Enhances Pontryagin-Thom theorem for manifold maps.
problem Identifying map spaces with moduli spaces of submanifolds.
method Space-level enhancement of Pontryagin-Thom theorem.
result Maps from manifolds to Thom spaces identified with moduli spaces of submanifolds.
Revisits Pontryagin's proof of stable stems 0, 1, and 2.
problem Computing stable stems in dimensions 0, 1, and 2.
method Introduction of homotopy theory concepts, framed cobordism, and Pontryagin-Thom construction.
result Corrects the result of \(\pi_2^S\) from \cite{Pont2} and computes stable stems in dimensions 0, 1, and 2.
We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.
Study cobordisms of nested manifolds and their invariants.
problem Understanding cobordisms of nested manifolds and their invariants.
method Identify a nested analog of the Pontryagin-Thom construction and find spaces homotopy equivalent to nested Pontryagin-Thom spaces.
result Discover nested cobordism invariants and provide an alternative proof of Wall's splitting result.
Maps self-duality in little disks operad to framed manifolds.
problem Self-duality of little disks operad.
method Configuration space level Pontryagin--Thom constructions.
result Existence of compatible self-duality map for framed manifolds.
The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and t…
Geometric model for Hodge filtered complex cobordism constructed.
problem Constructing a geometric model for Hodge filtered complex cobordism.
method Refinement of Pontryagin-Thom construction to create an explicit isomorphism.
result Explicit isomorphism between geometric and abstract models for complex manifolds.
We will present proofs for two conjectures stated in arXiv:1808.08073. The first one is that for an arbitrary manifold W, the homotopy classes of proper maps W×Rn→Rk+n stabilise as n→∞, and the second one is that in a stable range there is a Pontryagin--Thom type bijection for …
We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in En-algebra.
This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the fundamental class. We present a short and direct proof of the followin…
We use the generalized Pontryagin-Thom construction to analyze the effect of attaching a bypass on the homotopy class of the contact structure. In particular, given a 3-dimensional contact manifold with convex boundary, we show that the bypass triangle attachment changes the homotopy class of the contact structure rela…
Survey on manifolds with positive scalar curvature, focusing on obstructions and constructions.
problem Which manifolds admit positive scalar curvature metrics?
method Topological obstructions and geometric constructions (surgery/bordism theorem)
result The answer depends on the bordism class of the manifold, with complete solutions for simply connected manifolds.
In this paper we classify the homotopy classes of proper maps E→Rk, where E is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps Rn→Rk. We find a stability range of such maps. We conclude with some remarks…
We provide an alternative proof that Koschorke's κ-invariant is injective on the set of link homotopy classes of n-component homotopy Brunnian links BLM(n). The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techni…
Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homot…
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed norm…
Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.
problem Chas-Sullivan products on homology of loop spaces.
method Morse theory with differential graded coefficients, functorial properties, K{ü}nneth formula, Pontryagin-Thom construction.
result Chain-level description of Chas-Sullivan products.
Introduces quasi-holomorphic maps and their properties.
problem Understanding singularities and stratifications in non-complex manifolds.
method Pontryagin--Thom construction, cobordism groups, Thom polynomials.
result Thom polynomials determine cohomology classes of quasi-holomorphic maps.
Generalizes Pontryagin's construction for proper maps in stable dimensions.
problem Mapping submanifolds to homotopy classes of proper maps.
method Introduces a new bijection between cobordism sets and homotopy classes for proper maps.
result Provides a bijection for cobordism sets of submanifolds embedded in WimesRn. 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 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.
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.
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.
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.
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.
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.
Several proofs of Fáry--Milnor theorem are presented.
problem Fáry--Milnor theorem
method Sketches several proofs
result Proofs of Fáry--Milnor theorem
Reidemeister's theorem proved using smooth functions and transversality.
problem Proving Reidemeister's theorem
method Using smooth functions and transversality
result Reidemeister's theorem proved
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.
This note explores comparison geometry concepts and theorems.
problem Exploring various comparison theorems in geometry.
method Analyzes Rauch and Toponogov theorems and their applications.
result Introduction of Gromov-Hausdorff convergence and Alexandrov Spaces.
Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.
problem Proving variations of the Tait-Kneser theorem for different conics.
method Using Lorentzian geometry to prove the theorem and its variations.
result Proof of the theorem and its variations concerning different conics.