Paper explores new Kähler metrics from old, aiming to solve YTD conjecture.
problem Extending classical extremal Kähler metrics to include new objects.
method Surveying recent works on weighted extremal Kähler metrics and the YTD conjecture.
result Survey of recent research on weighted extremal Kähler metrics.
In this paper, we prove the conic version of YTD conjecture on log Fano manifolds.
Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
problem Exploring connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
method Surveying and reviewing recent works, transforming complex Monge-Ampère equations, and proving the YTD conjecture.
result Established a transformation from irregular Sasaki-Einstein metrics to g-solitons on quasi-regular quotients. Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
problem Proving Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
method Analyzing Monge-Ampère equations corresponding to generalized and twisted Kähler-Ricci g-solitons, proving stability conditions.
result Existence of solutions is equivalent to equivariantly uniform Θ-twisted g-Ding-stability.
Effective Yau-Tian-Donaldson conjecture for spherical varieties.
problem Finding effective K-stability criteria for spherical varieties.
method Formulated an effective variant of the Yau-Tian-Donaldson conjecture and reviewed effective K-stability criteria for spherical varieties.
result Effective K-stability criteria can be computed given combinatorial data.
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
New stability concept for Poisson structures leads to constant curvature metrics.
problem Finding constant scalar curvature metrics in generalized Kähler geometry.
method Introducing Poisson K-stability and using infinite-dimensional momentum map techniques.
result Existence of constant scalar curvature symplectic generalized Kähler structures on Kähler-Einstein Fano manifolds.
Introduces Poisson K-stability for Kähler manifolds and proves existence of constant scalar curvature structures.
problem Stability conditions for Poisson structures on Kähler manifolds.
method Infinite-dimensional momentum map techniques.
result Existence of constant scalar curvature symplectic generalized Kähler structures on Kähler-Einstein Fano manifolds.
We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archime…
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.
As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD…
Researchers introduce new energies to study constant scalar curvature metrics.
problem Understanding constant scalar curvature metrics on compact Kähler manifolds.
method Introduced a family of Kβ energies using Berman's quantization and intersection theory. Combined with non-Archimedean techniques, provided a uniform Yau-Tian-Donaldson correspondence. result Uniform Yau-Tian-Donaldson correspondence characterizes the existence of a unique constant scalar curvature Kähler metric.
Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.
problem Characterizing and approximating non-Archimedean metrics on pseudoeffective classes.
method Extending Ross-Witt Nyström correspondence to relative case, introducing flag configurations.
result Non-Archimedean finite energy metrics are approximable by flag configurations, and very general Ding energies are continuous.
A blindfolded LLM trading framework validates market signals without ticker memorization.
problem Ensuring LLMs trade based on genuine market understanding, not memorized data.
method Anonymize tickers and company names, verify signals through reasoning embeddings, and use PPO-DSR policy.
result Achieved Sharpe ratio of 1.40 +/- 0.22 across 20 seeds, robust in volatile markets.
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.
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…
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.
We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which tog…
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.
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 \(n=4\).
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.
In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward several new conjectures which will be shown equivalent to Conjecture 2. In particula…
In this paper, we generalize the Cosmetic Surgery Conjecture to an n-cusped hyperbolic 3-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For n=1 and 2, we show them without the assumption.
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.
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.
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.
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…
Verifies a conjecture for the figure eight knot.
problem Relates A-ideal and recurrence ideal of knots.
method Uses quantum A-ideals, q-holonomicity, and AJ conjecture.
result Strong AJ conjecture verified for figure eight knot.
Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.
problem Fundamental group of nonnegative curvature manifolds.
method Observation in dimension 4.
result Fukaya-Yamaguchi conjecture holds in 4D.
Proved a combinatorial conjecture in machine learning.
problem None explicitly stated in the abstract.
method Binomial and multinomial sums identities.
result Proved a combinatorial conjecture.
In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey--Vaugon conjec…
There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Propert…
Paper proves contractible fake surfaces up to complexity 6 are deformable.
problem Stable Andrews-Curtis conjecture and contractible fake surfaces.
method Induction scheme proving contractibility up to complexity 6.
result Contractible fake surfaces up to complexity 6 are 3-deformable.
Paper analyzes Bartnik's quasi-local mass conjectures and their validity.
problem Understanding the validity of Bartnik's quasi-local mass conjectures.
method Developed a framework to analyze Bartnik's static vacuum extension conjecture.
result Proved the static vacuum extension conjecture is not true in general.
This memoire consists of two main results. In the first one we describe Ricci flow theory and we give an educative way for proving Elliptization Conjecture and then we prove Poincare conjecture which is the second proof of Perelman for Poincare conjecture. In the second one which is the main propose of our memoire, we …
Proves Stolz conjecture for specific types of manifolds.
problem String manifolds with positive Ricci curvature and Witten genus.
method Analyzes toric string Fano manifolds and string torus manifolds with invariant metrics.
result Proves conjecture for specified types of manifolds.