This paper solves minimal surface equations near Hardt-Simon foliations.
problem Minimal surfaces near Hardt-Simon foliations.
method Uses gluing methods to construct minimal surfaces.
result Constructs minimal surfaces over Hardt-Simon surfaces and near quadratic cones.
Paper constructs flows converging to cones and foliations.
problem Understanding mean curvature flow convergence to cones and foliations.
method Constructs a family of mean curvature flows converging to cones and foliations under specific conditions.
result Flow converges to area minimizing, strictly stable hypercone and Hardt-Simon foliation of the cone.
Partial boundary regularity for area-minimizing currents at tangential boundary points.
problem Boundary regularity of co-dimension one area-minimizing currents.
method Proof closely follows Hardt and Simon's boundary regularity result.
result Partial regularity of tangent cones, uniqueness of tangent cone.
New special Lagrangian submanifolds with multiple conical singularities found.
problem Existence of special Lagrangian submanifolds with multiple conical singularities.
method Extending Caffarelli-Hardt-Simon perturbation argument to special Lagrangian setting and proving a bridge principle.
result Existence of conically singular special Lagrangian submanifolds with prescribed regular tangent cones.
The study examines singularities in flows with curvature bounds and identifies unique tangent flows.
problem Analyzing singularities in mean curvature flows with curvature bounds.
method Examines tangent flows and uses stationary and area-minimizing cones to identify unique flows.
result For flows with H∈L∞Llocp, the tangent flow is unique when p=∞ and C is a regular cone. Proves a principle for one-phase Bernoulli problem minimizers.
problem One-phase Bernoulli problem minimizers.
method Strong maximum principle, Alt-Caffarelli functional, Hardt-Simon-type foliation.
result Constructs a foliation for global minimizers.
Low-entropy surfaces can be flowed into spheres and cylinders.
problem Proving mean curvature flow for low-entropy hypersurfaces.
method Low-entropy density drop argument and recent work on hypersurfaces.
result Closed hypersurfaces with entropy ≤ 2 can be flowed into spherical and cylindrical shapes.
Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
Extends a Liouville theorem for stable minimal hypersurfaces.
problem Stable minimal hypersurfaces in cylindrical cones.
method Analyzes the density and foliation of minimal hypersurfaces.
result Shows that stable minimal hypersurfaces are cylindrical.
Smooth approximations near singularities of constant mean curvature surfaces are found.
problem Finding smooth approximations for constant mean curvature surfaces near singular points.
method Proving the existence of sequences of smooth CMC hypersurfaces converging to a given one in a ball centered at the singularity.
result Smooth approximations exist in a ball centered at the singularity of a CMC hypersurface.
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on Sn+1 precludes linearly stable tangent cones for area-minimizing boundaries. Paper proves minimal surfaces near quadratic cones have specific smooth structure.
problem Characterize minimal surfaces near quadratic cones.
method Analyzes n-varifolds in the unit ball close to a minimizing quadratic cone. result Singularities modeled on these cones determine the local structure of nearby minimal surfaces.
Machine learning improves searching for polynomial proofs.
problem Automatically searching for proofs of polynomial inequalities.
method Deep reinforcement learning guiding inference rules in semi-algebraic proof systems.
result Reduces the size of linear programs by several orders of magnitude.
Several proofs of Fáry--Milnor theorem are presented.
problem Fáry--Milnor theorem
method Sketches several proofs
result Proofs of Fáry--Milnor theorem
New proof of isoperimetric inequality using Steiner's formula.
problem Proving the isoperimetric inequality in the plane.
method Direct proof using Steiner's formula, bypassing domain existence.
result Establishes the isoperimetric inequality directly.
Machine learning automates proof generation in Coq proof assistant.
problem Automating proof construction in proof assistants.
method Developed ASTactic, a deep learning model generating tactics as ASTs.
result ASTactic can generate effective tactics and prove new theorems.
New proof for Gromov's theorem on almost flat manifolds.
problem Proving Gromov's theorem on almost flat manifolds.
method Inductive proof on dimension.
result New proof for Gromov's theorem.
Corrects a flawed proof of the Kropholler Conjecture.
problem Kropholler Conjecture in relative group theory.
method Explains the flaws and proposes possible ways to correct the proof.
result Provides a clearer understanding of the conjecture's proof difficulties.
Simplified proof of gluing formula for analytic torsion forms.
problem Proving the gluing formula for analytic torsion forms.
method Extending prior work to the family case, presenting a new proof.
result Simplified proof of the gluing formula.
A simplified proof of the Alexander-Conway polynomial exists.
problem Existence of the Alexander-Conway polynomial for links in 3D space.
method Presented an accurate detailed exposition of the proof.
result Existence of the Alexander-Conway polynomial proved.
Computer-generated proofs led to a mathematical result.
problem Discovering a mathematical result through computer-generated proofs.
method Combining computer-generated, human-readable proofs with mathematical abstraction.
result Abstracted lemma leading to an interesting mathematical result.
We give a direct proof that the Freed-Lott differential analytic index is well defined and a condensed proof of the differential Grothendieck-Riemann-Roch theorem. As a byproduct we also obtain a direct proof that the R/Z analytic index is well defined and a condensed proof of the R/Z Grothendieck-Riemann-Roch theorem.
Simplified proof for a theorem about graphs.
problem Proving a generalized Conway--Gordon--Sachs theorem for complete graphs.
method Provided a shorter proof over integers.
result A simpler proof of the generalized theorem.
We offer a simplified proof for Expected Shortfall's dual representation.
problem The dual representation of Expected Shortfall.
method Basic properties of quantile functions.
result New proof of Expected Shortfall's subadditivity.
Proof given for SGD convergence in a concise manner.
problem Convergence of Stochastic Gradient Descent (SGD)
method Self-contained proof
result SGD convergence proven
The Goldman-Parker Conjecture classifies the complex hyperbolic C-reflection ideal triangle groups up to discreteness. We proved the Goldman-Parker Conjecture in [Ann. of Math. 153 (2001) 533--598] using a rigorous computer-assisted proof. In this paper we give a new and improved proof of the Goldman-Parker Conjecture.…
Transformers improve logical reasoning on longer proofs but struggle with length.
problem Understanding systematic generalization in neural proof generation.
method Soft theorem-proving using Transformer models, evaluating logical consistency and inference accuracy.
result Transformers improve generalization with longer proofs but have difficulty with length.
Paper provides a rigorous proof of the index theorem for economists.
problem Lack of a rigorous proof in textbooks for economists.
method Constructs a readable proof of the index theorem under specific assumptions.
result Provides a gap-free proof of the index theorem.
Simplified proof of Gaussian concentration inequality using covariance.
problem Gaussian concentration inequality proof
method Covariance representation based on characteristic functions
result Elementary proof of Gaussian concentration inequality
Formally proves machine learning for simple classifiers.
problem Proving PAC learnability for decision stumps.
method Formal proof in Lean, separating deterministic and probabilistic proofs.
result Formal proof of PAC learnability for decision stumps.
Simple proof for special surface classification.
problem Classifying surfaces with specific curvature properties.
method Elementary proof for parallel mean curvature surfaces.
result A lemma is proven for surfaces in complex space forms.
Simplified proof of a famous geometry result for students.
problem Mostow Rigidity in geometry and topology
method Self-contained, accessible proof for grad students
result Clean and analytically light proof accessible to students
Three proofs show all 3D shapes are parallelizable without complex tools.
problem Proving all 3D shapes are parallelizable without advanced tools.
method Minimal proofs using no spin structures or Stiefel-Whitney classes.
result Three proofs show all 3D shapes are parallelizable.
A proof of Gauss--Bonnet theorem avoids triangulations using complex structures.
problem Proving the Gauss--Bonnet theorem without triangulations.
method Using complex structures to provide an intrinsic proof.
result A proof of the Gauss--Bonnet theorem achieved without triangulations.
We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.
GamePad uses machine learning to help prove theorems in Coq proof assistant.
problem Automating theorem proving in interactive proof assistants.
method Synthesizes proofs and trains models for tactic prediction and position evaluation.
result Trained models predict proof steps and tactics in Coq theorem proving.
First geometric proof of the flyping theorem.
problem Proving Tait's flyping conjecture.
method Geometric proof using Greene's characterization, Menasco's crossing ball structures, and isotopy/re-plumbing moves.
result First entirely geometric proof of Menasco-Thistlethwaite's flyping theorem.
We provide yet another proof of the existence of calibrated forecasters; it has two merits. First, it is valid for an arbitrary finite number of outcomes. Second, it is short and simple and it follows from a direct application of Blackwell's approachability theorem to carefully chosen vector-valued payoff function and …
New topological proof for a theorem about orbits in a specific group.
problem Characterizing orbits of dense subgroups in a specific group.
method Topological approach, using dynamics of unipotent flows.
result Orbits are either finite or dense, proving a theorem.
Two proofs show that removing a loop from a plane circuit splits the plane.
problem Proving the Weak Jordan Theorem about plane circuits.
method Detailed presentation of Thomassen's and Filippov's proofs.
result The complement of any loop in a plane circuit is disconnected.
Notes a flaw in a proof about embedding graphs.
problem A flaw in proving Sachs' conjecture about graph embeddings.
method Analyzing Stanfield's proof for gaps.
result Identifies a significant error in the proof.
Combinatorial proof of grid homology properties.
problem Properties of double-point enhanced grid homology.
method Purely combinatorial proof, extended to Z coefficients. result Skein exact sequence obeyed by grid homology.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
New proof for higher rank subvarieties in genus three.
problem Classifying higher rank invariant subvarieties in genus three.
method Uses recent techniques developed by Apisa and Wright.
result Short and simplified proof of classification.
A simplified proof for satellite knot signatures.
problem Proving Litherland's formula for satellite knots.
method Uses linear algebra and basic knot theory.
result A new, elementary proof of the signature formula.
A short proof for curve lengths on hyperbolic surfaces.
problem Proving a theorem about curve lengths on hyperbolic surfaces.
method Presented a concise proof for the theorem.
result A pair of curves has length at least half the perimeter of a specific polygon.
Improved the convergence proof of ADAM-Optimizer.
problem Incorrect convergence proof of ADAM-Optimizer.
method Provided an improved convergence proof.
result Corrected the convergence proof of ADAM-Optimizer.