Mathematical Theorems

Question 1

Mergelyan’s Theorem

Answer 1

Let 𝐾 ⊆ᶜᵒᵐᵖᵃᶜᵗ ℂ such that ℂ ∖ 𝐾 is connected. Every continuous function 𝑓 : 𝐾 → ℂ whose restriction to the interior of 𝐾 is holomorphic can be uniformly approximated by polynomials.

Question 2

Shell Theorem

Answer 2

A thin spherical shell exerts no gravitational influence on internal objects and attracts external objects as though its mass were concentrated at its center point.

Question 3

Rank-Nullity Theorem

Answer 3

Let 𝑉 and 𝑊 be vector spaces over a field 𝔽, with 𝑉 finite-dimensional, and let 𝑇 : 𝑉 → 𝑊 be a linear transformation. Then dim im 𝑇 + dim ker 𝑇 = dim 𝑉.

Question 4

Hyperplane Separation Theorem

Answer 4

If 𝐴 and 𝐵 are two disjoint convex subsets of ℝⁿ, then there exist 𝐯 ∈ ℝⁿ and 𝑐 ∈ ℝ such that 𝐱ᵀ𝐯 ≥ 𝑐 for all 𝐱 ∈ 𝐴 and 𝐲ᵀ𝐯 ≤ 𝑐 for all 𝐲 ∈ 𝐵.

Question 5

Robertson–Seymour Theorem

Answer 5

The set of (isomorphism classes of) finite undirected graphs is well-partial-ordered by the graph minor relation.

Question 6

Cook–Levin Theorem

Answer 6

The Boolean satisfiability problem is 𝖭𝖯-complete.

Question 7

Max-Flow Min-Cut Theorem

Answer 7

Let 𝐺 be a finite nonnegative-edge-weighted directed graph, and let 𝑠, 𝑡 ∈ 𝑉(𝐺) be distinct. The maximum value of an 𝑠-𝑡 flow in 𝐺 equals the minimum weight of an 𝑠-𝑡 edge cut in 𝐺.

Question 8

Brouwer’s Invariance of Domain Theorem

Answer 8

Let 𝑓 be a continuous injection from an open subset of ℝⁿ to ℝⁿ. The image of 𝑓 is open, and 𝑓 is a homeomorphism onto its image.

Question 9

Brouwer’s Invariance of Dimension Theorem

Answer 9

If 𝑈 ⊆ᵒᵖᵉⁿ ℝᵐ is homeomorphic to 𝑉 ⊆ᵒᵖᵉⁿ ℝⁿ, then 𝑚 = 𝑛.

Question 10

Novikov–Boone Theorem

Answer 10

There exists a finitely presented group with algorithmically undecidable word problem.

Question 11

Adian–Rabin Theorem

Answer 11

All Markov properties of finitely presented groups are algorithmically undecidable. In particular, it is undecidable whether a given finite presentation defines the trivial group.

Question 12

Apéry’s Theorem

Answer 12

𝜁(3) is irrational.

Question 13

Banach–Schröder–Bernstein Theorem

Answer 13

Let 𝐺 ↷ 𝑋 and 𝐴, 𝐵 ⊆ 𝐺. If 𝐴 is 𝐺-equidecomposable with a subset of 𝐵 and 𝐵 is 𝐺-equidecomposable with a subset of 𝐴, then 𝐴 and 𝐵 are 𝐺-equidecomposable.

Question 14

Uniformization Theorem

Answer 14

Every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere.

Question 15

Gelfand–Naimark Theorem

Answer 15

Every C*-algebra is *-isometric to an algebra of bounded operators on a complex Hilbert space.

Question 16

Wedderburn’s Little Theorem

Answer 16

Every nontrivial finite ring without zero divisors is a field.

Question 17

Frobenius’ Theorem

Answer 17

Every finite-dimensional associative division algebra over ℝ is isomorphic to ℝ, ℂ, or ℍ.

Question 18

Bott–Milnor–Kervaire Theorem

Answer 18

Every finite-dimensional division algebra over ℝ is isomorphic to ℝ, ℂ, ℍ, or 𝕆.

Question 19

Lindemann–Weierstrass Theorem

Answer 19

If α₁, …, αₙ are algebraic numbers linearly independent over ℚ, then exp(α₁), …, exp(αₙ) are algebraically independent over ℚ.

Question 20

Rosser’s Theorem

Answer 20

𝑝ₙ > 𝑛 log 𝑛; improved by Dusart in 1999 to 𝑝ₙ > 𝑛 log 𝑛 + 𝑛 log log 𝑛 - 𝑛.

Question 21

Szemerédi’s Theorem

Answer 21

Any subset of the natural numbers with positive upper density contains arbitrarily long arithmetic progressions.

Question 22

Central Limit Theorem

Answer 22

Suppose (𝑋₁, 𝑋₂, …) is a sequence of IID random variables with finite mean 𝜇 and variance 𝜎². As 𝑛→∞, the scaled sample averages (𝑆ₙ − 𝜇)√𝑛 converge in distribution to 𝑁(0, 𝜎²), i.e., their CDFs converge pointwise.

Question 23

Church–Rosser Theorem

Answer 23

λ-calculus is confluent under α-conversion, β-reduction, and η-conversion. That is, if a λ-expression 𝑥 can be reduced in two ways to 𝑦₁ and 𝑦₂, then there exists a λ-expression 𝑧 to which both 𝑦₁ and 𝑦₂ can be reduced.