Captain: Henry Yuen
Fundamental Theorem of AlgebraTextbook
Show that every nonconstant complex polynomial has a complex root.
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Algebra
9 missions · 6 completed
The study of algebraic structures — groups, rings, and fields — and, through algebraic geometry, the geometry of the solution sets of polynomial equations. Using commutative algebra to describe these varieties, the field provides a common language of symmetry and structure that underlies much of modern mathematics.
Missions
Captain: ShouqiaoWang
Symplectic Modules Free over an Abelian NilradicalResearch Paper
## Motivation Polynomial representations provide a concrete way to study modules over Lie algebras: the underlying vector space is a polynomial ring, while the Lie generators act by explicit multiplication, shift, and differential operators. Chen and Tan classify a family of modules over the symplectic Lie algebra $\mathfrak{sp}_{2\ell}(\mathbb C)$ that are free of rank one over the universal enveloping algebra of an abelian nilradical. Their paper determines the family, its isomorphism classes, its weight and simplicity criteria, its finite-length behavior at exceptional parameters, and an application to Hamiltonian Lie algebras. This mission packages those headline results into one common Lean target, corresponding to Theorems 1.1--1.3 of [Chen--Tan](https://doi.org/10.1016/j.jalgebra.2026.02.022). The common-family formulation matters. The source does not assert three unrelated existence theorems: one explicit two-parameter family $\tau(C,\Phi)$ carries all of the classification, simplicity, finite-length, and Hamiltonian consequences. The Lean goal therefore quantifies that family once and requires all headline properties of the same witness. ## Setting Fix $\ell\ge2$ and the complex symplectic Lie algebra $\mathfrak{sp}_{2\ell}(\mathbb C)$. The relevant maximal parabolic subalgebra has an **abelian nilradical** $\mathfrak n$. Its enveloping algebra is a polynomial algebra in the root generators, represented formally by a multivariate polynomial ring. A rank-one free $U(\mathfrak n)$-module can consequently be modeled on that polynomial ring. The definition bundle presents the simple Chevalley generators and their action by explicit operators depending on a scalar $C\in\mathbb C$ and a polynomial parameter $\Phi$. Rather than assuming that these formulas already form a representation, the target asks for a generator presentation satisfying the symplectic Lie relations and for a representation family $\tau(C,\Phi)$ realizing the formulas. It also formalizes module equivalence, weight spaces, simplicity, Noetherian and Artinian conditions, finite composition factors, and the Shen--Larsson construction for a Hamiltonian Lie algebra. ## Formalization targets ### Common polynomial-module family Prove that for every $\ell\ge2$ there is one generator presentation and one family $$ (C,\Phi)\longmapsto \tau(C,\Phi) $$ of $\mathfrak{sp}_{2\ell}(\mathbb C)$-representations on the polynomial ring, free of rank one over the abelian nilradical, satisfying the explicit generator formulas. Prove the source's classification and isomorphism criteria, including that $\tau(C,\Phi)$ is a weight module exactly when $\Phi$ is constant, and the stated simplicity criterion outside the exceptional arithmetic set $$ \left\{\frac{\ell+1}{2}-\frac{n}{2}:n\in\mathbb Z_{>0}\right\}. $$ For exceptional $C$, prove the Noetherian/Artinian and finite-composition-series conclusions and the weight/nonweight classification of the composition factors. Finally, prove that the canonical Hamiltonian Shen--Larsson construction has the exact degree-weight spaces and the source's simplicity and weight-module consequences. All clauses must be witnessed by the same family $\tau$. ## Significance The result gives a complete algebraic description of a large concrete class of non-highest-weight modules. It separates the generic simple regime from an exceptional finite-length regime and shows how nonweight symplectic modules generate weight modules over an infinite-dimensional Hamiltonian Lie algebra. The explicit formulas make the family suitable for calculation, while the classification prevents duplicate parameter choices from being mistaken for genuinely different modules. Formalization adds checks that are easy to blur in prose. In particular, the generator formulas cannot be called a Lie representation until the defining relations have been verified, and the same witness must support every later theorem. A completed proof will contribute reusable Lean infrastructure for symplectic root data, polynomial representations, module-theoretic finiteness, exact weight-space descriptions, and Hamiltonian Lie-algebra functors. The paper's proofs are known; the open task is their machine-checked reconstruction. ## Difficulty The first obstacle is structural rather than computational. Checking formulas on individual generators is insufficient: all Chevalley and Serre relations must hold with the correct operator order and signs, after which the action must extend to the full Lie algebra. Classification then requires controlling arbitrary rank-one-free modules, not merely verifying that the displayed examples exist. The exceptional parameters introduce a second layer. Generic simplicity and exceptional finite length are logically different claims, and the composition-factor statement must be tied to the same parameterized representation. The Hamiltonian application adds another algebra and a tensor construction; exact weight spaces and simplicity cannot be obtained by treating the functor as an opaque interface. The Lean goal deliberately keeps these obligations inside one theorem so that separate convenient witnesses cannot satisfy different portions. ## Formalization scope The mission works over $\mathbb C$ with natural rank $\ell\ge2$. The definition bundle uses concrete multivariate polynomials, matrices and linear maps, a presented symplectic Lie algebra, Lie representations, submodules, and tensor products. The exceptional set is expressed with complex coercions, so no accidental natural-number division is involved. The nilradical action, freeness, parameter equivalence, weight-space equalities, simplicity, finite-length properties, and Hamiltonian brackets are transparent propositions in the bundle. The final theorem is a single conjunction under one existentially quantified presentation and one existentially quantified family $\tau$. Several convenient corollaries can be projected from it, but they are not independent targets and do not permit different witnesses. The bundle contains no custom axioms or opaque semantic assumptions, and the only admitted term is the main theorem's `sorry`. Contributions may split the proof into source-numbered lemmas about generator relations, classification, exceptional submodules, or the Shen--Larsson application, provided the shared-family quantifier structure is preserved. ## Selected references - Yang Chen and Haijun Tan, *Simple $\mathfrak{sp}_{2\ell}(\mathbb C)$-modules which are free over an abelian nilradical*, Journal of Algebra 697 (2026), 341--372, Theorems 1.1--1.3 (formal Theorems 3.7, 3.8, 4.7, 4.9, and 5.2). [DOI](https://doi.org/10.1016/j.jalgebra.2026.02.022) - G. Shen, foundational work on mixed-product constructions for modules over Lie algebras of Cartan type, cited in the source paper for the Shen--Larsson functor.
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Captain: ShouqiaoWang
Arbitrary Torsion in Moment-Angle Homology and Loop HomologyResearch Paper
## Motivation Moment-angle complexes are central objects in toric topology. They convert the combinatorics of a simplicial complex into a topological space assembled from disks and circles, allowing face structure to influence homotopy and homology. When the simplicial complex triangulates a sphere, the resulting space is a moment-angle manifold. Torsion in the integral homology of these manifolds is difficult to realize in low simplicial dimension, and torsion in the homology of their based loop spaces is even more constrained. Yang Han and Keke Li's Theorem 1.7 asserts that dimension four is already universal: every finitely generated abelian group can occur as a subgroup of both homology theories for one and the same simplicial $4$-sphere. This mission formalizes that headline existence statement. It is not restricted to a chosen finite list of groups or primes, and it requires a common simplicial sphere rather than permitting separate witnesses for ordinary and loop homology. ## Setting Let $L$ be an abstract simplicial complex on a finite vertex set $[m]$. Its geometric realization $|L|$ is formed from probability vectors whose supports are faces of $L$. The condition that $L$ is a **simplicial $4$-sphere** means that this realization is homeomorphic to the unit sphere $S^4\subset\mathbb R^5$. For each face $\sigma\in L$, assign a copy of the closed disk $D^2$ at vertices in $\sigma$ and the boundary circle $S^1$ at vertices outside $\sigma$. The associated moment-angle complex is $$ \mathcal Z_L =\bigcup_{\sigma\in L} \prod_{i=1}^{m}Y_i(\sigma), \qquad Y_i(\sigma)= \begin{cases} D^2,&i\in\sigma,\\ S^1,&i\notin\sigma. \end{cases} $$ The all-ones point is a canonical basepoint. Write $\Omega\mathcal Z_L$ for the based loop space with the compact-open topology. For a space $X$, the mission uses total integral singular homology $$ H_*(X;\mathbb Z)=\bigoplus_{q\ge0}H_q(X;\mathbb Z) $$ as an additive abelian group. Saying that an abelian group $G$ is a subgroup means that there is an injective additive homomorphism $G\hookrightarrow H_*(X;\mathbb Z)$. ## Formalization targets ### Arbitrary torsion in one moment-angle manifold For every finitely generated abelian group $G$, prove that there are an integer $m$ and a simplicial complex $L$ on `Fin m` such that $|L|\cong S^4$ and there are injective homomorphisms $$ G\hookrightarrow H_*(\mathcal Z_L;\mathbb Z), \qquad G\hookrightarrow H_*(\Omega\mathcal Z_L;\mathbb Z). $$ The quantifier order matters: the same $m$ and the same $L$ must support both embeddings. The target concerns additive subgroups of total graded homology; it does not require the two embeddings to land in the same degree or to preserve multiplicative structures. ## Significance The theorem gives a universality statement for moment-angle manifolds over simplicial $4$-spheres. It says that no classification by a bounded list of torsion primes or exponents can describe all such homology and loop-homology groups. Requiring both embeddings for a single $L$ connects the ordinary topology of the manifold to its based-loop topology rather than proving two unrelated existence results. Formalizing the theorem requires reusable foundations in several areas: finite abstract simplicial complexes, geometric realization, polyhedral products, based loop spaces, integral singular homology, graded direct sums, and additive embeddings. The published article presents a human proof; this mission records its intended main theorem as an open Lean target. The definitions do not assume the existence of the required sphere or embeddings, so a solver must supply the mathematical construction and all homological consequences. ## Difficulty The assertion ranges over arbitrary finitely generated abelian groups, including free parts and prime-power torsion of unbounded exponent. A finite check of selected groups cannot establish the target. The same finite simplicial object must simultaneously control two different homology theories, one of which is applied to an infinite-dimensional function space. Standard library support is strongest for singular homology as a functor, while concrete calculations for moment-angle spaces and loop spaces require additional bridges. There is also a substantial representation boundary between combinatorics and topology. The face data of $L$, the union of disk-circle products, the homeomorphism $|L|\cong S^4$, and the induced maps on homology must all refer to compatible spaces and basepoints. A formal solution cannot replace “simplicial sphere” by a mere Boolean flag or replace homology by an arbitrary group-valued field. ## Formalization scope Lean represents $L$ using `AbstractSimplicialComplex (Fin m)`. Because Mathlib's structure includes singleton faces automatically, the auxiliary face predicate explicitly restores the conventional empty face where the moment-angle union needs it. The geometric realization is the standard support-restricted probability simplex, and the sphere condition is an actual homeomorphism to the Euclidean unit $4$-sphere. The moment-angle space is a subtype of `(Fin m → ℂ)` defined by the literal disk/circle coordinate condition. The loop space consists of based continuous paths with matching endpoints and carries the compact-open topology inherited from Mathlib's path construction. Homology is `singularHomologyFunctor` with coefficients in $\mathbb Z$, and total homology is a direct sum over all natural degrees. The statement permits the two embeddings to occupy different degrees and makes no ring-embedding claim; these choices match the source phrase “contain $G$ as a subgroup.” It rules out vacuity by requiring an actual simplicial complex, an actual sphere homeomorphism, and injective additive maps. Contributions that isolate degree-specific refinements, compute homology of standard polyhedral products, or formalize reusable loop-space equivalences are welcome, provided they reconnect to the stated root theorem. ## Selected references - Yang Han and Keke Li, *Moment Angle Manifolds Corresponding to $S^4$ Whose Homology and Loop Homology May Have Arbitrary Torsion*, International Mathematics Research Notices 2026(4), 1--7, 2026. [DOI](https://doi.org/10.1093/imrn/rnag024) - A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, *The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces*, Advances in Mathematics 225(3), 2010, 1634--1668. [DOI](https://doi.org/10.1016/j.aim.2010.04.009)
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Captain: tianyipeng
Hefferon Linear Algebra V: Jordan Canonical FormTextbook
Chapter Five of Jim Hefferon's *Linear Algebra* is one long search for a canonical form for matrix similarity, and Theorem IV.2.8 ends it: over the complex numbers every square matrix is similar to a matrix in Jordan form. That is the goal theorem of this mission and the capstone of the book. Mathlib carries the generalized eigenspace decomposition but has no Jordan canonical form, so this is a genuine target rather than a wrapper around an existing lemma; the Jordan block and the block-diagonal Jordan matrix are supplied as a mission definition. The milestones are the three results the proof is assembled from: diagonalizability as the existence of an eigenbasis, Cayley-Hamilton, and the canonical form of a nilpotent map, which is Jordan form applied to $t - \lambda$ on each generalized eigenspace.
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Captain: tianyipeng
Hefferon Linear Algebra III: Maps, Representation and Change of BasisTextbook
Chapter Three of Jim Hefferon's *Linear Algebra* is about maps between spaces and how matrices represent them. The goal theorem is where the chapter arrives: two matrices represent the same transformation with respect to different bases exactly when they are similar. That is the hinge of the whole book — it converts the search for a canonical form under similarity into the search for the basis in which a map looks simplest, which is the programme of Chapter Five. The milestones are the chapter's landmarks: dimension classifies spaces up to isomorphism, rank plus nullity recovers the dimension of the domain, matrix multiplication is exactly composition, and Gram-Schmidt splits a space into a subspace and its orthogonal complement.
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Captain: tianyipeng
Hefferon Linear Algebra II: Dimension and RankTextbook
Chapter Two of Jim Hefferon's *Linear Algebra* builds the vector space vocabulary — spanning, independence, basis — and turns it into a theory of dimension. The goal theorem is the chapter's most striking result, that the row rank and the column rank of a matrix always agree, which is the bridge between the matrix-of-numbers view of Chapter One and the vector space view of Chapter Two. The milestones are the two pillars it stands on: that any two bases of a space have the same size, so dimension is well defined at all, and that any linearly independent set can be extended to a basis.
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Captain: tianyipeng
Hefferon Linear Algebra I: Gauss's Method and the Solution SetTextbook
Chapter One of Jim Hefferon's *Linear Algebra* develops Gauss's method and asks what row reduction actually preserves. The answer arrives as the Linear Combination Lemma: row operations change the rows of a matrix but never the subspace those rows span, and that invariant is complete. The goal theorem is that completeness — two matrices are row equivalent exactly when they have the same row space — which is what makes reduced echelon form a genuine canonical form. The milestones are the two results the chapter builds on the way: that row operations leave a system's solution set alone, and that a solution set is always one particular solution translated by the solutions of the associated homogeneous system.
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Captain: Community (Bot)
The Jacobian ConjectureOpen Problem
First raised for two variables by Ludwig Kraus in 1884 and stated in full generality by Ott-Heinrich Keller in 1939, the Jacobian conjecture asks something that sounds almost like freshman calculus: if a polynomial map from complex n-space to itself has a Jacobian determinant equal to a nonzero constant, must it be invertible by another polynomial map? That constant-Jacobian condition is precisely the algebraic shadow of the inverse function theorem, yet producing a polynomial — not merely analytic — inverse has resisted every attack for over eighty years. Shreeram Abhyankar championed the problem because it can be stated 'using little beyond a knowledge of calculus,' and Stephen Smale placed it sixteenth on his 1998 list of problems for the new century. Its notoriety is sharpened by a graveyard of published 'proofs' that later collapsed. Deep reductions exist — Bass, Connell, and Wright showed in 1982 that the general case reduces to maps of degree three — and the problem is equivalent, through work of Tsuchimoto, Belov-Kanel, and Kontsevich, to the Dixmier conjecture on the Weyl algebra. A formal statement anchors this famously slippery problem so that progress can be verified rather than merely believed.
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Captain: wenxinzhang
Existence of complete sets of mutually unbiased basesOpen Problem
# Motivation Two orthonormal bases of C^d are mutually unbiased when every transition amplitude has squared modulus 1/d. At most d+1 such bases can coexist, and complete families are known in prime-power dimensions through finite-field constructions. Dimension six is the smallest famous composite case where existence of the complete seven-base family remains unknown. This mission turns CUHK-Shenzhen AI Math Problem 16, **Existence of complete sets of mutually unbiased bases**, into an auditable Lean campaign. The objective is not merely to transcribe notation: it is to expose the mathematical model, the capstone, and a smaller attack surface as separate artifacts that other formalizers can inspect and reuse. ## Setting Construct seven 6 by 6 unitary-column matrices whose every distinct pair has all transition amplitudes of squared modulus 1/6. The baseline milestone constructs three pairwise mutually unbiased bases, a known lower bound that tests all matrix conventions. ## Significance Solving this target would settle the precise finite or analytic core represented by the Lean statement and would create reusable infrastructure in quantum information theory, mutually unbiased bases, finite fields, Hilbert spaces. Even a rigorous disproof is valuable: several entries in this collection deliberately ask whether an attractive extrapolation is true, and Lean forces a counterexample to satisfy every side condition. The mission therefore treats theorem proving and model criticism as equally legitimate research outcomes. ## Difficulty The equations are a large coupled system of polynomial equalities over complex phases, modulo substantial gauge symmetry. Numerical near-solutions do not certify exact existence, while nonexistence would require a global obstruction beyond currently known bounds. Dimension six lacks the finite-field structure that supplies complete prime-power constructions. ## Suggested attack route Formalize standard gauge reductions: fix the first basis to the identity and dephase transition Hadamard matrices. Verify a three-basis tensor-product construction. Then encode additional bases through complex Hadamard matrices and study algebraic constraints, Gröbner-style eliminations, semidefinite bounds, or exact certificates. Computational searches may guide conjectures, but uploaded proofs must convert numerical evidence to exact algebraic identities or certified inequalities. ## Formalization scope The Lean target is exact: column orthonormality is U-adjoint times U equals identity, and mutual unbiasedness uses Mathlib complex norm squared. Seven bases are indexed by Fin 7. No quotient by phase, permutation, or global unitary is built into the statement, since these symmetries preserve the predicate and can be used within proofs. The natural-language source remains authoritative for motivation, while the Lean declaration is authoritative for what Prove2Me will verify. The mission description calls out restrictions where the current formal target is a finite-dimensional core, a fixed interpretation of informal terminology, or one sharpened subquestion from a broader classification problem. Those restrictions should not be silently generalized in a proof claim. ## Milestones Publish an exact three-basis construction in dimension six, then formalize dephasing and obstruction lemmas for extending a partial family. The capstone is marked as the mission's main item and is never duplicated as a milestone. Definitions precede theorem statements in the proposal order. A milestone is considered complete only when its own exact statement is proved; proving a nearby theorem with stronger-looking prose but mismatched quantifiers, signs, supports, dimensions, or asymptotic constants does not complete it. ## Timeline and literature status The CUHK-Shenzhen AI Math Problems page added this problem on June 23, 2026. At the drafting date, August 31, 2026, the status and target corrections described above were checked against the source page and the cited primary material. ## Acceptance criteria A contribution may prove the displayed theorem or refute it by constructing data satisfying every Lean hypothesis while negating the conclusion. Informal changes of model do not count: any proposed correction must be submitted as a separately reviewed statement with an explanation of which source ambiguity or false implication it repairs. Definitions must remain computational or mathematically constrained; fields that simply assume the desired conclusion are not acceptable. Every proof must compile against the mission's pinned Mathlib revision, use no `sorry`, and expose a top-level `theorem solution` when submitted to Prove2Me. The main theorem is intentionally separated from a smaller milestone. Contributors should preserve that dependency order, publish reusable lemmas rather than monolithic tactics, and report whether a lemma is analytic, algebraic, combinatorial, or infrastructure-only. Numerical evidence, external computer algebra, and exhaustive search are welcome for discovery, but a final certificate must be replayable by Lean. If an external result is invoked, its hypotheses must be represented in the formal statement or proved in the dependency tree. ## Formal verification policy The files were built locally with Lean 4.30.0 and Mathlib revision `c5ea00351c28e24afc9f0f84379aa41082b1188f`, the supported Prove2Me environment at drafting time. The mission definition file is ordered before all theorem files, and each theorem imports exactly that public definition module or Mathlib. Independent blind read-backs accompany the draft items so reviewers can compare what the Lean code literally says with this mathematical description. Human confirmation remains required before the public proposal can be submitted for moderation. ## Selected references - [Original problem](https://rybindmitry.github.io/problems/16.html) - [Durt et al., review of MUBs](https://arxiv.org/abs/1004.3348)
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