Problem Directoryv1.0

Morera's Theorem (Giacinto Morera)

A continuous function with vanishing closed-loop integrals is holomorphic. Morera's theorem acts as a converse to Cauchy's integral theorem, allowing…

math.CV · Complex Variables · math.CA

📚 Undergraduate · Complex Analysis

Monotone Convergence Theorem

Every monotonic sequence that is bounded converges to its least upper bound or greatest lower bound. It is the fundamental bridge between order and completenes…

math.CA · Classical Analysis and ODEs · math.RA

📚 Undergraduate · Real Analysis

Maximum Modulus Principle (Cauchy)

The modulus of a non-constant holomorphic function on a connected open domain cannot attain a local maximum inside the domain. Step through the replay; discuss…

math.CV · Complex Variables · math.AP

📚 Undergraduate · Complex Analysis

Liouville's Theorem (Cauchy/Liouville)

An entire function that is bounded must be constant; global constraints on growth force a function's derivative to vanish. Step through the replay to see …

math.CV · Complex Variables · math.AP

📚 Undergraduate · Complex Analysis

Lebesgue Number Lemma (Henri Lebesgue)

In any compact metric space, every open cover admits a Lebesgue number: a positive constant such that any subset of smaller diameter must be contained within a…

math.GN · General Topology · math.MG

📚 Undergraduate · Topology

Lagrange's Theorem (subgroup order divides group order)

If H is a subgroup of a finite group G, then |H| divides |G|. The first structural theorem of group theory; its proof partitions G into equal-sized cosets. Ste…

math.GR · Group Theory

📚 Undergraduate · Algebra

Krull's Intersection Theorem (Wolfgang Krull)

In a Noetherian local ring, the intersection of the powers of a proper ideal annihilates the intersection itself. Step through the proof; see how the Artin-Ree…

math.RA · Rings and Algebras · math.AC · math.AG

📚 Graduate · Algebra

König's Theorem (König)

König's theorem proves that the sum of a sequence of cardinals is strictly less than their product, provided each component satisfies the strict inequalit…

math.LO · Logic · math.GM

📚 Undergraduate · Set Theory

König's Lemma (Dénes Kőnig)

Every infinite, finitely branching tree contains an infinite branch. This fundamental result bridges finite and infinite combinatorics. Step through the replay…

math.LO · Logic · math.CO · math.GM

📚 Graduate · Set Theory

Kleene's Normal Form Theorem (Kleene)

Every partial computable function is expressible as a primitive recursive extraction from the result of a minimal search over a primitive recursive computation…

math.LO · Logic · math.GM

📚 Graduate · Logic

Jordan–Brouwer Separation Theorem

An embedding of S^{n-1} in R^n divides the space into exactly two connected components: one bounded and one unbounded. Step through the replay to see how the w…

math.AT · Algebraic Topology · math.DG

📚 Graduate · Topology

Intermediate Value Theorem (Bolzano)

For a continuous function f on [a, b], any value u between f(a) and f(b) is attained by some c in (a, b). Continuity prevents the function from skipping values…

math.CA · Classical Analysis and ODEs · math.RA

📚 Undergraduate · Real Analysis

Hurewicz Theorem (Hurewicz)

The Hurewicz homomorphism identifies homotopy and homology in the first dimension where they are non-zero. The theorem bridges discrete homotopy with abelianiz…

math.AT · Algebraic Topology

📚 Graduate · Topology

Homotopy Lifting Property (Hurewicz, Eilenberg, Steenrod)

Covering spaces are defined by their path lifting, but they also extend homotopies. The Homotopy Lifting Property ensures any homotopy of a map into the base s…

math.AT · Algebraic Topology · math.GT

📚 Graduate · Topology

Hilbert's Nullstellensatz (Hilbert)

The Nullstellensatz establishes a perfect dictionary between affine varieties and radical ideals over an algebraically closed field. The proof relies on the in…

math.RA · Rings and Algebras · math.AG · math.AC

📚 Graduate · Algebra

Heine–Borel Theorem (Heine, Borel)

A subset of Euclidean space is compact if and only if it is closed and bounded. The proof uses the nested interval property to find a point that contradicts th…

math.CA · Classical Analysis and ODEs · math.FA · math.MG

📚 Undergraduate · Real Analysis

Hartogs' Lemma (Hartogs)

For every set A, there exists an ordinal \alpha such that no injection from \alpha into A exists. This demonstrates that the class of ordinals is unbounded in …

math.LO · Logic · math.GM

📚 Graduate · Set Theory

Hahn–Banach Theorem (Hahn, Banach)

A bounded linear functional on a subspace can be extended to the whole space while preserving its norm, or more generally, its domination by a sublinear functi…

math.FA · Functional Analysis · math.LO

📚 Graduate · Real Analysis

Gödel's Second Incompleteness Theorem (Gödel)

A consistent, sufficiently strong formal system cannot prove its own consistency statement Con(T). A cornerstone of modern logic; step through this formalizati…

math.LO · Logic · math.GM

📚 Graduate · Logic

Gödel's First Incompleteness Theorem (Gödel)

Any consistent, effectively axiomatized formal system $F$ capable of expressing basic arithmetic contains a sentence $\sigma$ such that neither $\sigma$ nor $\…

math.LO · Logic

📚 Graduate · Logic

Fundamental Theorem of Cyclic Groups

Every subgroup of a cyclic group is cyclic. If a finite cyclic group has order n, for each divisor k of n there exists exactly one subgroup of order k. Step th…

math.GR · Group Theory · math.RA

📚 Undergraduate · Algebra

First Isomorphism Theorem (Noether)

The kernel of a group homomorphism captures the redundancy of the map. By quotienting the domain by the kernel, we recover an isomorphic image in the codomain.…

math.GR · Group Theory · math.RA

📚 Undergraduate · Algebra

Extreme Value Theorem (Weierstrass)

Continuous functions on a closed, bounded interval [a, b] always attain their absolute maximum and minimum. This result bridges the gap between boundedness and…

math.CA · Classical Analysis and ODEs · math.FA

📚 Undergraduate · Real Analysis

Excision Theorem (Eilenberg-Steenrod)

The Excision Theorem allows computing relative homology by removing subspaces that do not contribute to the homology of a pair. It relies on barycentric subdiv…

math.AT · Algebraic Topology · math.GT

📚 Graduate · Topology