# The College of All Minds — The Truly Hardest Exam in the World
## Paper II — Mathematics

1. State the Riemann Hypothesis precisely and explain why its truth constrains the distribution of primes.
2. Prove that there is no continuous surjection from a compact interval onto a square, or explain why the naïve intuition here fails.
3. Give an account of what independence results (e.g., the Continuum Hypothesis) show about mathematical truth.
4. Sketch the structure of a proof that the Navier–Stokes existence-and-smoothness problem remains open in three dimensions; identify the precise obstruction.
5. What does the P versus NP question ask, and why is a proof so elusive?
6. Explain the role of the axiom of choice in a theorem of your choosing, and what is lost without it.
7. Prove that the algebraic numbers are countable.
8. What is a motive, and why did Grothendieck want one?
9. Give a rigorous statement of Gödel's second incompleteness theorem and explain its hypotheses.
10. Construct a non-measurable set, or explain what forbids it constructively.
11. Why is the classification of finite simple groups considered complete, and what would undermine that claim?
12. Explain the sense in which the exponential of a matrix solves a linear system.
13. What is the Langlands correspondence trying to unify?
14. Prove the fundamental theorem of Galois theory, or state it and give its content precisely.
15. In what sense is the Banach–Tarski paradox not a paradox?
16. Define a scheme and motivate the definition against classical varieties.
17. State and interpret the spectral theorem for self-adjoint operators.
18. What does it mean for a PDE to be well-posed in the sense of Hadamard?
19. Explain why the halting problem is undecidable, and connect this to Gödel.
20. What is the significance of the Atiyah–Singer index theorem?
21. Give an example of a statement true in the standard model of arithmetic but unprovable in Peano Arithmetic.
22. Describe the role of compactness in first-order logic and give one striking consequence.
23. What is homotopy type theory, and what does it propose to found?
24. Prove that π is irrational, or outline the strategy honestly.
25. Explain the difference between pointwise and uniform convergence and why it matters analytically.
26. State the Hodge Conjecture and explain what kind of object it concerns.
27. Why is the Birch–Swinnerton-Dyer conjecture a statement about arithmetic, despite its analytic form?
28. What is a topos, and in what sense is it a generalised space?
29. Give a probabilistic argument for a purely combinatorial fact.
30. Is there a mathematical statement you regard as true but expect never to be proved? Defend the attitude.

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*Website v1.3 · Examination lineage: concept v1.0 → naming v1.1 → motto v1.2 → expansion v2.0 → amendment v2.1 → expansion v3.0. Questions reproduced verbatim; none altered or renumbered.*