Simplified proof of topological Tverberg conjecture for prime powers, with counterexamples for others.
problem Proving the topological Tverberg conjecture for all integers r, d > 1.
method Combining combinatorics, algebra, and topology to provide a simplified proof.
result The conjecture holds for prime powers r but not for others.
The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in th…
We found stronger counterexamples to the topological Tverberg conjecture.
problem Proving the topological Tverberg conjecture for non-prime power values of r and large enough d.
method Using higher-dimensional counterexamples and generalizations of Mabillard-Wagner and Özaydin theorems.
result We produced counterexamples for almost r-embeddings in higher dimensions.
New method proves topological Tverberg problem for all q, not just primes.
problem Establish sufficient conditions for simplicial complexes to map q points to the same point.
method General method that yields results beyond prime powers.
result Proves previously conjectured upper bounds for topological Tverberg problem for all q.
Motivated by topological Tverberg-type problems and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without triple, quadruple, or, more generally, r-fold points. Specifically, we are interested in maps f from K to R^d…
The history and proofs of Topological Tverberg Theorem are discussed.
problem Understanding the proofs and properties of Topological Tverberg Theorem.
method Historical overview and detailed cell structure analysis of classifying space.
result Clarification of the cell structure of the classifying space K(Sr,1). A new Tverberg theorem with negative coefficients.
problem Proving a Tverberg type theorem with negative coefficients.
method General position set and partition into r sets with affine combination constraints.
result The unique point in the intersection of affine hulls can be expressed as a weighted sum with exactly k negative weights.
Short proofs for complex Tverberg theorems using prime powers.
problem Proving Tverberg-type theorems for cell complexes.
method Short proofs using prime powers and continuous maps.
result Pairwise disjoint faces of a complex intersect under continuous maps.
The paper improves a counterexample to the topological Tverberg conjecture.
problem Conditions for finite simplicial complexes to map to Rd without higher-multiplicity intersections. method Algebraic criterion in codimension 2 for almost r-embeddings.
result Improved counterexample to topological Tverberg conjecture.
Study the discontinuity of functions not embeddable in Euclidean space.
problem Understanding discontinuity of non-embeddable functions.
method Define a modulus of discontinuity and establish lower bounds.
result Quantified nonembeddability results and topological Tverberg theorem.
Proves existence of equivariant maps avoiding diagonal in n-space.
problem Existence of equivariant maps between spaces.
method Different approach to vanishing all obstructions.
result Vanishing of all equivariant obstructions.
Here are two of our main results: Theorem 1. Let X be a normal space with dim X=n and m\geq n+1. Then the space C*(X,R^m) of all bounded maps from X into R^m equipped with the uniform convergence topology contains a dense G_δ-subset consisting of maps g such that \bar{g(X)}\capΠ^d is at most (n+d-m)-dimensional for eve…
New criterion for mapping simplicial complexes without higher intersections.
problem Mapping simplicial complexes into R^d without higher-multiplicity intersections.
method Generalized Haefliger-Weber embeddability criterion to a new metastable range.
result A well-known necessary condition is sufficient for almost r-embeddings in a new range of dimensions.
Study of f-neighbors in Riemannian manifolds, proving infinite set of distances.
problem Exploring variations of Hopf theorem in Riemannian manifolds.
method Investigates continuous maps of compact Riemannian manifolds to Rm and introduces f-neighbors. result Set of distances realized as visual f-neighbors is infinite. Polyhedra volume conjecture supports Stoker conjecture weakly.
problem Proving the Stoker conjecture for polyhedra.
method Using an extension of Montcouquiol and Weiss' result on polyhedra angles as local coordinates.
result Volume Conjecture for polyhedra implies a weak version of the Stoker conjecture.
Survey on two non-Kähler geometry conjectures.
problem Constant holomorphic sectional curvature and Fino-Vezzoni conjectures in non-Kähler geometry.
method Survey and discussion of historical and recent developments.
result Discussion of conjectures without new results.
The paper generalizes a surgery conjecture and proves it under the Zilber-Pink conjecture.
problem Generalizing the Cosmetic Surgery Conjecture to n-cusped hyperbolic 3-manifolds. method Proves the generalized conjecture under the assumption of the Zilber-Pink conjecture, and without assuming it for n=1 and 2. result Proves the generalized Cosmetic Surgery Conjecture for n=1 and 2 without assuming the Zilber-Pink conjecture. Numerical study confirms Brennan's conjecture for a counterexample to Thurston's K=2 conjecture.
problem Thurston's K=2 conjecture and Brennan's conjecture in planar domains. method Numerical analysis of a specific counterexample to Thurston's conjecture.
result The counterexample does not contradict Brennan's conjecture.
The non-vanishing conjecture implies the abundance conjecture in certain cases.
problem Abundance conjecture in algebraic geometry.
method Proof of the abundance conjecture under specific conditions.
result The abundance conjecture holds in dimensions ≤ 5 when κ ≥ 0 and ν ≤ 1.
Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
problem Gromov's flat corner domination conjecture and Stoker's conjecture for convex polyhedra.
method Same techniques applied to prove conjectures.
result Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…
The Burghelea conjecture is proven for many groups, but not all, with counter-examples provided.
problem Computing the periodic cyclic homology of complex group rings.
method Analyzing groups of finite asymptotic dimension and constructing counter-examples.
result The Burghelea conjecture holds for many classes of groups but not for all, with specific counter-examples provided.
Paper discusses conjectures and proves some related inequalities.
problem Unified generalization of BW and DDVV inequalities.
method Unified discussion and proofs of conjectures.
result Proves Conjecture 2 and obtains a new upper bound for Conjecture 3.
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
problem Integrality structure of framed knots' quantum invariants.
method Explicit formulas of colored HOMFLY-PT invariants of torus knots, verified in a limit form for any framed knots.
result Proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
New proof shows most thin knots satisfy Cabling Conjecture.
problem Cabling Conjecture for thin knots using Heegaard Floer homology.
method Heegaard Floer homology and immersed curves techniques.
result Almost all thin knots satisfy the Cabling Conjecture.
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
Equivalent conjectures proved for group presentations.
problem Balanced presentations of the trivial group.
method Equivalence of cyclic and satellite versions, restrictive cancellative version, stabilizations consideration.
result Original conjecture equivalent to cyclic version.
Metric SYZ conjecture proved using non-archimedean geometry.
problem Proving the metric SYZ conjecture in large generality.
method Using non-archimedean geometry to support the conjecture.
result Metric SYZ conjecture can be proved in large generality.
Proved Burghelea Conjecture for specific groups.
problem Proving Burghelea Conjecture for certain groups.
method Used additional cohomological properties to prove the conjecture.
result Proved Burghelea Conjecture for specified groups.
Proves Simon's conjecture for 3-manifolds.
problem Simon's conjecture for 3-manifolds.
method Proof for 3-manifolds.
result Proved Simon's conjecture for 3-manifolds.
Paper confirms Chen's biharmonic conjecture for hypersurfaces in 5D.
problem Chen's conjecture on biharmonic submanifolds in Euclidean spaces.
method Analyzes hypersurfaces in \(\mathbb{R}^5\) for \(n=4\).
result Chen's conjecture is confirmed for hypersurfaces in \(\mathbb{R}^5\) when \(n=4\).
Proved Farrell-Jones conjecture for free-by-cyclic groups.
problem Proving the Farrell-Jones conjecture for a specific class of groups.
method Geometric methods for establishing the Farrell-Jones Conjecture.
result Proved the Farrell-Jones conjecture for free-by-cyclic groups.
Counterexample disproves recent Penrose conjecture variant.
problem A variant of the Penrose conjecture.
method Provided a counterexample.
result The conjectured variant of the Penrose inequality is false.
In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
Proves isomorphism conjecture for braid groups.
problem Proving isomorphism conjecture for braid groups.
method Fibered Isomorphism Conjecture of F. T. Farrell and L. E. Jones.
result Proved K and L theoretic versions of the conjecture. Survey on Novikov conjecture and its applications.
problem Novikov conjecture in topology.
method Survey and recent developments.
result Applications to topological rigidity and non-rigidity.
Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between A-polynomial and colored Jones polynomial. method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between A-polynomial and colored Jones polynomial potential function. Akbulut and Kirby conjectured that two knots with the same 0-surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.
Counterexample disproves conjectures about log canonical thresholds.
problem Conjectures about log canonical thresholds were disproved.
method Provided a counterexample to both conjectures.
result Conjectures about log canonical thresholds are false.
Study confirms conjecture on Hermitian manifolds with bounded mass.
problem Morse-type integrals in nef (1,1) classes on compact Hermitian manifolds with bounded mass.
method Analyzes conjecture using bounded mass property on compact Hermitian manifolds.
result Confirms Demailly-Păun and Tosatti-Weinkove's conjectures.
Paper disproves conjectures about hyperbolic knots.
problem Conjectures about hyperbolic knots and their properties.
method Analyzes and refutes several conjectures about hyperbolic knots.
result Proves the conjecture about hyperbolic knots contradicts other plausible conjectures.
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
Reformulated Markov's conjecture in combinatorial terms.
problem Markov's uniqueness conjecture in integral necklaces.
method Geometric reformulation and combinatorial description.
result Explicitly described set of lengths on modular torus.
Proves Vol-Det Conjecture for many alternating links using Mahler measures.
problem Vol-Det Conjecture relating hyperbolic link volumes and determinants.
method Exact computations of Mahler measures of two-variable polynomials.
result Proves Vol-Det Conjecture for many infinite families of alternating links.
We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…