Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.
problem Contractibility of spaces of invariant positive scalar curvature metrics.
method Combining equivariant Morse theory with conformal deformations and local flexibility properties.
result Spaces of invariant positive scalar curvature metrics are contractible.
We present a Donaldson-Witten type field theory in eight dimensions on manifolds with Spin(7) holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi…
We developed a perturbation model for affine gravity theories.
problem Cosmological perturbations in theories without metric.
method Segregated perturbations into symmetric and antisymmetric components, decomposing into irreducible elements.
result Fully addressed gauge freedom in affine gravity theories.
Develops a Kaluza-Klein theory in affine spaces without metric.
problem Formalizes a geometric theory of electromagnetic fields in affine spaces.
method Formulates dimensional reduction using principal fiber bundles and Ehresmann connections.
result Shows that non-integrability of horizontal distribution implies nontrivial electromagnetic fields.
The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
It is shown that there are infinitely many compact orientable smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2 chi > 3 |tau|. The examples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems fro…
Existence of Ricci flat metric on Kummer K3 surface proven.
problem Proving existence of Ricci flat metric on Kummer K3 surface.
method General strategy of Donaldson's gluing construction, compact elliptic theory on usual Hölder and Sobolev spaces, explicit isometry to Gibbons-Hawking ansatz.
result Existence of a Ricci flat metric on the Kummer K3 surface.
Global rigidity theorem for metrics on Σ×S1 without conjugate points.
problem Classifying metrics without conjugate points on Σ×S1.
method Two independent proofs: Busemann functions and Riccati equation, curvature operator analysis.
result Metrics on Σ×S1 without conjugate points are Riemannian products.
New 4-manifolds found without certain Einstein metrics, using Seiberg-Witten theory.
problem Constructing 4-manifolds without specific Einstein metrics.
method Using Seiberg-Witten theory and constructing solutions on noncompact manifolds.
result Infinitely many examples of 4-manifolds without cusped asymptotically hyperbolic Einstein metrics.
The paper develops glueing theory for topological spaces and applies it to compactifications.
problem Developing a theory for gluing topological spaces and its applications.
method Developed the theory of Artin-Wraith glueings for topological spaces and applied it to compactifications.
result The space of ends of coarse equivalent metric spaces are the same.
Disease classification is a crucial element of biomedical research. Recent studies have demonstrated that machine learning techniques, such as Support Vector Machine (SVM) modeling, produce similar or improved predictive capabilities in comparison to the traditional method of Logistic Regression. In addition, it has be…
New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
We develop methods to efficiently approximate data in metric spaces without additional assumptions.
problem Efficiently approximating data in metric spaces without imposing structural assumptions.
method Identify discrete modulus of continuity, investigate consistency, propose algorithm, and develop approximation theory.
result Consistent approximation of data in metric spaces without structural assumptions.
The paper solves linearized Ricci curvature equations on compact manifolds.
problem Linear analysis of Ricci curvature equations on general compact Riemannian manifolds.
method Established solvability and uniqueness conditions using cohomology of a cochain complex.
result Vanishing theorems for cohomology under geometric assumptions on boundary and error term.
New proof of Riemannian Penrose Inequality for manifolds with corners
problem Riemannian Penrose Inequality for asymptotically flat manifolds with corners
method Unified argument based on approximate monotonicity
result Positive Mass Theorem and Riemannian Penrose Inequality
Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this end, we study analytic paths of metrics that induce isospectral Laplace-Beltram…
In a previous work, the authors introduced the notion of `coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss-Bonnet formulas on coherent tangent bundles on 22-dimensional manifolds were proven, and several applications to…
In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same…
We study solutions of the Bogomolny equation on R^2\times S^1$ with prescribed singularities. We show that Nahm transform establishes a one-to-one correspondence between such solutions and solutions of the Hitchin equations on a punctured cylinder with the eigenvalues of the Higgs field growing at infinity in a particu…
This paper extends geometric structure theory to infinite type structures.
problem Calculating characteristic class relations in complex Cartan geometries.
method Improves representation theory for infinite type structures.
result Direct calculation of characteristic class relations from structure group representation.
Study shows no hidden symmetries in specific spacetime metrics.
problem Demonstrating the absence of Killing tensors in Koutras-McIntosh spacetimes.
method Geometric theory of overdetermined PDEs and Cartan prolongation-projection method.
result No Killing tensors of low degrees in Wils metrics and generic pp-waves.
We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Ise…
New theory approximates functions between metric spaces using random probability measures.
problem Building universal functions approximators between arbitrary metric spaces.
method Using elementary functions between Euclidean spaces, randomization to output discrete probability measures over target space.
result Very general qualitative guarantees and quantitative guarantees for Hölder-like maps.
We propose some axioms for hierarchical clustering of probability measures and investigate their ramifications. The basic idea is to let the user stipulate the clusters for some elementary measures. This is done without the need of any notion of metric, similarity or dissimilarity. Our main results then show that for e…
New Ricci flow solutions found with rotational symmetry and cone-like singularities.
problem Finding Ricci flow solutions with specific symmetry and singularity properties.
method Rotationally symmetric Ricci flow with scaling-invariant curvature bounds, using approximation method.
result Complete Ricci flow solution with cone-like singularity at the origin.
ZDP detects drift in large language models without labels, proving key theorems and metrics.
problem Detecting drift in large language models without task labels or output evaluations.
method Zero-Direction Probing (ZDP) framework based on null directions of transformer activations, proving theoretical guarantees.
result Proves the Variance--Leak Theorem, Fisher Null-Conservation, Rank--Leak bound, and logarithmic-regret guarantee.
Develops Lefschetz theory for noncompact manifolds.
problem Lefschetz fixed-point theory for noncompact manifolds.
method Introduces uniform bounded cohomology and develops obstruction theory.
result Uniform Lefschetz class vanishes if and only if map is homotopic to a strongly fixed-point free map.
The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.
problem Transforming Vaisman metrics into Kähler-Einstein structures.
method Using the transverse Kähler-Ricci flow on the canonical foliation of a closed Vaisman manifold.
result Direct proof of short time existence of transverse Kähler-Ricci flow on Vaisman manifolds.
Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.
problem Understanding Hilbert geometry over general valued fields and their limits.
method Developed a theory of Hilbert geometry over general ordered valued fields, proving ultralimit results.
result Ultralimit of rescaled real Hilbert geometries is isometric to a non-Archimedean Hilbert metric space.
This paper classifies Calabi-Yau manifolds near cones.
problem Classifying Calabi-Yau manifolds near cones.
method Complete classification of smooth complete Calabi-Yau manifolds asymptotic to a given cone.
result Classification of all smooth complete Calabi-Yau manifolds near a given cone.
We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation theory to prove the existence of a infinite number of conformal classes with at leas…
Modeling wormhole creation without singularities in relativity.
problem Creating wormholes without singularities in classical relativity.
method Topological surgery and Morse theory to construct a nonsingular wormhole.
result Wormholes can be created nonsingularly in classical relativity.
Metric measure boundary vanishes on certain spaces without boundary.
problem Existence of infinite geodesics on Alexandrov spaces without boundary.
method Solving conjecture by showing metric measure boundary vanishes on mRCD(K,N) spaces. result Metric measure boundary vanishes on mRCD(K,N) spaces without boundary. New method identifies latent relationships in deep models without additional constraints.
problem Latent representations in deep latent variable models are not statistically identifiable.
method Identifies relationships between latent variables (distances, angles, volumes) under mild model conditions.
result Empirically demonstrates more reliable latent distances without additional labeled data.
Reconstructs Riemannian geometry from diffusion properties.
problem Recovering Riemannian geometry from diffusion data.
method Intrinsic reconstruction from diffusion semigroup and calculus.
result Reveals Riemannian structure from diffusion properties.
IRT metrics improve model evaluation by assessing latent characteristics.
problem Limitations of classic metrics like precision and F1.
method Introducing psychometric metrics like Item Response Theory (IRT).
result IRT complements classical metrics, offering new insights.
A geometric theory explains loss functions for robust representation learning.
problem Treats robustness, domain adaptation, and sensor drift as separate literatures.
method Estimates covariance Sigma_task and uses it to pin Jacobian penalties.
result Proves optimality and necessity of range coverage for penalty matrices.
The Nirenberg problem yields multiple conformal metrics for a given scalar curvature function.
problem Finding multiple conformal metrics with a given scalar curvature on spheres.
method Morse theoretical methods and counting index formulae, leveraging subcritical approximation and blowing-up solutions.
result Arbitrarily many metrics can be found that are conformally equivalent to the standard sphere and have the desired scalar curvature.
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
Neural models price financial options without assuming underlying price forms.
problem Pricing financial options under flexible price processes.
method Apply neural SDEs as universal approximators, use Wasserstein distance for training.
result Error in option prices bounded by Wasserstein distance used for training.
We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any nondegeneracy assumptions except that the critical locus must have only finitely many connected components.
Defines operations in non-Archimedean metrics theory.
problem No specific problem stated; focuses on theory development.
method Defines operations using non-Archimedean metrics theory.
result Establishes the envelope conjecture holds in the theory.
We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kähler surfaces. This is used to prove that for any scalar-flat Kähler ALE surface, all small deformations of complex structure also admit scalar-flat Kähler ALE metrics. A local moduli space of scalar-flat Kähler ALE metrics is then constr…
Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M. Under suitable conditions, we show that they are almost conformally isometric in an Lp sense. Assume also that M carries a Riemannian metric with parallel Ricci curvature. Then an equation of Ricci type, is in …
This paper proposes an active metric learning method for clustering with pairwise constraints.
problem Clustering with pairwise constraints and improving clustering performance.
method Active metric learning method that queries informative instance pairs and updates the learned metric sequentially.
result The proposed method enhances clustering performance and provides a tighter error bound.
Derives curvature conditions for spatial isotropy without field equations.
problem Conditions for spatial isotropy in cosmological models.
method Geometric derivation of curvature conditions independent of field equations.
result Local isometry between space and Robertson-Walker space-time.
DFNNs predict non-Euclidean responses from Euclidean predictors.
problem Regression with non-Euclidean responses.
method Deep Fréchet neural networks (DFNNs) approximating conditional Fréchet means.
result DFNNs consistently outperform existing methods in empirical studies.
New convergence rates for shuffling gradient methods without strong convexity.
problem Theoretical gap between shuffling gradient methods' empirical success and established convergence rates.
method Proved last-iterate convergence rates for shuffling gradient methods using function value gap.
result First last-iterate convergence rates for shuffling gradient methods without strong convexity.