Elementary Prerequisites
This reference contains fundamental mathematical concepts assumed in the main articles. The entries are kept concise and serve as a quick reference – not a substitute for a textbook, but a tool for targeted lookup.
A. Logic and Proof Techniques
| Topic |
Content |
| Propositional Logic |
Propositions, truth values, conjunction, disjunction, negation |
| Implication and Equivalence |
"A ⟹ B" ≡ "¬A ∨ B", contraposition, biconditional |
| Types of Proof |
Direct proof, proof by contradiction, induction, counterexample |
| What Is a Proof? |
Axioms, definitions, theorems, lemmas, conjecture vs. theorem |
B. Arithmetic and Numbers
| Topic |
Content |
| Fractions |
Addition, multiplication, reducing, expanding, common denominator |
| Equations and Equivalent Transformations |
Equality sign, permitted operations, equivalent transformations |
| Inequalities |
Order on ℝ, computation rules, absolute value, triangle inequality |
| Divisibility and GCD |
Division with remainder, greatest common divisor, Euclidean algorithm |
| Modular Arithmetic |
Congruence, computing modulo n, residue classes |
C. Sets and Structures
| Topic |
Content |
| Sets and Set Operations |
Notation (∈, ⊂, ∪, ∩, ∅), subsets, power sets |
| Mappings (Functions) |
Domain/codomain, injective, surjective, bijective |
| Number Systems |
ℕ, ℤ, ℚ, ℝ, ℂ – chain of extensions |
D. Geometry
E. Algebra
F. Geometry and Analysis (Advanced)
In-depth concepts for the differential geometry behind the
Poincaré conjecture – manifolds, curvature,
vector calculus, and parabolic differential equations.
| Topic |
Content |
| Manifolds – an Intuitive View |
charts, atlas, smooth structures, examples \(S^n\), \(T^n\), \(\mathbb{RP}^n\) |
| Tangent Space and Tensors |
\(T_p M\), vector fields, tensor fields, Riemannian metric |
| Curvature of Surfaces (Gauss) |
principal curvatures, Theorema Egregium, Gauss–Bonnet |
| Vector Calculus in a Nutshell |
gradient, divergence, Laplace–Beltrami, integral theorems |
| The Heat Equation – Intuition |
\(\partial_t u = \Delta u\), smoothing, maximum principle, heat kernel |