New solutions found for a free boundary problem using refined methods.
problem Free boundary problem with two component boundaries.
method Refined Lyapunov-Schmidt reduction method and variational arguments.
result Global minimizers of energy functional found in dimension 8.
The paper proves unique blow-down for critical points of a Yang-Mills-Higgs functional.
problem Proving uniqueness of blow-down for critical points of a Yang-Mills-Higgs functional.
method Using an Allard-type improvement of flatness to establish co-dimension-two analogue of Savin's theorem.
result Entire critical points have unique blow-down, two-dimensional in ambient dimensions 2-4 or any dimension assuming local minimizer.
New, shorter proofs for varifolds and flows with improved decay of flatness.
problem Proving regularity theorems for varifolds and flows with bounded first variation and forcing.
method Decay of flatness via weighted monotonicity formulas and viscosity approach.
result Improved proofs with decay of flatness and characterization of blow-ups.
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
Proves regularity for Brakke flow near stationary half-plane.
problem Regularity of Brakke flow near stationary half-planes.
method Viscosity techniques from Savin applied to Brakke's flow.
result Proves C1,α regularity for flows close to stationary half-planes. Graphical theory of binary Hamiltonian forms and their values.
problem Understanding values of binary Hamiltonian forms.
method Graphical theory, maximal orders, Busemann distances, equivariant cellulation.
result Combinatorial description of binary Hamiltonian form values.
We prove the existence of the analog of Lawson's minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the ca…
Proves curvature estimates for singular submanifolds with bounded mean curvature.
problem Estimating curvature for singular submanifolds of bounded mean curvature.
method Introducing a curvature notion for arbitrary closed sets, proving ABP-type estimates, and deriving weak-Harnack-type estimates.
result Generalizes results for viscosity solutions of the minimal surface equation.
Study on minimizing singular capillary cones with stability and instability results.
problem Minimizing singular capillary cones with free boundary.
method Stability criterion à la Jerison-Savin, Simons-type inequality for convex, homogeneous, symmetric functions of principal curvatures, boundary condition specific to capillary setting.
result Minimizing cones with non-sign-changing mean curvature are flat in dimensions up to 4, and non-trivial axially symmetric cones are unstable in dimensions up to 6.
Study confirms C1 regularity for convex functionals with bounded degeneracy set.
problem Confirming C1 regularity for minimizers of convex functionals with small degeneracy set. method Building on previous work, confirms C1 regularity when D2F is positive and bounded away from finitely many points. Constructs a counterexample in R4 where F is strictly convex but D2F degenerates on a Simons cone intersection. result Confirms C1 regularity for minimizers of convex functionals with small degeneracy set. Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.
problem Understanding phase transitions through entire solutions of the Allen-Cahn equation.
method Proving minimality of the zero level set with respect to a perimeter functional with density and showing zero mean curvature.
result The zero level set of entire solutions of the Allen-Cahn equation has zero mean curvature and is minimal.
New method reduces errors in pricing and sensitivities for discontinuous payoffs.
problem Errors in pricing and sensitivities for discontinuous payoffs in digital and barrier options.
method Alternative methods for estimating sensitivities, including likelihood ratio and hybrid methods.
result New methods substantially reduce test errors in prices and sensitivities.
Three counterexamples show SLCE's limitations in 2D and 3D.
problem Demonstrate limitations of the special Lagrangian curvature equation.
method Construct counterexamples in 2D and 3D, showing non-C1 solutions and unbounded curvature. result SLCE's a priori estimates are sharp and require convexity and critical phase.
The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus appl…
Deep learning accelerates Heston model calibration.
problem Calibrating stochastic volatility models is computationally expensive.
method Differential Machine Learning (DML) technique to train neural networks on differentials of features and labels.
result DML reduces Heston model calibration time significantly.
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.
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.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
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.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
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.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
New measure proves Poncelet-type theorems.
problem Proving Poncelet-type theorems.
method Introducing a new invariant measure on the circle.
result Simple proof of Emch closing theorem.
New theorem for doodles on sphere, similar to Markov's.
problem Understanding doodles on a sphere.
method Description of twins with equivalent closures.
result Analogous to Markov's theorem for doodles.
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.
Analyzes Saito vanishing theorem using L2 methods.
problem Proving the Saito vanishing theorem.
method Uses L2-methods to prove the theorem. result Analytic proof of the Saito vanishing theorem.
Extends calculus theorem to higher dimensions.
problem Calculus theorem limitations in higher dimensions.
method Type θ Stokes' theorem for type θ k-chains. result Extends fundamental theorem of calculus.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
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 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.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
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 explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
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.
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
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.
Abstracts a theorem for non-smooth maps in infinite dimensions.
problem Generalizing inverse mapping theorem for non-smooth maps.
method Introduces property A and applies it to non-smooth maps.
result Generalized inverse mapping theorems for non-smooth maps.
A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps.
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
Paper proves compactness and rigidity of λ-surfaces in 3D.
problem Compactness and rigidity of λ-surfaces in R3. method Developed a compactness theorem for λ-surfaces with uniform λ, genus, and area growth. result Proved a rigidity theorem for convex λ-surfaces. Notes on a theorem with broad applications in dynamical systems.
problem The Oseledets Multiplicative Ergodic Theorem and its applications.
method Introduction and lectures on the theorem and its generalizations.
result Introduction to the theorem and its applications.
The paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
The paper proves sphere theorems for submanifolds in Kähler manifolds.
problem Sphere theorems for submanifolds in Kähler manifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler manifolds, especially in complex space forms.
result Proves sphere theorems for submanifolds in Kähler manifolds.
Several proofs of Fáry--Milnor theorem are presented.
problem Fáry--Milnor theorem
method Sketches several proofs
result Proofs of Fáry--Milnor theorem