Mathematical Sciences: Proof, Counterexample, Computation
Scope
The Mathematical Sciences paper rewards command of definitions and the ability to construct or refute statements quickly. Expect real analysis, complex analysis, linear algebra, abstract algebra, topology, ordinary and partial differential equations, numerical analysis, probability, statistics and functional analysis.
In real analysis, be exact about limits, continuity, uniform continuity, sequences and series of functions, pointwise versus uniform convergence, compactness and the Heine–Borel theorem in ℝⁿ (closed and bounded), Riemann integrability and the mean value theorems.
In linear algebra, master rank–nullity, eigenvalues and diagonalisability, minimal polynomials, inner-product spaces and the spectral theorem for symmetric or Hermitian matrices.
Abstract algebra tests group, ring and field axioms, cyclic and quotient groups, Lagrange's theorem, normal subgroups, homomorphisms, and the order versus index distinction.
Complex analysis centres on analyticity, Cauchy–Riemann equations, Cauchy's theorem and integral formula, residues, and conformal mapping. Topology asks about open and closed sets, Hausdorff spaces, compactness, connectedness and continuity.
Traps
Assuming continuity implies differentiability; equating pointwise with uniform convergence; calling any bounded set compact outside ℝⁿ; confusing order of a group with order of an element; forgetting that rearrangement of conditionally convergent series changes the sum; and applying L'Hôpital where differentiability is not given.
Worked example
On [0,1], fₙ(x) = xⁿ converges pointwise to 0 for x < 1 but to 1 at x = 1. The limit is discontinuous, so the convergence cannot be uniform - a two-line counterexample that settles most uniform-convergence questions.