The Math of an Invasion Front Has a Counterintuitive Price Tag
A mathematical model of invasive spread finds a unique control strategy—and shows why faster fronts can carry a surprisingly uneven cost.
Math
A Random Walk’s Endpoint Hides on a Measure-Zero Boundary
On a random hyperbolic tessellation, each walk’s endpoint law hides on a zero-measure set—even though averaging landscapes gives a uniform boundary.
A New Map Could Help Mathematicians Find the Lemma They Need
A theorem-by-theorem map links millions of mathematical claims across papers and Lean proofs, offering new ways to search—and revealing limits.
A Prime-by-Prime Test Solves a Surprisingly Global Matching Puzzle
A divisor-matching puzzle has a surprisingly local solution: prime-by-prime counts guarantee a global coprime pairing.
How Gradual Cooling Could Help Samples Escape Energy Traps
A mathematical proof shows how annealing may help sampling cross energy barriers, while strict assumptions reveal what still stands in the way.
Each Ripple Count Has Just One Bound State in a Logarithmic Wave
A long-standing uniqueness puzzle is solved: each node count selects one radial, finite-energy bound state for a logarithmic wave equation.
The Zero-Sum Mystery: Why Two Piles Set the Toughest Benchmark
A 1992 conjecture asks how few zero-sum groups can hide in a sequence. A new proof shows two piles are the hardest arrangement to beat.
A New Counting Trick Catches Rare Spikes in the Zeta Function
A new method counts unusually large values of the Riemann zeta function, nudging a long-standing sixth-moment bound in a better direction.
A Local Rule Reveals the Covering Density of Enormous Random Trees
A canonical covering rule turns local subtree patterns into large-scale packing and covering densities across diverse random-tree models.
Why a Randomly Growing Cluster Can’t Settle Into a Smooth Shape
A new DLA theorem rules out smooth bulk shapes and shows why random particle growth must preserve a sprawling, exposed boundary.
The 16-Vertex Graph That Revealed a Hidden Parity Trap in Graph Theory
A 16-vertex cubic graph breaks a decades-old path-partition conjecture through a parity mismatch hidden at a single bridge.
The 27-Dimensional Tensor That Needed One Extra Building Block
A 27-dimensional tensor exposes a one-term gap between ordinary and symmetric rank, settling a long-running question in multilinear algebra.
How Thermodynamics Turns Blood-Vessel Remodeling Into a Testable Theory
A thermodynamic framework turns local vessel remodeling into testable predictions about flow, metabolic budgets, and hidden network signals.
How Topology Finds a Level Square on an Uneven Mathematical Floor
A topological reframe turns table balancing into a parity argument, proving when a square can sit level on an uneven graph and what else follows.
Can One Rule Tell You Everything About a Strange Operation?
An equation may hide consequences—or admit a counterexample. Meet a proof-search system that checks both paths and reveals where its search stalls.
The Physics Trick That Could Make Neural Networks Less Mysterious
Physics offers a surprising map for neural-network learning: solvable models, scaling laws, and shared patterns point toward a new mathematical science.
When Equations Resist Proof, Build a World Where They Fail
A new solver tackles abstract algebra from both sides: deriving consequences or constructing counterexamples, then asking Lean to verify every answer.
One Extra Step Makes a Directed Network Far Easier to Cover
A graph-theory result shows how one extra step can shrink the independent set needed to reach every vertex in a directed network.
The Online Secretary Problem’s Hidden Map of Dependencies
A new matroid secretary result secures a classic selection guarantee despite hidden dependencies, but leaves efficient computation as the next hurdle.
How Hidden Rings Turn a Wave Continuum Into Dense Bound States
A faint landscape of distant rings overturns expectations about quantum waves, replacing a traveling-state continuum with dense bound states.