Proves Rudyak's conjecture for low-dimensional simply connected spin manifolds.
problem Rudyak's conjecture on the relationship between the Lusternik-Schnirelmann category of manifolds.
method Analyzes simply connected spin manifolds of dimensions up to 8.
result Proves the conjecture for n-dimensional simply connected spin manifolds for n≤8. Summarizes connections between Euler characteristic theorems and conjectures.
problem Vanishing of the Euler characteristic
method Diagrammatic summary of connections
result Connections between various theorems and conjectures
Paper finds non-positive Weyl connections on Lie groups, confirming a conjecture.
problem Finding non-positive invariant Weyl connections on Lie groups.
method Investigation of completely solvable Lie groups, focusing on SOL group.
result Only SOL admits non-positive Weyl connections, confirming a conjecture.
This paper discusses partial answers and a proof for conjectures about Gauduchon connections on Hermitian manifolds.
problem Conjectures about Gauduchon connections and Hermitian metrics on compact manifolds.
method Analyzes partial answers to conjectures and provides a proof for a related conjecture, discovering a duality phenomenon.
result Proof of the second conjecture about two Kähler-like Gauduchon connections implying a Kähler metric.
We show that, under some technical conditions, the Strong Slope Conjecture proposed by Kalfagianni and Tran is closed under connect sums and cabling. As an application, we establish the Strong Slope Conjecture for graph knots.
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. 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.
Verify conjecture for special Hermitian manifolds.
problem Conjecture about space forms for canonical metric connections.
method Verify conjecture for complex nilmanifolds and Bismut torsion-parallel manifolds.
result Verify conjecture for two special types of Hermitian manifolds.
Proves a connectivity conjecture for free groups, showing homotopy type of spheres.
problem Establishing a connectivity conjecture for free groups.
method Provided homotopy-equivalent models of the common basis complex using free factors and sphere systems.
result The common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3.
We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.
Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.
problem Determining the structure of Kauffman bracket skein module of connected sums of solid tori.
method Used algebraic methods over the ring of Laurent polynomials to prove the conjecture.
result Proved a conjecture about the Kauffman bracket skein module of connected sums of genus one handlebodies.
The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.
problem The Double Soul Conjecture for non-simply connected manifolds.
method Analysis of double disk bundles and counterexamples.
result Infinitely many counterexamples to the generalized Double Soul Conjecture.
The AJ conjecture is verified for certain connected sums of torus knots.
problem Verifying the AJ conjecture for specific connected sums of torus knots.
method Analyzing recurrence polynomials and their factorization properties.
result The AJ conjecture requires a modification for certain connected sums of torus knots.
The paper confirms Arnold's conjecture about hyperbolic polynomials.
problem The number of connected components of hyperbolic polynomials increases linearly with degree.
method Constructive proof using homotopy invariance of the index of a curve and properties of homogeneous polynomials.
result Exact number of connected components of Hyp(D) is determined and representatives for each component are provided. Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-manifold. We prove the K-theoretic Farrell-Jones Conjecture (up to dimension one) and the L-theoretic Farrell-Jones Conjecture for G, where we allow coefficients in additive G-categories with (involution).
Study proves a conjecture for certain spin manifolds.
problem Proving a conjecture about positive scalar curvature metrics.
method Connected sum construction and Gromov-Lawson area enlargement.
result Connected sum of spin manifolds admits no complete metric of positive scalar curvature.
Introduces stability for families of K-polystable varieties and connects it to optimal symplectic connections.
problem Forming correct moduli for fibrations and understanding their geometry.
method Introduces a new stability condition for fibrations and relates it to the existence of optimal symplectic connections.
result Proves that the existence of an optimal symplectic connection implies semistability of the fibration.
The Strominger conjecture is confirmed for compact Hermitian manifolds in 2D and special higher dimensions.
problem Determining conditions for a Hermitian metric to be Kähler based on the Strominger connection's curvature.
method Analyzing the Strominger connection's holomorphic sectional curvature in compact Hermitian manifolds.
result The Strominger conjecture is confirmed in 2D and special higher dimensions.
We extend Garsia's conjecture about surface embeddings.
problem Realizing conformal classes of genus-1 surfaces via embeddings.
method Formulated within the framework of connections on principal bundles, solutions provided in several cases.
result Novel parameterizations of the moduli space of conformal classes of compact surfaces of genus 1.
Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
problem Proving the finitely generated nature of the Goeritz group for genus-3 Heegaard splittings of the 3-sphere.
method Establishing the connectivity of reducing sphere complexes for the genus-3 case.
result Confirmation of the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.
We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian Kähler manifold consists of isometries unless the metric has constant positive holomorphic curvature.
The article confirms a complex geometry conjecture for a specific type of manifold.
problem Compact Hermitian manifolds with constant holomorphic sectional curvature.
method Restricting to pluriclosed manifolds and confirming the conjecture for Strominger Kähler-like manifolds.
result The conjecture is confirmed for a specific type of Hermitian manifold.
The volume conjecture for surface diffeomorphisms connects volumes to polynomial evaluations.
problem Connecting surface volumes to polynomial evaluations of quantum invariants.
method Developing combinatorial and algebraic techniques to compute isomorphisms between representations.
result Numerical evidence supports a conjecture linking surface volumes to polynomial evaluations.
The paper confirms a conjecture about Hermitian manifolds with constant mixed curvature.
problem Compact Hermitian manifolds with non-zero constant mixed curvature must be Kähler.
method Verification for specific types of Hermitian manifolds including complex nilmanifolds, solvmanifolds, and Lie algebras.
result Partial evidence supporting Kai Tang's conjecture.
We prove the A-theoretic Farrell-Jones Conjecture for virtually solvable groups. As a corollary, we obtain that the conjecture holds for S-arithmetic groups and lattices in almost connected Lie groups.
Classifies connected shelves up to order six.
problem Classifying finite right-distributive binary algebraic structures called shelves.
method Symbolic computations with Python to classify shelves up to isomorphism, exploring group structure, and defining shelf polynomials.
result Classified all connected shelves with order less than six up to isomorphism.
Penrose conjecture proven for specific initial data sets.
problem Proving Penrose conjecture for certain types of initial data sets.
method Used σ-inverse mean curvature flow and a monotonicity formula.
result Penrose conjecture established for 2-convex initial data sets.
Complete classification of Hermitian manifolds with flat Gauduchon connections.
problem Classifying compact Hermitian manifolds with flat Gauduchon connections.
method Analyzing properties of Hermitian manifolds and using Gauduchon connections.
result Established a conjecture about Kähler-like conditions and flatness.
The paper explores meromorphic connections over Frobenius manifolds.
problem Existence and uniqueness of meromorphic connections.
method Holomorphic bundles with meromorphic connections, conjecture proof.
result Proof of conjecture in 2D cases.
Proves a conjecture for a specific group using spectral sequences and homology.
problem Proves the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4.
method Used the Adams spectral sequence and detection theorems to compute connective real k-homology.
result Determines differentials of the Adams spectral sequence and studies the cap structure of relevant sub-hopf algebras.
Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in S3 do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.
A consequence of the Cabling Conjecture of Gonzalez-Acuña and Short is that Dehn surgery on a knot in S3 cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of th…
The paper proves a quantum modularity conjecture for 3-manifolds.
problem Quantum invariants of 3-manifolds at roots of unity.
method Formulates and proves a strong version of the conjecture for geometric 3-manifolds.
result The conjecture holds for Brieskorn homology spheres and some other examples.
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…
The study refines known counterexamples in 4D to satisfy certain inequalities.
problem Addressing counterexamples in Gromov's and Rosenberg's conjectures.
method Analyzing simply connected and non-simply connected four manifolds up to homeomorphism.
result Gromov's and Rosenberg's conjectures hold for simply connected four manifolds up to homeomorphism.
I explain an open conjecture by Braverman/Milatovic/Shubin (BMS) on the positivity of square integrable solutions f of (−Δ+1)f≥0 on a geodescially complete Riemannian manifold, and its connection to essential self-adjointness problems of covariant Schrödinger operators. The latter conjecture has remained open f…
Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.
Around 1988, Floer introduced two important theories: instanton Floer homology as invariants of 3-manifolds and Lagrangian Floer homology as invariants of pairs of Lagrangians in symplectic manifolds. Soon after that, Atiyah conjectured that the two theories should be related to each other and Lagrangian Floer homology…
In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…
A well-known conjecture in knot theory says that the percentage of hyperbolic knots amongst all of the prime knots of n or fewer crossings approaches 100 as n approaches infinity. In this paper, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjectur…
In this paper, by use of techniques associated to cobordism theory and Morse theory,we give a simple proof of Poincare conjecture, i.e. Every compact smooth simply connected 3-manifold is homeomorphic to 3-sphere.
The abstract discusses connections between K-stability, heights, and rational points on Fano varieties.
problem Understanding the density of rational points on Fano varieties.
method Exploring connections between height bounds, K-stability, and Peyre's conjecture.
result Established an analog of height inequalities for real points, related to Kähler-Einstein metrics.
The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds'',…
Compact method proves Brown-York mass positivity and connects to major conjectures.
problem Proving positivity of Brown-York's mass and its connections to conjectures.
method Compact approach to proving mass positivity and exploring connections.
result Proved the positivity of Brown-York's mass and its relation to conjectures.
The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.
problem Understanding the relationship between Kähler-Ricci shrinkers and Fano fibrations.
method Using birational algebraic geometry, the paper proves properties of Kähler-Ricci shrinkers and formulates conjectures relating them to Fano fibrations.
result The existence of Kähler-Ricci shrinkers is conjectured to be related to K-stability of polarized Fano fibrations.
Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups Γ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions…
The study proposes a conjecture about the monodromy group of singular hyperbolic metrics and provides evidence and confirmations.
problem Understanding the monodromy group of singular hyperbolic metrics on Riemann surfaces.
method Using meromorphic differentials and affine connections, the study examines the monodromy group and confirms the conjecture for specific Riemann surfaces.
result The monodromy group of the singular hyperbolic metric is Zariski dense in PSL(2, R) and cannot be contained in certain Lie subgroups.
We prove the existence of a map of spectra τA:kA→lA between connective topological K-theory and connective algebraic L-theory of a complex C∗-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural e…