New approach uses isomorphic vector fields for stability of relative equilibria.
problem Stability of relative equilibria in nonlinear systems.
method Introduced and used isomorphic vector fields to study stability.
result Obtained a criterion for nonlinear stability of relative equilibria.
Introduces relative stability conditions on triangulated categories.
problem Stability conditions in triangulated categories.
method Definition and deformation of relative stability conditions.
result Deformation of relative stability conditions via gluing stability conditions.
In this paper we study the relative Chow and K-stability of toric manifolds in the toric sense. First, we give a criterion for relative K-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford cr…
The paper classifies stable toric Fano manifolds and compares K-stability and Ding stability.
problem Classifying stable toric Fano manifolds in low dimensions.
method Using Mabuchi constants calculated from moment polytopes and Bott tower structure.
result List of uniform relative Ding stability for toric Fano manifolds up to four dimensions.
Characterizes geometric actions on graphs with flexible stabilizers.
problem Understanding geometric actions on flexible stabilizers.
method Defining generalized fine actions and proving relative quasi-convexity criteria.
result Characterizes Bowditch boundary points in relatively geometric actions.
For a polarized algebraic manifold (X,L), let T be an algebraic torus in the group of all holomorphic automorphisms of X. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by taking T to be trivial, we see that asymptotic Chow-stability follows from stron…
Extremal metrics linked to stability in complex geometry.
problem Existence and uniqueness of extremal metrics in complex geometry.
method Proving asymptotic relative Chow stability implies extremal metrics existence and uniqueness.
result Existence and uniqueness of extremal metrics in any polarization.
The study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.
The study proves Kähler manifolds with extremal Kähler metrics are relatively K-stable.
problem Existence of extremal Kähler metrics on Kähler manifolds.
method Introducing relative K-stability and proving it for Kähler manifolds with extremal Kähler metrics.
result Kähler manifolds admitting extremal Kähler metrics are relatively K-stable.
Mabuchi's metric correlates with a specific stability condition for Fano manifolds.
problem Characterizing Fano manifolds with Mabuchi's soliton metric.
method Proving Mabuchi's metric corresponds to relative D-stability.
result Fano manifolds admit Mabuchi's metric if and only if they are uniformly relatively D-stable.
Generalizes energy-momentum method for non-autonomous Hamiltonian systems.
problem Stability analysis of non-autonomous Hamiltonian systems with symmetries.
method Develops a new approach to relative equilibrium points and stability conditions for non-autonomous systems.
result Conditions ensuring stability of relative equilibrium points in non-autonomous Hamiltonian systems.
In this paper, we discuss the relative K-stability and the modified K-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative K-stability and the properness of modified K-energy. In …
Paper solves PDEs for optimal investment strategies in volatile markets.
problem Finding optimal investment strategies in volatile markets.
method Numerical methods using time-changed Bessel bridges.
result Solves PDEs for relative arbitrage opportunities in volatility-stabilized markets.
The paper improves asymptotic polybalanced kernels for extremal Kaehler metrics.
problem Stability and metrics on algebraic manifolds.
method Asymptotic polybalanced kernels associated to extremal Kaehler metrics.
result Stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds.
The study examines necessary conditions for Mabuchi solitons on Fano manifolds and their relation to Ding stability.
problem Existence of Mabuchi solitons on Fano manifolds.
method Investigates the inner product of C∗-actions on equivariant test-configurations and uses convex-geometry descriptions. result Uniformly relative Ding stability implies a necessary condition for the existence of Mabuchi solitons.
We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.
Proves stability pulls back under proper actions, with applications to mapping class groups and free groups.
problem Stability of subgroups in mapping class groups and free groups.
method Proves stability pulls back under proper actions on metric spaces.
result Stability of convex cocompact subgroups in mapping class groups and free groups.
reval package selects best clustering solutions via stability-based validation.
problem Challenges in determining best clustering solutions due to lack of validation methods.
method Stability-based relative clustering validation methods.
result Determines best clustering solutions that generalize to unseen data.
Kim-Milman flow map stable under regular target measures
problem Stability of Kim-Milman flow map under target measure variations
method Stability in relative entropy and 2-Wasserstein distance result Lipschitz stability up to logarithmic factor
Study Mabuchi solitons on toric Fano varieties, linking stability and energy.
problem Existence and stability of Mabuchi solitons on toric Fano varieties.
method Algebraic stability notion (relative Ding stability) and variational approach.
result Partial coercivity and singular Mabuchi solitons in non-uniformly stable cases.
Study quantized extremal Kähler metrics for algebro-geometric stability.
problem Stability of extremal Kähler metrics and manifolds.
method Quantized extremal Kähler metrics and equivariant Riemann-Roch theorem.
result Proves weak relative Chow polystability and K-semistability.
New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds (M,L), a series of weighted balanced metrics ωm, m≫1, called polybalanced metrics, are obtained from complete linear systems ∣Lm∣ on M. Then the asymptotic behavior of the weights as m→∞ will be stud…
Theory of relatively Anosov representations using flow methods.
problem Developing a theory for relatively Anosov representations.
method Using the contracting flow on a bundle to define and study relatively Anosov representations.
result Definition and study of uniformly relatively Anosov representations and a stability result.
Survey on stability of Minkowski spacetime in relativity.
problem Nonlinear stability of Minkowski spacetime in general relativity.
method Decay assumptions, geometric foliations, energy identities, and gauge choices.
result Understanding of decay, dispersion, and geometry-analysis interplay.
Paper defines a new invariant for 4-manifolds with boundary.
problem Defining a new invariant for 4-manifolds with boundary.
method Defined the relative L-invariant by measuring path lengths in the cut complex of a trisection surface.
result Shows that if a rational homology ball has a zero relative L-invariant, it must be the 4-ball.
Consider a group G and a family A of subgroups of G. We say that vertex finiteness holds for splittings of G over A if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in A. We show vertex finiteness when G…
We present an equivariant Liapunov stability criterion for dynamical systems with symmetry. This result yields a simple proof of the energy-momentum-Casimir stability analysis of relative equilibria of equivariant Hamiltonian systems.
We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.
Extremal metrics exist if uniformly K-stable over models.
problem Existence of extremal metrics on complex projective varieties.
method Uniform K-stability over models of extremal tori. result Extremal metrics exist if uniformly K-stable. The paper proves a new stability condition for certain Fano varieties.
problem Stability conditions for Fano varieties with Gorenstein singularities.
method Using toric test configurations and combinatorial criteria.
result Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties.
Local minimality proven for stable free-boundary minimal hypersurfaces.
problem Proving local minimality for stable free-boundary minimal hypersurfaces.
method Using relative current setting and strict stability, proving local minimality among relative cycles.
result Local minimality of stable free-boundary minimal hypersurfaces in a small tubular neighborhood.
The paper extends trisection theory to 4-manifolds with multiple boundary components.
problem Trisection theory for 4-manifolds with multiple boundary components.
method Extending relative trisection theory to include multiple boundary components and providing conditions for gluing.
result Definition of a category Tri whose objects are 3-manifolds with open book decompositions and morphisms are relatively trisected cobordisms.
Groups' boundary actions are stable under small perturbations.
problem Stability of group actions on their boundaries.
method Proving semi-conjugacy of perturbed actions to the standard action.
result Perturbed actions are globally semi-conjugate to the standard action.
Stability of weighted extremal manifolds proven through blowups.
problem Stability of weighted extremal manifolds.
method Blowup technique to analyze weighted extremal Kähler manifolds.
result Proves weighted extremal manifolds are relatively weighted K-polystable.
We prove a criterion for stability of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map. This generalizes a theorem of G.W. Patrick. The method of the proof is also useful in studying the bifurcation of relative equilibria.
Defines new stability conditions for Sasaki manifolds and extremal metrics.
problem Stability conditions for Sasaki manifolds and extremal metrics.
method Combining weighted K-stability with Sasaki extremality theory.
result Weighted K-stability is necessary for extremal Sasaki metrics.
Exotic diffeomorphism survives stabilizations on a contractible 4-manifold.
problem Finding exotic diffeomorphisms on contractible 4-manifolds.
method Developed a Pin(2) × Z_2-equivariant refinement for computing Seiberg-Witten Floer homotopy types.
result Constructed an exotic diffeomorphism that survives two stabilizations.
The Martin boundary of certain groups is stable under specific conditions.
problem Stability of Martin boundaries in relatively hyperbolic groups.
method Extending Floyd-Ancona inequalities, defining spectral degenerescence, and proving stability criteria.
result The Martin boundary of admissible symmetric finitely supported probability measures on geometrically finite Kleinian groups of dimension at most 5 is always strongly stable.
Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. The paper proves volume stability for hyperbolic manifolds and applies it to general relativity.
problem Volume stability of hyperbolic manifolds and its implications in general relativity.
method Sharp volume-stability theorem for closed hyperbolic three-manifolds, tensorial \(C^0\)-convergence.
result Near-equality in the sharp hyperbolic volume bound forces tensorial \(C^0\)-convergence to the hyperbolic metric.
The paper proves stability of the positive mass theorem using intrinsic flat convergence.
problem Stability of the positive mass theorem in mathematical relativity.
method Intrinsic flat convergence of points and applications to stability.
result Revisits and strengthens the stability results for graphical hypersurfaces of Euclidean space.
RELTA-SGLD stabilizes nonconvex SGLD updates with a lighter taming scheme.
problem Stabilizing superlinear stochastic-gradient updates in nonconvex optimization.
method Threshold-based taming with relative-growth principle for stability.
result Polynomial moment stability and first-order stationary accuracy in nonconvex SGLD.
The study defines K-stability for Sasakian manifolds and finds obstructions to extremal metrics.
problem Finding obstructions to Sasaki-extremal metrics in Sasakian manifolds.
method Defining K-stability relative to a maximal torus of automorphisms and computing the Sasaki cone.
result The existence of a Sasaki-extremal metric implies K-semistability, and the Lichnerowicz obstruction is extended.
In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected 2-complex X with a linear homological isoperimetric inequality, a bound on the length of attaching maps of 2-cells and finitely many 2-cells adjacent to any edge must …
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by Ck,α stability proofs. result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
New model predicts stock performance in large equity markets.
problem Predicting stock performance in large equity markets over long time horizons.
method Rank-based volatility stabilized models calibrated to empirical data.
result The model exhibits relative arbitrage and statistically fits empirical features.
Study proves quantitative results for isoperimetric problem outside convex bodies in the plane.
problem Quantitative estimates for the relative isoperimetric problem outside convex bodies in the plane.
method Flow approach and Łojasiewicz estimates to prove quantitative stability for minimizers.
result Explicit constants and optimal exponents/rates for Łojasiewicz estimates and rates of convergence for gradient flow.