and give one striking consequence.
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23.
What is homotopy type theory, and what does it propose to found?
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24.
Prove that π is irrational, or outline the strategy honestly.
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25.
Explain the difference between pointwise and uniform convergence and why it matters analytically.
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26.
State the Hodge Conjecture and explain what kind of object it concerns.
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27.
Why is the Birch–Swinnerton-Dyer conjecture a statement about arithmetic, despite its analytic form?
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28.
What is a topos, and in what sense is it a generalised space?
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29.
Give a probabilistic argument for a purely combinatorial fact.
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30.
Is there a mathematical statement you regard as true but expect never to be proved? Defend the attitude.
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31.
Prove that no equal division of the octave contains a just perfect fifth, and say what the continued-fraction expansion of log₂(3/2) recommends in its place.
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32.
A rhythm and its retrograde can be indistinguishable to the autocorrelation. Exhibit two distinct rhythmic patterns sharing an interval content, and say what the ambiguity implies for identifying a piece by its statistics.
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33.
Prove that no plane map of the sphere preserves all distances, and say what Mercator, the equal-area projections, and the geodesic-preserving projections each surrender instead.
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34.
How long is the coast of Britain? State the Richardson relation, define the dimension it implies, and decide whether the length is undefined or merely scale-relative.
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35.
Is Ramsey’s theorem the pigeonhole principle on stilts? Exhibit the principle as the theorem’s one-dimensional case, derive the finite two-colour theorem from it by iteration, and then let the calibrations judge: say what the growth of the Ramsey numbers and the reverse-mathematical strength of the theorem for pairs each establish that counting alone does not.
Questions added by Examination Expansion v5.0, which extended the examination to music and geography. All earlier questions stand verbatim and unrenumbered.
Question 35 was added by Examination Amendment v5.2. All earlier questions stand verbatim and unrenumbered.