Mathematics
Abelian-Surface Modularity Moves Beyond Potentiality
A 2025 theorem proves modularity for a positive proportion of abelian surfaces, while the 2026 ICM account explains the hypotheses and remaining gap.
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203 articlesMathematics
1–24 of 50 articles
Mathematics
When pressure and velocity tails prevent a kinetic singularity
A July 2025 theorem replaces an entropy hypothesis with pressure and moment bounds, sharpening what can keep a non-cutoff Boltzmann gas smooth.
Mathematics
Borel Games Enter a Proof Assistant: What Lean’s 2026 Determinacy Formalization Establishes
A 2026 formalization puts Borel determinacy into Lean. Explore infinite games, winning strategies, and the precise boundary between existence and algorithms.
Mathematics
Separating solitons from radiation at critical regularity
A September 2026 preprint studies critical-space soliton resolution for the Calogero–Moser equation, with spectral and solution assumptions kept explicit.
Mathematics
How random directions illuminate spiky convex bodies
A 2026 revision confirms Hadwiger’s illumination bound for cap bodies in every dimension, combining random directions with controlled computation.
Mathematics
Carleson’s Theorem in Lean: Making Almost-Everywhere Convergence Explicit
The September 2026 Carleson formalization paper explains a completed Lean proof, its continuous-function corollary, and the meaning of almost-everywhere convergence.
Mathematics
The Conway Knot’s Infinite Order: A New Question Beyond Sliceness
A June 2026 preprint, revised in September, shows why no positive connected sum of Conway knots bounds a smooth disk, extending the famous sliceness result.
Mathematics
Why Zero Volume Does Not Allow Uncountably Many Hamiltonian Copies
A May 2026 theorem limits disjoint Hamiltonian copies of a closed Lagrangian, exposing how symplectic constraints survive when ordinary volume vanishes.
Mathematics
Dense sum-product growth reaches an optimal constant
A 2026 preprint studies the sharp tradeoff between addition and multiplication in dense prime fields, where the answer becomes a question of occupied space.
Mathematics
Checking the Ramsey Breakthrough Twice: Isabelle’s 2025 Formalization
Isabelle’s 2025 account of the diagonal Ramsey formalization shows how graph arguments, asymptotic bounds, and certified numerical estimates fit together.
Mathematics
Directed graph limits finally keep track of their spectra
A result revised in January 2026 connects directed graph limits with normalized eigenvalues, with careful conditions on convergence and multiplicity.
Mathematics
Covering points once changes the geometry
A SWAT 2026 paper separates hard point-covering problems from sparse tractable cases, while exact-cover results show why disjointness changes the question.
Mathematics
Smooth Euler implosions and the mathematics of coming back out
Two 2026 preprints connect smooth non-isentropic Euler implosions, conditional stability and reflected shocks, while carefully specifying the solution class.
Mathematics
Geometry Behind the Jacquet–Langlands Transfer
A September 2026 preprint relates arithmetic cohomology through Igusa stacks, proving many cases of a geometric conjecture while retaining explicit limits.
Mathematics
A Gröbner Walk Through the Space of Polynomial Orders
The 2026 GroebnerWalk.jl publication brings geometric order conversion into OSCAR, where term orders can determine the cost and usefulness of exact algebra.
Mathematics
From colliding spheres to fluid equations: what Hilbert’s sixth problem now means
A close look at the 2025 hard-sphere derivation of fluid equations, the long-time collision argument behind it, and the precise limits of the result.
Mathematics
Short Geodesics Can Also Twist: Density of Complex Systoles
A 2025 construction controls both the length and rotational holonomy of shortest hyperbolic geodesics, revealing more than short loops alone.
Mathematics
Why packing bounds need a definition of density
A 2025 preprint extends linear programming bounds to homogeneous spaces, showing why hyperbolic packing needs invariant probability and careful averaging.
Mathematics
Why hypergraph Turán densities can fill whole intervals
A May 2026 preprint finds intervals among hypergraph Turán densities. Its result depends on allowing infinite forbidden families, a crucial distinction.
Mathematics
Kakeya after the breakthrough: why the 2026 shorter proof matters
A shorter 2026 Kakeya proof clarifies how thin tubes force full fractal dimension in three dimensions, while keeping volume and dimension distinct.
Mathematics
Why Fisher information stops the homogeneous Landau equation blowing up
Fisher information connects collision geometry to global smoothness, while 2025 rough-data results show why assumptions still matter for the Landau equation.
Mathematics
Lighting a smooth shape from a lattice
Lenny Fukshansky’s 2025 theorem bounds how far lattice-based light sources must sit from a smooth convex body, with geometry and translation kept explicit.
Mathematics
What a Kernel Fix Teaches Us About Mathematical Certainty
Lean’s July 2026 kernel fix illustrates the distinct roles of logical rules, proof terms, axiom audits, and independent checking in formal mathematics.
Mathematics
Rank or tolerance? What librla changes about matrix compression
The 2026 librla preprint puts rank and tolerance choices into a common factorization library. Understanding those choices takes more than counting stored values.