A Map With No Local Folds Still Sends 3 Points to the Same Place
A counterexample to the 1939 Jacobian conjecture shows how a map with no local folds can still send several points to one place.
algebraic geometry
Two Nine-Line Patterns Look the Same—Until Algebra Looks Closer
Two nine-line arrangements share the same intersection map, yet their equations reveal different hidden algebra—challenging a proposed mathematical rule.
A 30-Dimensional Function Breaks a 64-Year-Old Math Conjecture
A pair of five-by-five matrices hides a determinant sign that defeats every finite polynomial recipe for a continuous piecewise-quadratic function.