Study shows (p,q)-cables of non-trivial knots are not thin.
problem Proving (p,q)-cables of non-trivial knots are not thin. method Bordered Floer theory of Lipshitz-Ozsváth-Thurston and Zemke's theorem.
result Proves (p,q)-cables of non-trivial knots are not thin. Defines a new Upsilon torsion function for knot Floer homology.
problem Obtaining constraints on knot cobordisms.
method Defines a one-parameter family of Heegaard Floer torsion invariants.
result Provides new obstructions related to the Gordian distance between knots.
New examples show clasp numbers can be zero yet four-genus can be arbitrarily large.
problem Understanding the relationship between clasp numbers and four-genus for knots.
method Constructing knots with specific clasp numbers and four-genus values.
result Examples of knots with zero clasp numbers but arbitrarily large four-genus.
Study ribbon homology concordances using link Floer homology.
problem Understanding ribbon homology concordances and their effects on link Floer homology.
method Combining results from Daemi, Lidman, Vela-Vick, Wong, and Zemke, using link Floer homology and torsion submodules.
result Ribbon homology concordances induce split injections on HFL−. Paper calculates homology and intersection form of trisected 4-manifolds with boundary.
problem Calculating homology and intersection form of trisected 4-manifolds with boundary.
method Uses relative trisection diagrams to calculate homology and intersection form.
result Describes a representative of the second Stiefel-Whitney class using relative trisection diagrams.
Heegaard Floer homology study confirms composition maps match up to homotopy.
problem Verifying consistency of composition maps in Heegaard Floer homology.
method Using Auroux and Zemke's results, proving agreement up to homotopy.
result Proves consistency of composition maps in Heegaard Floer homology.
Study shows infinite-rank summand in homology cobordism group.
problem Identifying infinite-rank summands in homology cobordism groups.
method Introduced an algebraic variant of the involutive Heegaard Floer package.
result Demonstrated an infinite-rank summand in the homology cobordism group.
Tangle replacements help in understanding knot properties.
problem Understanding the minimum number of tangle replacements to unknot a knot.
method Using knot Floer homology torsion to find lower bounds.
result Torsion order gives a lower bound on the minimum number of tangle replacements.
New invariants detect corks obstructing homology ball extensions.
problem Detecting corks obstructing homology ball extensions.
method Adapting Heegaard Floer homology formalism for local equivalence, using monotonicity and surgery.
result Established new families of corks and proved various examples are strongly non-extendable.
Study concordance of alternating torus knots to L-space knots.
problem When are linear combinations of alternating torus knots concordant to L-space knots?
method Proved Allen's conjecture for alternating torus knots and established a necessary condition.
result Linear combinations of alternating torus knots are concordant to L-space knots if and only if they are a single torus knot.
Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.
problem Proving a conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres.
method Using Zemke's isomorphism between lattice and Heegaard Floer homology, the paper proves Némethi's conjecture.
result The conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres is proven.
New cobordism maps for instanton knot homology help compute surgeries on knots.
problem Computing surgeries on knots using instanton knot homology.
method Construct cobordism maps for instanton knot homology.
result Compute the framed instanton Floer homology of certain knots.
New insights into knot fusion numbers via cabling.
problem Understanding fusion numbers of ribbon knots and their behavior under cabling.
method Utilizing knot Floer homology and cabling formulas to analyze fusion numbers.
result The fusion number and strong homotopy fusion number of (p,1)-cable knots are preserved.
The paper extends Khovanov homology results to sl(n) homologies and provides bounds on knot properties.
problem Extending Khovanov homology results to sl(n) homologies and knot properties. method Spectral sequence arguments and Levine-Zemke's ribbon concordance obstruction.
result Bounds on the alternation number and Turaev genus of knots.
Study non-orientable link cobordisms using Floer homologies to prove inequalities.
problem Prove inequalities involving Euler characteristic and local maxima in non-orientable cobordisms.
method Use unoriented instanton and knot Floer homology to introduce unoriented versions of band unknotting number and refined cobordism distance.
result Show that the difference between unoriented refined cobordism distance of a knot from the unknot and non-orientable slice genus can be arbitrarily large.
Proves module structure on odd Khovanov homology and applies to ribbon 2-knots.
problem Proving module structure on odd Khovanov homology.
method Geometric description of module structure and link cobordism.
result Combinatorial proof of odd invariant for ribbon 2-knots.
Study shows how Khovanov homology behaves for split links and cobordisms.
problem Understanding Khovanov homology's sensitivity to cobordisms between split links.
method Proved that Khovanov homology map is determined by individual components of cobordism, not linking.
result Khovanov homology detects non-split links from split links.
New conditions prevent non-trivial relations in local equivalence group.
problem Understanding non-trivial relations in homology cobordism groups.
method Filtered instanton Floer homology and rs-invariants. result Certain conditions prevent non-trivial relations in the local equivalence group.
This paper shows hyperbolic knots can have arbitrarily large torsion in knot Floer homology.
problem Understanding the torsion order in knot Floer homology for hyperbolic knots.
method Unified approach using Upsilon torsion function.
result Arbitrarily large torsion orders realized by hyperbolic knots, most of which are twisted torus knots.
Projective naturality proved for Heegaard Floer homology.
problem Proving naturality of Heegaard Floer invariants under diffeomorphisms.
method Showed Heegaard Floer invariants yield functors to transitive systems in a projectivized category of Z[U]-modules. result Established Heegaard Floer invariants as functors to transitive systems in a projectivized category.
Classifies L-space surgeries on all two-bridge links
problem Classifying L-space surgeries on two-bridge links method Introduces a sufficient diagrammatic condition for links in S3 to be persistently foliar, defines a simplified model for Heegaard Floer homology, and uses Turaev torsions for computations result Determines L-space surgeries in the case of generalised L-space links 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.
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.
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.
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.
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.
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.
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.
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 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 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.
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