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.
Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
Introduces stability conditions and their connection to Artin groups.
problem Understanding the relationship between stability conditions and Artin groups.
method Explains the connection between Bridgeland stability conditions and the K(π,1) conjecture. result Establishes a link between stability conditions and Artin groups.
We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to β-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stabilit…
It is shown that the original Andrews--Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis made in 1966. We also…
New examples challenge Geroch conjecture stability.
problem Stability of the Geroch conjecture in warped products.
method Constructing warped-product manifolds with specific curvature properties.
result First counterexample to Sormani's conjecture on stability.
The paper refines the stability index for a specific minimal hypersurface and verifies Yau's conjecture.
problem Stability of minimal hypersurfaces in spheres and eigenvalue multiplicity.
method Analytical and numerical methods to study eigenvalues and stability indices.
result The multiplicity of the eigenvalue for the Carlotto-Schulz minimal embedding is at least 2n+1+n^2.
The paper proves a stability conjecture for manifolds with zero Euler characteristic.
problem Stability of manifolds with zero Euler characteristic under certain curvature conditions.
method Analyzes manifolds with dimensions 5 or more, proving stability under specific curvature and completeness conditions.
result 2006 Rosenberg's S1-stability holds for manifolds with zero Euler characteristic. C. Gordon conjectured that a connected sum of two Heegaard splittings is stabilized if and only if one of the two factors is stabilized (Problem 3.91 in Kirby's problem list). In this paper, we shall prove this conjecture.
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
A combinatorial proof of the Gordon Conjecture: The sum of two Heegaard splittings is stabilized if and only if one of the two summands is stabilized.
We study the Fibered Isomorphism conjecture of Farrell and Jones for groups acting on trees. We show that under certain conditions the conjecture is true for groups acting on trees when the stabilizers satisfy the conjecture. These conditions are satisfied in several cases of the conjecture. We prove some general resul…
The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.
problem Proving the geometric stability conjecture for scalar curvature on manifolds conformal to tori.
method Reduction via the Yamabe problem and handling sequences of manifolds conformal to flat tori or constant negative scalar curvature.
result Proves the geometric stability conjecture for certain sequences of manifolds conformal to flat tori.
Proves a conjecture about manifolds and scalar curvature.
problem Determining when a manifold admits a positive scalar curvature metric.
method Uses a geometric bound to measure discrepancies between vector fields.
result Proves the conjecture in codimension two.
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the C∞ compact manifold (M,g) with quasi-convex universal covering and divergent geodesic rays. result Proved the C1-stability conjecture for geodesic flows of compact manifolds. The paper proves stability of minimal embeddings in spheres and relates it to Yau's conjecture.
problem Stability of minimal embeddings in spheres and Yau's conjecture.
method Analyzes stability index and solves differential equation to relate to Yau's conjecture.
result Stability index of minimal hypersurfaces is at least n^2+4n+3 and Yau's conjecture holds under specific conditions.
Proves conjecture about sphere widths under rotational symmetry.
problem Width stability of rotationally symmetric metrics.
method Proof of conjecture and extensions to higher dimensions.
result Stability of min-max width under rotational symmetry.
In this paper we conjecture the stability and vanishing of a large piece of the unstable rational cohomology of SL_n Z, of mapping class groups, and of Aut(F_n).
We formulate a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra, and verify it for all torus knots and all simple Lie algebras of rank 2. Our conjecture is motivated by a structure theorem for the degree …
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact moduli space, are proven to be K-stable although it is well known that they are \t…
Finite p-group actions on manifolds have limited stabilizer subgroups.
problem Understanding the structure of stabilizer subgroups in group actions on manifolds.
method Bounding the index of a subgroup H in a finite p-group G acting on a compact manifold M, ensuring a controlled number of stabilizers.
result The existence of a subgroup H with a controlled index and limited stabilizers.
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.
Stability result for nearly isometric subspaces and Finsler surfaces.
problem Stability of normed spaces and Finsler surfaces under near-isometric conditions.
method Refined topological argument and explicit quantification using Banach-Mazur distance.
result A 2-dimensional surface with near-monochromatic Finsler metric is approximately Riemannian.
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.
K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical not…
Accelerated gradient method's stability deteriorates exponentially with steps.
problem Algorithmic stability of Nesterov's accelerated gradient method.
method Analysis of two notions of algorithmic stability for Nesterov's accelerated gradient method.
result Stability of Nesterov's accelerated method deteriorates exponentially with the number of gradient steps.
Proves stability in Weyl polytopes using optimal transport.
problem Stability of Weyl polytopes under optimal transport.
method Optimal transport stability for reflexive Weyl polytopes.
result Weak metric SYZ conjecture holds for Delzant reflexive Weyl polytopes.
Proposes a method to prove closing of periodic orbits in dynamical systems.
problem Proving generic dynamical systems have finite periodic orbits.
method Perturbation method for Cr closing of periodic orbits. result Validates the conjecture for generic dynamical systems.
New findings on maximizing noise stability in partitions of Gaussian space.
problem Maximizing noise stability in partitions of Gaussian space.
method Analyzing the correlation between sets and their noise stability, proving conditional conjectures and hardness results.
result Hyperstable partitions maximize noise stability and have specific properties.
Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.
problem Proves Yau-Tian-Donaldson conjecture for a specific class of manifolds.
method Uses holomorphic actions of compact Lie groups and combinatorial conditions.
result Equivalence of K-uniform stability and K-stability for spherical varieties.
Proves finitely generated associated graded rings for valuations on log Fano pairs.
problem Stability thresholds of log Fano pairs.
method Proves finite generation of associated graded rings for valuations.
result Log Fano pairs are uniformly K-stable if their stability threshold is less than a certain value.
We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates thi…
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…
We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature Kähler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the use of pluripotential theory and the interpretation of K-stability in terms of non-…
New Einstein metrics identified on a specific full flag manifold.
problem Investigating Einstein metrics on a full flag manifold.
method Analyzing G-stability of Einstein metrics on M=G/K. result Identified four new Einstein metrics on F(5), confirming pairwise non-homotheticity. 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.
Proves results on K-stability using arcs and Mabuchi functional.
problem K-stability in Fano manifolds and uniform K-polystability.
method Arcs and numerical criterion for stability of pairs.
result Characterizes coercivity of Mabuchi functional in terms of K-polystability.
Non-asphericity of strata of genus-one differentials
problem Strata of genus-one differentials
method Non-asphericity
result Infinitely many counterexamples to conjectures
Proves stability of geodesic flows on closed surfaces.
problem Stability of geodesic flows on closed surfaces.
method Generic Riemannian metrics and Reeb flows.
result Proves C2-stability conjecture for geodesic flows. In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown to be linearly stable, leading to the conjecture that all solvsolitons are line…
Proves uniform K-stability is open in Kähler cone.
problem Stability of Kähler metrics in complex geometry.
method Introduced new norm on test configurations and estimates for non-archimedean energy functionals.
result Uniform K-stability is an open condition in the Kähler cone.
Paper confirms conjecture for projective manifolds in supercritical phase.
problem Stability condition for deformed Hermitian-Yang-Mills equation.
method Establishes stability result not involving uniform constants.
result Confirms conjecture for projective manifolds in supercritical phase.
We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log K-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjectu…
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.
In this paper, we establish a general relationship between the nonvanishing of GW invariants with the existence of the closed orbits of a Hamiltonian system. As an application, we completely solved the stabilized Weinstein conjecture.
Surveying conjectures on compactness and scalar curvature.
problem Understanding sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces.
method Analyzing progress and related examples towards conjectures.
result Survey of current progress and related examples.