INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.
problem Evaluating theorem proving agents' ability to generalize to unseen theorems.
method INT benchmark based on a theorem generation and proof procedure with adjustable knobs for measuring 6 types of generalization.
result MCTS can help agents prove new theorems.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Proves an analytic Bertini theorem, generalizing previous work.
problem Generalizing previous results in algebraic geometry.
method Analytic Bertini theorem proof.
result Generalizes previous results in algebraic geometry.
Proves a theorem for 3D Poincaré duality pairs.
problem No specific problem stated; focuses on proving a theorem.
method Analogous to Johannson's theorem for PD3 pairs.
result Proves a theorem for 3D Poincaré duality pairs.
W. Simon proved a conformal positive mass theorem, which was used to prove uniqueness of black holes later. In this note, we will generalize Simon's conformal positive mass theorem in two directions. First we will consider spacetime version of conformal positive mass theorems on asymptotically flat initial data set. Ne…
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Proves Gannon-Lee theorem for C1 spacetimes.
problem Classical singularity theorems for C1 spacetimes. method Proves theorem for C1 spacetimes, shows geodesic properties. result Gannon-Lee theorem holds for C1 spacetimes. Proves curvature comparison theorem for manifolds with conical singularities.
problem Comparing scalar mean curvature of manifolds with conical singularities.
method Uses Dirac operator and index theory to prove curvature comparison theorem.
result Proves curvature comparison theorem without knowing the index of the twisted Dirac operator.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
Proves spacetime positive mass theorem in all dimensions.
problem Proving the spacetime positive mass theorem in arbitrary dimensions.
method Using Brendle--Wang's Riemannian positive mass theorem approach.
result Proves the spacetime positive mass theorem for all dimensions.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
The paper proves symplectic neighbourhood theorems for stratified subspaces.
problem Finding symplectic neighbourhoods of stratified subspaces.
method Analogy with Weinstein's neighbourhood theorem, strong version of Moser's trick, and tubular neighbourhood theorem.
result Generalization of existing constructions for exotic Lagrangians.
Proves positive mass theorems for specific types of curved spaces.
problem Analyzing mass in curved spaces with boundaries.
method Proves positive mass theorems for specific types of curved spaces with boundaries.
result Establishes conditions under which mass is positive in these spaces.
In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportuni…
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
In this paper, we prove an equivariant Kastler-Kalau-Walze type theorem for spin manifolds without boundary. For 6 dimensional spin manifolds with boundary, we also give an equivariant Kastler-Kalau-Walze type theorem. Then we generalize this theorem to the general n dimensional manifold. An equivariant Kastler-Kal…
The paper defines positivity for singular metrics on vector bundles and proves related theorems.
problem Positivity of singular Hermitian metrics for holomorphic vector bundles.
method The method of Berndtsson and Lempert, along with a Berndtsson-type positivity theorem for holomorphic vector bundles.
result Sharp L2 extension theorem for holomorphic vector bundles. Proves a generalized table theorem for odd Euler characteristic surfaces.
problem Proving a generalized table theorem for surfaces with odd Euler characteristic.
method Using the square peg problem for smooth curves, the result is generalized to real valued functions on Riemannian surfaces with odd Euler characteristic.
result Proves the table conjecture for even functions on the two sphere.
The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.
The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.
problem Extending the Bakry-Émery-Ricci tensor and proving comparison theorems.
method Generalizations of the drifted Laplacian and Bakry-Émery-Ricci tensor, mean curvature comparison theorem, Myers-type theorem, Cheeger-Gromoll splitting theorem.
result Proved a version of the mean curvature comparison theorem and its consequences.
Paper proves a Cohen-Dimca-Orlik type theorem for Z-local systems of hyperplane arrangements.
problem Proving a Cohen-Dimca-Orlik type theorem for Z-local systems. method Analyzing local system cohomology groups of hyperplane arrangements complements.
result Proves a Cohen-Dimca-Orlik type theorem for Z-local systems. The L-move for classical braids extends naturally to trivalent braids. We follow the L-move approach to the Markov Theorem, to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this L-Move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we …
Lean Copilot uses LLMs to assist theorem proving in Lean, improving efficiency and automation.
problem Challenges in using existing neural theorem provers to prove novel theorems autonomously.
method Introduces Lean Copilot, a framework for integrating LLMs into Lean's theorem proving process.
result Lean Copilot automates 74.2% of proof steps on average, significantly improving over existing methods.
In this paper we first give a one-move version of Markov's braid theorem for knot isotopy in S3 that sharpens the classical theorem. Then a relative version of Markov's theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov's theorem for knot isotopy in knot complements. Finally …
We prove Minding's Theorem for C2-immersions with constant negative Gauss curvature. As a Corollary we also prove Minding's Theorem for C1M-immersions in the sense of \cite{DS}.
We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Com…
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
Study on Kähler Finsler manifolds with curvature bounds, proving theorems.
problem Understanding Kähler Finsler manifolds with curvature constraints.
method Analyzing partial parallelism of complex structure, proving theorems.
result Generalized comparison theorem for positively curved Kähler Finsler manifolds.
Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.
problem Division Theorem in Skoda's context
method Degeneration approach inspired by B. Berndtsson and L. Lempert's L2 extension theorem result Simplified and extended proof of L2 extension theorem Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves a theorem for complex flat vector bundles using differential forms.
problem No specific problem stated; focuses on proving a theorem.
method Uses differential forms to prove the Riemann-Roch-Grothendieck theorem.
result Proves the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles.
Proves Hamilton's theorem using mean curvature flow.
problem Compactness of pinched hypersurfaces with bounded curvature.
method Mean curvature flow to prove Hamilton's theorem.
result Rigorous proof of Hamilton's theorem.
Proves positive mass theorem for non-spin weighted manifolds.
problem Proving the positive mass theorem for non-spin weighted manifolds.
method Establishing density theorem and generalizing Geroch conjecture.
result Proves positive weighted mass theorem for non-spin weighted manifolds.
Proves density and mass theorems for specific initial data sets.
problem Initial data sets with boundary in spacetime.
method Harmonic asymptotics and dominant energy condition.
result Spacetime positive mass theorem for initial data sets with apparent horizon boundary.
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a K-theoretic formula for an index-type invariant. result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
The study proves fixed-point theorems for groups acting on CAT(0) spaces.
problem Finding fixed points for groups acting on CAT(0) spaces.
method Bootstrapping technique with Helly-type theorems to prove intersections of fixed-point sets.
result Lower bounds on the smallest dimension for groups to act on CAT(0) spaces without global fixed points.
Proves better rigidity theorems for special solitons.
problem Understanding rigidity properties of specific solitons.
method Refined point-wise estimates for mean curvature.
result Stronger rigidity results for Lagrangian and symplectic translating solitons.
We prove a Z-set unknotting theorem for Nobeling spaces. This generalizes a result obtained by S. Ageev for a restricted class of Z-sets. The theorem is proved for a certain model of Nobeling spaces.
Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.
problem Proving convergence of Yang-Mills flow on ALE gravitational instantons.
method Noncompact version of the 'parabolic gap theorem'.
result Sharp convergence theorem for Yang-Mills flow on ALE 4-manifolds.
Proves an equivariant version of index theorem for geometric families.
problem Index theorem for geometric families with group action.
method Apply equivariance --> families principle to Clifford module bundles.
result Equivariant version of Bismut's families index theorem.
We first proved a compactness theorem of the Kähler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed by Chen-Li-Wang to prove the small energy theorems of t…
The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
Proves positive mass theorem on conical manifolds with small angles.
problem Proving the positive mass theorem on conical manifolds with small cone angles.
method Analyzes conical manifolds with small cone angles, assuming spin structure and locally conformal flatness.
result Proves the positive mass theorem under specified conditions.
The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with S1-actions, using a ring of formal power series. result Found an appropriate form of the index theorem for loop spaces.
Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.
problem Vanishing theorems for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.
method Derived blow-ups, intrinsic blow-up theory, Kiem-Li-Savvas blow-up theory, virtual localization theorem, desingularization theorem, resolution of diagonal.
result Generalized vanishing theorem for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.