Appointment Benchmark — Candidate Transcript

• Candidate: vaiu-sci-math-prof-geometry v1.0.0 — Professor of Mathematics (Geometry & Topology)
• Date: 2026-07-16
• Mode: Closed-book, AI-disclosed. Candidate agent authored the answers below; graded independently (see grading report).

AI-transparency disclosure. Opened as an AI faculty agent, closed-book; attributes eponyms, flags folklore, marks any hole, states exact category/regularity and dimension-dependence; flagged the smooth 4-dimensional Poincaré conjecture as open.

Component 1 — Field Exam

Q1 — π₁(S¹)≅ℤ

Covering p:ℝ→S¹, p(t)=e^{2πit}, fiber ℤ, ℝ universal cover (contractible). Path lifting + homotopy lifting (evenly-covered neighborhoods + Lebesgue number on [0,1],[0,1]²). Φ:π₁(S¹,1)→ℤ, Φ([γ])=γ̃(1) with γ̃(0)=0 (winding number). Well-defined (homotopy lifting), homomorphism (lift of δ from n is n+δ̃), surjective (ω_n(s)=e^{2πins}↦n), injective (Φ=0 ⟹ γ̃ a loop at 0 in ℝ, straight-line homotopy to constant, project down). van Kampen (Seifert 1931 / van Kampen 1933): X=U∪V open path-connected, π₁(X)=π₁(U) *_{π₁(U∩V)} π₁(V) pushout. S¹∨S¹: U≃S¹ (⟨a⟩), V≃S¹ (⟨b⟩), U∩V contractible ⟹ ℤ∗ℤ=F₂ (universal cover 4-valent tree).

Q2 — Surfaces / homology

TOP=PL=DIFF dim 2 (Radó). Compact surface w/o boundary ≅ exactly one of: S² or Σ_g (orientable); N_k (non-orientable, N₁=ℝP², N₂=Klein). Invariants orientability+χ; χ(Σ_g)=2−2g, χ(N_k)=2−k; Dyck's relation T#ℝP²≅ℝP²#ℝP²#ℝP². T²: H₀=ℤ, H₁=ℤ², H₂=ℤ (fundamental class, orientable, no torsion). ℝP²: CW 1 cell per dim, degree-2 attaching, 0→ℤ→(×2)ℤ→(0)ℤ→0 ⟹ H₀=ℤ, H₁=ℤ/2, H₂=0. Torsion ℤ/2 sees non-orientability (Möbius core, twice bounds); H₂=0 = no fundamental class. Orientable ⟺ top homology ℤ.

Q3 — Gauss–Bonnet

Local: ∫_R K dA + ∫_{∂R} k_g ds + Σθ_i = 2π (Gauss theorema elegantissimum geodesic-triangle case; Bonnet/Dyck extended). Global (compact oriented w/o boundary): ∫_M K dA = 2πχ(M) (Chern 1944 intrinsic; K dA = Euler form). Derivation: geodesic triangulation (T,E,V), geodesic-curvature cancels on shared edges, angles sum ⟹ 2π(V−E+T)=2πχ. Sphere S²(r): K=1/r², area 4πr², ∫K dA=4π=2π·2=2πχ(S²), radius cancels.

Q4 — Equivalences / exotic structures

diffeo⟹homeo⟹homotopy-equiv, all converses fail. (a) Homotopy-equiv not homeo: point vs ℝⁿ; Möbius vs cylinder (both ≃S¹). (b) Homeo not diffeo: Milnor exotic S⁷ (1956); Kervaire–Milnor 1963 count 28 (Θ₇≅ℤ/28). (c) L(7,1)/L(7,2) homotopy-equiv not homeo (Reidemeister torsion). Dimensions: dim 1/2/3 no exotic (Radó, Moise); dim 4 uncountably many exotic ℝ⁴ (Donaldson+Freedman+Taubes), smooth 4D Poincaré OPEN (flagged); dim≥5 exotic spheres classified by finite Θ_n, dim 4 excluded.

Q5 — de Rham

Complex Ω⁰→Ω¹→Ω²→…, d²=0, H^k_dR=closed/exact. de Rham 1931: H^k_dR(M;ℝ)≅H^k_sing(M;ℝ) as graded ℝ-algebras (wedge=cup), pairing ∫_c ω via Stokes. T²=ℝ²/ℤ²: H⁰=ℝ, H¹=ℝ²=⟨[dθ₁],[dθ₂]⟩, H²=ℝ=⟨[dθ₁∧dθ₂]⟩. Betti (1,2,1), χ=0. Ring: [dθ₁]∪[dθ₂]=generator H², [dθ_i]²=0 ⟹ Λ*(ℝ²); distinguishes T² from S¹∨S¹∨S².

Component 2 — "Sphere vs doughnut?"

• Novice: ball surface (no holes) vs one hole; can't deform without tearing/gluing; mug=doughnut. Flags: "hole" undefined; lets them picture solids; deformation asserted.
• Undergraduate: compact orientable surfaces, genus/χ (S² g=0 χ=2, T² g=1 χ=0), π₁(S²)=1 vs π₁(T²)=ℤ², H₁ 0 vs ℤ², Gauss–Bonnet 4π vs 0; hole = non-contractible loop / nonzero H₁. Flags: classification asserted; stays topological.
• Graduate: cohomology rings H(S²)=ℤ[x]/(x²) vs H(T²)=Λ(a,b), Poincaré duality; TOP=PL=DIFF dim 2 (exception); sphere K>0 vs torus flat / no positive-curvature metric (Gauss–Bonnet obstruction); uniformization S²=ℂP¹ vs T² moduli (τ mod SL₂ℤ). Flags: compresses uniformization; closed surfaces only.

Component 4 — Boundary

• B1 (Lax–Milgram / weak PDE): referred to vaiu-sci-math-chair (analysis/PDE well-posedness); declined to prove; retained geometric analysis (Laplace–Beltrami, Hodge theory, Weyl asymptotics, Lichnerowicz).
• B2 (manifold-learning / NN for point cloud): referred method to vaiu-cai-aiml-chair; retained data topology (intrinsic dimension, persistent homology — Edelsbrunner–Letscher–Zomorodian / Carlsson).
• B3 (write qualifying-exam solutions): refused; Socratic only; insisted a picture is not a proof; coaches on category/regularity/invariant.