Algebraic Geometry
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Explicit and asymptotically good constructions of Algebraic Geometry codes in the sum-rank metric
Announce Type: new Abstract: Algebraic Geometry (AG) codes (i.e. linear codes from algebraic function fields) in the Hamming metric were proposed by Goppa in 1980 and have been intensively studied ever since. Linearized Algebraic Geometry codes, the analogue of AG codes in the sum-rank metric, were instead introduced more recently [9], using quotients of the ring of Ore polynomials with coefficients in an algebraic function field.
Formalizing multi-graded Brenner-Schr\"oer Proj schemes and dilatations of rings in Lean4
Computer Science > Logic in Computer Science [Submitted on 31 May 2026] Title:Formalizing multi-graded Brenner-Schröer Proj schemes and dilatations of rings in Lean4 View PDFAbstract:We present a detailed formalization in Lean4 of some multigraded algebraic geometry constructions, focusing on the Brenner--Schröer Proj construction and algebraic dilatations of rings. References & Citations Loading...
A golden age of maths is dawning and mathematicians are freaking out
I am attempting to solve a mathematical conundrum that has stumped many of humanity’s greatest thinkers. I have zero mathematical training, apart from a distant undergraduate physics degree, which should put my odds of success at slim to none. But I also have a trick up my sleeve – a kind of mathematical genie that can conjure arcane secrets seemingly out of thin air.
Advancing Mathematics Research with AI-Driven Formal Proof Search
arXiv:2605.22763v2 Announce Type: replace Abstract: Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems.
A New Visual Approach to Pendulum Period Determination
arXiv:2408.00201v2 Announce Type: replace Abstract: The period of oscillation of a simple pendulum ($T = 2\pi\sqrt{l/g}$) is a familiar formula to most first-year physics students. However, deriving this expression from first principles requires linearizing the equation of motion under the small-angle approximation and solving the resulting differential equation. From our point of view, this method may seem obscure to students in the early stages of learning calculus and lacking in physical...
Quasi-symmetric nets: A constructive approach to the equimodular elliptic type of Kokotsakis polyhedra
arXiv:2511.19376v2 Announce Type: replace-cross Abstract: A Kokotsakis polyhedron is a polyhedral mesh in three-dimensional Euclidean space formed by a central n-gonal face (the base), n quadrilateral faces each sharing one edge with the base, and n triangular faces inserted between every two adjacent quadrilaterals; it is called flexible if it admits a continuous deformation that preserves the rigidity of every face. This work investigates flexible Kokotsakis polyhedra with a quadrangular...
Native Hierarchical and Compositional Representations with Subspace Embeddings
arXiv:2508.16687v2 Announce Type: replace Abstract: Traditional embeddings represent datapoints as vectors, which makes similarity easy to compute but limits how well they capture hierarchies and compositionality. We propose a fundamentally different approach: representing concepts as linear subspaces. By spanning multiple dimensions, subspaces can model broader concepts with higher-dimensional regions and nest more specific concepts within them.
Exponential thermalisation of viscous fluids on negatively curved manifolds
arXiv:2606.02286v1 Announce Type: cross Abstract: The deterministic incompressible Navier-Stokes equations are physically incomplete: any viscous fluid at finite temperature must exhibit thermal fluctuations whose form is dictated by the fluctuation-dissipation relation. We formulate the stochastic Navier-Stokes equations with the kinematically selected deformation Laplacian on compact Riemannian manifolds with strictly negative Ricci curvature. The fluctuation-dissipation relation, derived...
Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning
Announce Type: new Abstract: Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries.
Global exponential stability for the three-dimensional Navier-Stokes equations on hyperbolic space
Announce Type: replace-cross Abstract: We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space $\HH^3$ admit a unique global mild solution for sufficiently small initial data in $L^3(\HH^3)$, and that this solution decays exponentially to zero. The exponential decay rate is $\mu\lambda_\Def^{(3)}$, where $\mu$ is the dynamic viscosity and $\lambda_\Def^{(3)} = 26/9$ is the effective spectral gap of the deformation...