New proof for stable reduction theorem using Kähler-Einstein metrics.
problem Proving the stable reduction theorem for curves over punctured curves.
method Using Kähler-Einstein metrics on fibers to obtain limiting stable curves.
result A new analytic proof of the stable reduction theorem for curves over punctured curves.
Defines super stable maps and proves quotient superorbifolds for genus zero.
problem Defines stable supercurves and super stable maps of genus zero.
method Uses labeled trees and slice theorem for super Lie groups.
result Proves moduli space of stable supercurves and super stable maps are quotient superorbifolds.
In this paper we establish a uniform C2,θ estimate for level sets of stable solutions to the singularly perturbed Allen-Cahn equation in dimensions n≤10 (which is optimal). The proof combines two ingredients: one is the infinite dimensional reduction method which enables us to reduce the C2,θ estimate …
The paper extends stable minimal hypersurface results to δ-stable hypersurfaces in R^(n+1).
problem Extending stable minimal hypersurface results to δ-stable hypersurfaces.
method Regularity and compactness theorems for immersed δ-stable minimal hypersurfaces in R^(n+1).
result Optimal range of δ for δ-stable hypersurfaces.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving …
Let EG be a stable principal G--bundle over a compact connected Kaehler manifold, where G is a connected reductive linear algebraic group defined over the complex numbers. Let H⊂G be a complex reductive subgroup which is not necessarily connected, and let EH⊂EG be a holomorphic reduction of s…
We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.
We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having m=O(r~/ε2) rows. Here r~ is the maximum stable rank, i.e. squared ratio of Frobenius and op…
Let H be a semisimple algebraic group. We prove the semistable reduction theorem for μ--semistable principal H--bundles over a {\it smooth projective variety X} defined over the field $\bc$. When X is a {\it smooth projective surface} and H is simple, we construct the algebro--geometric Donaldson--Uhlenbeck…
The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
problem Investigating volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
method Applying similar techniques to derive local rigidity theorems for strictly stable Ricci flat manifolds.
result Derives local rigidity theorems for strictly stable Ricci flat manifolds with respect to σ2-curvature.
NROWAN-DQN improves stability and exploration in noisy networks.
problem Noisy networks struggle with stable exploration in complex tasks.
method Noise reduction and online weight adjustment for stable actions.
result NROWAN-DQN outperforms prior algorithms in stability and exploration.
Adapts a short argument to derive a stability theorem for smooth maps.
problem Proving stability of smooth proper maps.
method Adapting a short argument from Golubitsky and Guillemin to derive the Mather stability theorem.
result Derives the Mather stability theorem from the Mather stability theorem in [MaII].
Compressing data helps learn Mahalanobis metrics effectively.
problem Learning Mahalanobis metrics in high-dimensional spaces.
method Randomly compress data to train a full-rank metric in a reduced feature space.
result Theoretical guarantees on error for Mahalanobis metric learning, independent of ambient dimension.
The study proves that certain stable minimal hypersurfaces must be cylindrical.
problem Characterizing stable minimal hypersurfaces in Euclidean space.
method Analyzing the density at infinity and using stable area minimizing hypercone properties.
result Stable minimal hypersurfaces with specific conditions are cylindrical.
Study of symplectic and Poisson reduction, proposing Poisson implosion.
problem Understanding and generalizing symplectic reduction to Poisson manifolds.
method Recalled and reviewed symplectic and Poisson reduction, proved cross-section theorem for Poisson manifolds.
result Generalized Guillemin-Sternberg theorem for Poisson manifolds, identified Poisson transversals.
Generalizes symplectic reduction to cosymplectic groupoid actions.
problem Symplectic reduction for cosymplectic groupoid actions.
method Introduced cosymplectic groupoid actions and proved a theorem.
result Proved a theorem analogous to Mikami-Weinstein theorem.
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
New approach to proving Chen-Donaldson-Sun theorem with examples.
problem Proving Chen-Donaldson-Sun theorem for families of curves.
method Construction of a special metric on stable vector bundles over surfaces formed by families of curves.
result Demonstrates existence of a special metric related to one-dimensional cycles in moduli space.
The paper proves vanishing theorems for harmonic forms and spinors on stable minimal hypersurfaces.
problem Vanishing theorems for harmonic forms and spinors on stable minimal hypersurfaces.
method Positive curvature assumptions on the ambient manifold.
result Vanishing of L2-harmonic forms and spinors on stable minimal hypersurfaces. Corrects an earlier theorem, establishing new facts about information structures and non-anticipative aggregation.
problem The nature of information structures and their impact on non-anticipative aggregation.
method Local reduction of pricing to the natural price filtration, stability properties, and the establishment of new facts.
result Non-anticipative signals can reveal future information, requiring dependence among signals (masking relation) and not independence.
Strong Frankel theorem for shrinkers in all dimensions.
problem Intersection of shrinkers in large balls.
method Proof using strong Bernstein theorem for stable Gaussian surfaces.
result Shrinkers are connected in all large balls.
The paper proves the existence of stable spheres in asymptotically flat 3-manifolds.
problem Existence of stable spheres in asymptotically flat 3-manifolds.
method Lyapunov-Schmidt reduction
result Existence of an asymptotic foliation of (M,g) by stable constant mean curvature spheres. Develops theory for stable capillary minimal hypersurfaces in half-space.
problem Regularity and compactness of stable capillary minimal hypersurfaces.
method Integral curvature estimate and tilt excess function.
result Generalized Bernstein theorem for stable capillary minimal hypersurfaces.
We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the first reduction theorem in order two we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
Sparse random projection (RP) is a popular tool for dimensionality reduction that shows promising performance with low computational complexity. However, in the existing sparse RP matrices, the positions of non-zero entries are usually randomly selected. Although they adopt uniform sampling with replacement, due to lar…
Two reduction schemes for symplectic manifolds are shown equivalent.
problem Reduction of Hamiltonian systems on exact symplectic manifolds.
method Modified Marsden-Meyer-Weinstein reduction theorem for exact symplectic manifolds and contact manifolds.
result Reduction schemes are equivalent for exact symplectic manifolds and energy hypersurfaces.
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Δ^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where p>1 and n≥1. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an…
The paper explores density of stable mappings and their properties.
problem Density of stable mappings in different dimensions.
method Infinitesimal and algebraic methods to prove density of proper stable and topologically stable mappings.
result Density of topologically stable mappings holds for any pair (n,p), and for proper stable mappings if (n,p) is in nice dimensions.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
Let Y be a closed 3-manifold such that all flat SU(2)-connections on Y are non-degenerate. In this article, we prove a Uhlenbeck-type compactness theorem on Y for stable flat SL(2,C) connections satisfying an L2-bound for the real curvature. Combining the compactness theorem and a previous…
Estimates for stable minimal hypersurfaces in Euclidean space.
problem Deriving estimates for stable minimal hypersurfaces.
method Derivation of estimates related to Bernstein theorems.
result Indicates limitations of existing methods for n=6. Stable solutions to Yang-Mills-Higgs equations on spheres and tori identified.
problem Stable solutions to abelian Yang-Mills-Higgs equations on S2 and T2. method Reduction to vortex equations and application of Bourguignon-Lawson's method for stable SU(2) Yang-Mills connections. result Stable solutions to abelian Yang-Mills-Higgs equations on S2 and T2 are identified as satisfying vortex equations. Introduces new algebraic structures for relational groupoids and proves a reduction theorem.
problem Developing algebraic tools for relational groupoids.
method Introduces relational groupoids and convolution algebras, provides examples, and proves a reduction theorem.
result Establishes a reduction theorem recovering the usual convolution of Lie groupoids.
Study stability of surfaces in spacetimes, proving new estimates and theorems.
problem Stability of surfaces in spacetime and their applications.
method Variational techniques, Christodoulou-Yau estimate, Cohn-Vossen inequality, global theorem, capillary stability, area inequality, diameter estimate.
result Established new estimates and theorems for stable surfaces in spacetime.
In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, γ). This is the space consisting of triples, (F_{g,n}, φ, f), where F_{g,n} is a Riemann surface of genus g and n-boundary components, φis a parameterization of the boundary, and f : F_{g,n} \to X is a contin…
Study improves understanding of Ricci curvature in manifolds.
problem Understanding Ricci curvature in manifolds with specific assumptions.
method Exploring m-intermediate Ricci curvature and proving comparison theorems.
result Stable weighted slicing in manifolds with non-negative m-intermediate Ricci curvature has almost non-negative Ricci curvature.
We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of the Kirwan convexity theorem, the Meyer-Marsden-Weinstein symplectic reduction theorem, and the Duistermaat-Heckman theorem in this context.
The paper establishes a correspondence between Higgs torsors and connections on curves.
problem Establishing a correspondence between Higgs torsors and connections on curves.
method Introduced a stability condition on filtered Stokes local systems and used it to prove a one-to-one correspondence.
result One-to-one correspondence between stable meromorphic parahoric Higgs torsors and stable meromorphic parahoric connections.
It follows from a theorem of Gromov that the stable systolic category of a closed manifold is bounded from below by the rational cup-length of the manifold. In the paper we study the inequality in the opposite direction. In particular, combining our results with Gromov's theorem, we prove the equality of stable systoli…
The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ibaˊn~ez et al. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduct…
In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except 15,31,63,127 is obtained. It is proved that for n>127 in the stable homotopy group o…
New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.
problem Non-Euclidean norms in binary classification problems.
method Black-box reduction method using uniformly convex regularizers.
result Achieves optimal statistical risk bounds on excess risk for non-Euclidean norms.
The paper explores polysymplectic structures and their reductions in field theories.
problem Invariance of Lagrangian and Hamiltonian field theories under symmetry groups.
method Application of polysymplectic reduction theorem for both Lagrangian and Hamiltonian field equations.
result Identification and relation of polysymplectic structures through Routhian function and Legendre transformation.
Contradicts claims about Poincaré complexes and homology manifolds.
problem Claims about Poincaré complexes and homology manifolds are contradicted.
method Constructs a Poincaré complex with specific properties to contradict the claims.
result A Poincaré complex with vanishing periodic total surgery obstruction is not necessarily homotopy equivalent to a homology manifold.
The paper proves criticality criteria and spectral splitting theorems for manifolds with Ricci bounds.
problem Understanding criticality and splitting theorems for manifolds with spectral Ricci bounds.
method Proving criticality criteria and spectral splitting theorems for manifolds with more than one end and spectral Ricci bounds.
result New insights into Li-Wang's theory and applications to stable and δ-stable minimal hypersurfaces.
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
problem Understanding the homotopy types of free racks and quandles.
method Proved analogs of Milnor's theorem for racks and quandles and their pointed variants.
result Identified the homotopy types of free racks and quandles on spaces of generators.
Paper studies curvature of stable surfaces meeting at a common boundary.
problem Stable multiple junction surfaces and their curvature estimates.
method Derived Lp estimate of curvature for stable multiple junction surfaces. result Bernstein Theorem holds for stable multiple junction surfaces in certain cases.