counterexample to a conjecture: Search

197

A torsion-free group algebra that is not directly finite

Algebra 4 manuscripts, 93 pages Lean Reasoning summary

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽₂ is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.

Closest manuscript: A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic

340

A counterexample to the nearby Lagrangian conjecture

Differential geometry 1 manuscript, 28 pages

Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.

053

A counterexample to Pixton completeness in Chow

Algebraic and complex geometry 1 manuscript, 61 pages

Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.

156

Borsuk's conjecture fails in dimension nine

Combinatorics 1 manuscript, 18 pages Lean

Constructs a compact subset of ℝ⁹ that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ⁴, with the Frobenius metric.

155

A counterexample to periodic tiling in dimension three

Combinatorics 1 manuscript, 26 pages Lean

Constructs a finite translational tile in ℤ³ that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ³, even with arbitrary real translation vectors.

049

A stable-coordinate counterexample in four variables

Algebraic and complex geometry 2 manuscripts, 20 pages Lean

Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.

Closest manuscript: An explicit noncoordinate polynomial with affine three-space zero fibre

198

A counterexample to finitistic-dimension finiteness

Algebra 1 manuscript, 24 pages Lean

Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

196

A counterexample to Kaplansky’s zero-divisor conjecture

Algebra 1 manuscript, 26 pages Lean

Constructs a finitely presented torsion-free group G whose group algebra $\mathbb F_2[G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

249

A finitely generated Eilenberg–Ganea counterexample

Group theory 1 manuscript, 25 pages Lean

Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.

329

A counterexample to metric-entropy duality

Functional analysis 1 manuscript, 15 pages Lean

Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.

195

A counterexample to the small Cohen–Macaulay module conjecture

Algebra 1 manuscript, 17 pages

Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.

321

A counterexample to Wall's finite D(2) problem

Topology 1 manuscript, 10 pages

Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.

050

A counterexample to Griffiths’ positivity conjecture

Algebraic and complex geometry 1 manuscript, 18 pages Lean

Constructs ample rank-two bundles on $\mathbb P^1\times\mathbb P^1$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.

151

A C¹ counterexample to the entropy conjecture

Dynamical systems and ergodic theory 1 manuscript, 15 pages Lean

Constructs a noninvertible C¹ self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C¹ self-maps: homological growth need not force positive orbit complexity.

048

A characteristic-zero counterexample to Lipman–Zariski

Algebraic and complex geometry 1 manuscript, 37 pages

Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.

294

Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness

Operator algebras 1 manuscript, 16 pages Lean

Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C^∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C^∗-algebras can have a properly infinite minimal tensor product, with one factor $C_r^*(\mathbb F_2)$ .

088

Sharp projection-body inequalities and a counterexample to simplex maximization

Convex and metric geometry 2 manuscripts, 37 pages Lean

Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.

Closest manuscript: A product counterexample to the simplex maximum for projection-body volume

353

Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample

Differential geometry 2 manuscripts, 30 pages Lean

Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.

Closest manuscript: A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten

100

Cylinder coverings below the half-area bound

Convex and metric geometry 4 manuscripts, 83 pages Lean

Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.

Closest manuscript: Finite angular cylinder covers below the half-area bound

192

Boolean functions violate the square-root degree bound by arbitrary factors

Combinatorics 1 manuscript, 23 pages Lean

Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with $\sum_i\widehat f(\{i\})\gt C\sqrt{\deg(f)}$ . Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.

320

Nonhomeomorphic closed aspherical four-manifolds

Topology 1 manuscript, 60 pages

Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.

132

A superquadratic separation of sensitivity and block sensitivity

Theoretical computer science 1 manuscript, 11 pages Lean

Constructs total Boolean functions with block sensitivity $\mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha$ for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here $s(f)$ counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.

298

Two notions of free entropy differ even when both are finite

Operator algebras 1 manuscript, 15 pages Lean

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy $-\infty\lt \chi\lt \chi^*\lt \infty$ . This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.

351

Scalar curvature and finite-time Ricci-flow singularities

Differential geometry 3 manuscripts, 118 pages

Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.

Closest manuscript: A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup

189

Cycle–clique Ramsey numbers

Combinatorics 1 manuscript, 41 pages Lean

Proves the Erdős–Faudree–Rousseau–Schelp conjecture: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\ge n\ge3$ , except $R(C_3,K_3)=6$ . This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.

072

Brennan's conjecture and the integral-means spectrum

Real and complex analysis 2 manuscripts, 46 pages Lean

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $|\phi'|^s$ is area-integrable for $4/3\lt s\lt 4$ . The sharp universal integral-means identity is $B_{\mathcal S}(t)=|t|-1$ for t ≤ −2. A strict bound $B_b(-1)\lt 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.

Closest manuscript: Brennan's conjecture and sharp inverse-square integral means

161

Counterexamples to Sidorenko’s conjecture and the forcing conjecture

Combinatorics 1 manuscript, 51 pages Lean

Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.

162

Counterexamples to Ryser’s covering conjecture

Combinatorics 2 manuscripts, 79 pages Lean

Disproves Ryser's covering conjecture by constructing intersecting $(q+1)$ -partite, $(q+1)$ -uniform hypergraphs with covering number $q+1$ , rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.

Closest manuscript: A counterexample to Ryser's covering conjecture

157

Graph coloring, clique minors, and Colin de Verdière invariants

Combinatorics 3 manuscripts, 277 pages Lean

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $\chi_f(G)\gt h(G)$ , where $h(G)$ is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound $\chi_f(G)\le\mu(G)+1$ . In the positive direction, every finite nonempty graph satisfies $\chi_{\mathrm{list}}(G)\le C h(G)$ for a universal constant C.

Closest manuscript: A counterexample to Hadwiger's conjecture

209

Integral counterexamples to Gersten’s conjecture

Algebra 2 manuscripts, 35 pages

Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic $(0,5)$ have nonzero integral K-theory classes that vanish over their fraction fields.

Closest manuscript: An integral counterexample to Gersten's conjecture