querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2023, 3 Topology, covering spaces, Klein bottle, classification of surfaces

Let $𝑛 \geq 2$ . Classify (up to diffeomorphism) all connected surfaces that can be represented as $𝑛$ -fold covers of the Klein bottle.

Stanford Fall 2023, 3 Topology, distributions, Frobenius theorem, integrability

Show that

𝐷 = ker (d 𝑥_{3} - 𝑥_{1} d 𝑥_{2}) \cap ker (d 𝑥_{1} - 𝑥_{4} d 𝑥_{2}) \subset 𝑇 ℝ^{4}

is a smooth distribution of rank two, and determine whether $𝐷$ is integrable.

Stanford Fall 2023, 4 Topology, Morse theory, real projective space

Find the minimum number of critical points of a Morse function $𝑓 : ℝ ℙ^{𝑛} \to ℝ$ .

Stanford Fall 2023, 4 Topology, Betti numbers, Poincaré duality, CW complexes

Let $𝑏_{0}, 𝑏_{1}, \dots, 𝑏_{𝑛}$ be a sequence of positive integers satisfying $𝑏_{𝑘} = 𝑏_{𝑛 - 𝑘}$ for all $𝑘$ , and $𝑏_{𝑛 / 2}$ is even if $𝑛$ is even. Construct an $𝑛$ -dimensional closed, oriented manifold with these Betti numbers.

Stanford Fall 2023, 5 Topology, differential forms, fiber bundles, exterior derivative

Let $𝑀$ and $𝑁$ be smooth, connected manifolds and let $𝜋 : 𝑀 \times 𝑁 \to 𝑀$ be the standard projection. A tangent vector $𝑉$ to $𝑀 \times 𝑁$ is called vertical if $𝜋_{*} 𝑉 = 0$ . For any $𝑝$ -form on $𝑀 \times 𝑁$ we define the $(𝑝 - 1)$ -form $𝜄_{𝑉} 𝜔$ by

(𝜄_{𝑉} 𝜔) (𝑥_{1}, \dots, 𝑥_{𝑝 - 1}) = 𝜔 (𝑉, 𝑥_{1}, \dots, 𝑥_{𝑝 - 1}) .

Consider

\tilde{Ω} = {𝜔 \in Ω^{*} (𝑀 \times 𝑁) | 𝜄_{𝑉} 𝜔 = 0 and 𝜄_{𝑉} (d 𝜔) = 0 for all vertical vectors 𝑉} .

(a) Show that $\tilde{Ω}$ is closed under addition, wedge product and the exterior differential $d$ .

(b) Show that $𝜋^{*}$ induces a map $Ω (𝑀) \to \tilde{Ω}$ .

Stanford Fall 2023, 5 Topology, embeddings, isotopy, knot theory

Show that any two smooth embeddings $𝑓, 𝑔 : 𝑆^{1} \to ℝ^{4}$ are homotopic through smooth embeddings.

Stanford Fall 2024, 1 Topology, cotangent bundle, symplectic geometry, 1-forms

Let $𝑀$ be a smooth manifold, and $𝜋 : 𝑇^{*} 𝑀 \to 𝑀$ its cotangent bundle. Consider the 1-form $𝜆$ on $𝑇^{*} 𝑀$ whose value at a tangent vector $𝜉 \in 𝑇_{𝛼} 𝑇^{*} 𝑀$ at a covector $𝛼$ over a point in $𝑀$ is given by evaluating $𝛼 (d 𝜋 (𝜉))$ . Show that, for a section $𝑠 : 𝑀 \to 𝑇^{*} 𝑀$ defined by a 1-form $𝜌$ on $𝑀$ , the pullback $𝑠^{*} 𝜆$ agrees with $𝜌$ .

Stanford Fall 2024, 1 Topology, homotopy equivalence, spheres, tori

Recall that the 3-sphere may be described as the set of points in $ℂ^{2}$ , with coordinates $(𝑧, 𝑤)$ , satisfying ${| 𝑧 |}^{2} + {| 𝑤 |}^{2} = 1$ . Show that the complement of the union of the sets $| 𝑧 | = 0$ and $| 𝑤 | = 0$ is homotopy equivalent to a 2-torus.

Stanford Fall 2024, 2 Topology, distributions, vector bundles, real projective space

For which values of $𝑛$ does there exist a 1-dimensional distribution on $ℝ ℙ^{𝑛}$ ? Construct an example when one exists, and prove one cannot exist otherwise.

Stanford Fall 2024, 2 Topology, covering spaces, homology

Let $𝑋 \to 𝑌$ be a covering map, and $𝑑$ a positive integer not equal to 0 or 1. Give examples where the homology $𝐻_{𝑑} (𝑋, ℚ)$ is non-trivial and the map

𝐻_{𝑑} (𝑋, ℚ) \to 𝐻_{𝑑} (𝑌, ℚ)

(a) is an isomorphism,

(b) is surjective but not an isomorphism,

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