Paper II — Mathematics
30 QUESTIONS
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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.