Tackles variational problems with curvature and fractional Brezis-Nirenberg equations, overcoming lack of compactness.
problem Variational problems with curvature and fractional Brezis-Nirenberg equations.
method Refined techniques and Lyapunov-Schmidt method for ODE systems, blow-up analysis for elliptic equations.
result Existence and multiplicity of solutions with qualitative properties.
The classical Brody's theorem asserts the equivalence between two notions of hyperbolicity for compact complex spaces, one named after Kobayashi and one expressed in terms of lack of non constant holomorphic entire functions (compactness is only used to prove the harder implication). We extend this theorem to Deligne-M…
Lisa Jeffrey and Frances Kirwan developed an integration theory for symplectic reductions. That is, given a symplectic manifold with symplectic group action, they developed a way of pulling the integration of forms on the reduction back to an integration of group-equivariant forms on the original space. We seek an anal…
For a compact surface X0, Thurston introduced a compactification of its Teichmüller space T(X0) by completing it with a boundary PML(X0) consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space T(X0) of a noncompact Ri…
For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks fa…
Study compact PL 4-manifolds with special handle decompositions.
problem Existence of special handlebody decompositions for simply-connected closed PL 4-manifolds.
method Investigate colored triangulations inducing handle decompositions without 1-handles or 1- and 3-handles.
result Detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary that admit such decompositions.
New example shows non-compact manifolds can lack Lp-Calderón-Zygmund inequalities.
problem Exploring Lp-Calderón-Zygmund inequalities on non-compact manifolds. method Developed a concrete example using local deformations of metrics.
result Found a non-compact manifold without Lp-Calderón-Zygmund inequalities. Study eigenvalues of CROSS spaces with various metrics.
problem Eigenvalue of Laplace-Beltrami operator on CROSSes.
method Explicit formulae for the first eigenvalue of CROSSes with homogeneous metrics.
result Homogeneous metrics on CROSSes are isospectral if and only if they are isometric.
Minimal annuli constructed in PSL2 via variational method.
problem Constructing minimal annuli in a non-symmetric 3-manifold.
method Variational method, foliations by minimal surfaces, limit of compact minimal annuli.
result Existence of complete, embedded minimal annuli asymptotic to vertical planes.
Study shows how compact shapes can be rigidly mapped into complete manifolds.
problem Rigidity of isometric immersions in complete manifolds.
method Local quantitative rigidity estimates, reduced to Euclidean setting.
result Subsequence of immersions converges to an isometric immersion.
The abstract discusses compactness of manifolds with pinched Ricci curvature.
problem Prove that a complete Riemannian manifold with positively pinched Ricci curvature is compact.
method Detailed alternate proof using quasi-conformal maps and mean curvature flow.
result Provides a proof of Hamilton's result on compactness of convex hypersurfaces.
Study shows hypercomplex twistor spaces lack divisors and special metrics.
problem Characterizing properties of hypercomplex twistor spaces.
method Analyzing the general fiber's lack of divisors and curves, proving trascendental degree and absence of special metrics.
result Proves hypercomplex twistor spaces have no divisors, curves, Kähler, or pluriclosed metrics.
Paper proves non-zero generalization boost for equivariant models.
problem Improving generalization in machine learning models.
method Analyzes simplest case of linear models, focusing on invariant/equivariant properties.
result First provably non-zero improvement in generalization for invariant/equivariant models.
Localized Multidirectional Correction improves non-refusal target-response behavior in foundation models.
problem Controlled post-training refusal suppression in routed MoE and hybrid-MoE foundation models.
method Introduce Localized Multidirectional Correction (LoMC), a support-gated intervention framework.
result Substantially improves non-refusal target-response behavior while maintaining general capability under a compact intervention footprint.
Defines a functional for Riemann surfaces, proving a unique solution.
problem Lack of regularity and compactness in Donaldson functional.
method Develops analytical tools to apply variational approach to functional.
result Proves existence of a unique critical point and solution.
New formula for Lichnerowicz Laplacian on homogeneous spaces.
problem Finding new Einstein metrics on homogeneous spaces.
method Using Casimir operators to derive a new formula for the Lichnerowicz Laplacian.
result Derives many new Einstein metrics stable in the Einstein-Hilbert sense.
Toolbox provides datasets and metrics for state representation learning.
problem Lack of standard evaluation datasets, metrics and tasks for state representation learning.
method Provides a set of environments, data generators, robotic control tasks, metrics and tools.
result Facilitates iterative state representation learning and evaluation in reinforcement learning settings.
The study extends conserved quantities theory to non-compact boundary initial data sets.
problem Extending conserved quantities theory to initial data sets with non-compact boundaries.
method Analysis of scalar curvature and mean curvature in the interior and boundary.
result Rigidity/flexibility phenomena in positive mass theorems and Penrose inequalities.
Constructs metrics with Q-curvature on manifolds with singularities.
problem Positive singular Q-curvature problem on compact manifolds with punctures.
method One-parameter family solutions, perturbation methods, gluing techniques, linearized operator mapping properties.
result One-parameter family of solutions constructed for positive Q-curvature.
Paper presents a method to summarize HMC samples for neural networks, providing meaningful uncertainty estimates.
problem Lack of interpretable summary statistics for HMC samples in neural networks due to permutation symmetry.
method Introducing a transpositions metric to quantify permutations and using rebasin method to summarize HMC samples.
result Compact representation of HMC samples provides meaningful uncertainty estimates for each weight in a neural network.
The paper generalizes trisections to compact PL 4-manifolds with boundary and introduces gem-induced trisections.
problem Generalizing trisections to compact PL 4-manifolds with boundary and any 5-colored graph encoding simply-connected 4-manifolds.
method Introducing gem-induced trisections and analyzing their properties for compact PL 4-manifolds with and without boundary.
result Conditions for gem-induced trisections to realize the G-trisection genus and direct determination of the genus from the graph.
Method converts neural networks to function space for better uncertainty quantification.
problem Lack of uncertainty estimates and difficulty in incorporating new data in deep neural networks.
method Dual parameterization to convert from weight space to function space, enabling sparse representation.
result Compact and principled way to capture uncertainty and incorporate new data.
Maximal volume entropy rigidity extended to RCD spaces.
problem Volume entropy rigidity for RCD spaces with bounded Ricci curvature.
method Extending known results to RCD spaces, dealing with lack of smooth structure.
result Rigidity result for compact RCD spaces with specific curvature bounds.
We prove Zimmer's conjecture for C2 actions by finite-index subgroups of SL(m,Z) provided m>3. The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in SL(m,R) but new ideas are needed to overcome the lack of compactn…
LDF combines neural networks with probabilistic models for data fusion.
problem Combining limited primary data with readily available auxiliary data.
method Neural networks as conjugate mappings of auxiliary data for posterior analysis.
result Efficient inference and compact latent variable posterior distributions.
Study heat content in sub-Riemannian manifolds, obtaining asymptotic expansion.
problem Heat content in sub-Riemannian manifolds with non-characteristic domains.
method Fourth-order asymptotic expansion, combining rough boundary temperature and stochastic completeness.
result Obtained a fourth-order asymptotic expansion for relative heat content.
ConfHit provides valid guarantees for generative models without oracle access.
problem Reliable guarantees for novel candidate generation in generative models.
method Formalizes certification and refinement of generated sets, leveraging weighted exchangeability and density-ratio weighted conformal p-values.
result Consistently delivers valid coverage guarantees and compact certified sets across various generative tasks.
In this article, we examine the behavior of the Riemannian and Hermitian curvature tensors of a Hermitian metric, when one of the curvature tensors obeys all the symmetry conditions of the curvature tensor of a Kähler metric. We will call such metrics G-Kähler-like or Kähler-like, for lack of better terminologies. Such…
Study shows most hyperbolic knot complements lack hidden symmetries.
problem Proving most knot complements lack hidden symmetries.
method Using rational functions on varieties associated to a link.
result Most knot complements lack hidden symmetries.
A geometrical interpretation of the G-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained
On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler setting, one can seek to understand extensions of these characterizations to genera…
The article confirms a conjecture for solvmanifolds with complex commutator.
problem Confirming a conjecture about compact Hermitian manifolds with constant holomorphic sectional curvature.
method Analyzing solvmanifolds with complex commutator, extending results on nilmanifolds.
result The conjecture is confirmed for all solvmanifolds with complex commutator.
New theory for area of Legendrian surfaces, proving smoothness and variational results.
problem Understanding the area of Legendrian surfaces under constraints.
method Introducing PHSLVs, proving sequential compactness, regularity, and variational results.
result Generalized regularity theory for Legendrian surfaces, achieving variational minima.
Extends complex manifold structures to line bundles, revealing new projective manifolds.
problem Generalizing scalar-valued holomorphic structures to line bundles.
method Study of holomorphic p-contact and s-symplectic structures on complex manifolds with line bundles. result Holomorphic p-contact and s-symplectic manifolds can be projective. In many fields observations are performed irregularly along time, due to either measurement limitations or lack of a constant immanent rate. While discrete-time Markov models (as Dynamic Bayesian Networks) introduce either inefficient computation or an information loss to reasoning about such processes, continuous-time…
Geometric approach to majorizing measures for polyhedra and general compact objects.
problem Understanding the relationship between a space and its convex hull in geometric measure theory.
method Geometric approach using covering number relationships and volume ratios.
result Established a method to evaluate covering number and volume ratios for various spaces.
Improved RTM uses integer weights to reduce computation and increase interpretability.
problem Lack of interpretability in nonlinear regression models.
method Integer weighted RTM clauses, combined with a novel learning scheme.
result Significantly reduced computation cost with improved accuracy.
In this paper, we study the prescribed Q-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the Q-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topolo…
Distance metric learning (DML), which learns a distance metric from labeled "similar" and "dissimilar" data pairs, is widely utilized. Recently, several works investigate orthogonality-promoting regularization (OPR), which encourages the projection vectors in DML to be close to being orthogonal, to achieve three effect…
Let A⊂R2 be a smooth doubly connected domain. We consider the Dirichlet energy E(u)=∫A∣∇u∣2, where u:A→C, and look for critical points of this energy with prescribed modulus ∣u∣=1 on ∂A and with prescribed degrees on the two connected components o…
The paper solves the Nirenberg problem on high-dimensional half spheres with pinching conditions.
problem Finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on half spheres.
method Variational approach with pseudogradient and Morse theory to handle non-compactness.
result Existence results for the Nirenberg problem under various pinching conditions.
Study the energy distribution of harmonic 1-forms on Riemann surfaces with a short geodesic.
problem Energy distribution of harmonic 1-forms on Riemann surfaces with a short geodesic.
method Analyzes the Jacobian torus of Riemann surfaces with a short geodesic, considering both separating and nonseparating cases.
result Estimates the energy distribution in terms of geometric data of the surface.
Maximal representations in infinite dimensional Hermitian spaces are studied with boundary maps.
problem Characterizing maximal representations in infinite dimensional Hermitian symmetric spaces.
method Definition of Toledo number, study of boundary maps, geometric constructions.
result Existence and non-existence conditions for maximal representations.
Randomized neural networks improve function approximation on manifolds.
problem Slow learning in neural networks on manifolds.
method Random vector functional link networks for function approximation.
result Theoretical guarantees for function approximation on manifolds with high probability.
Study on minimizing network energy in R^d, introducing degenerate elastic networks.
problem Minimizing network energy in R^d with constraints on curves and junctions.
method Characterizing limits of sequences of networks bounded in energy, providing explicit representation of the relaxed problem.
result Explicit representation of degenerate elastic networks, a new concept involving only given class properties.
A new method to rescale ReLU neural networks based on path-lifting.
problem Lack of principled ways to leverage rescaling symmetries in ReLU neural networks.
method Introduces a geometrically motivated criterion to rescale neural network parameters, aligning a kernel in the path-lifting space with a chosen reference.
result Proposed method can speed up training and aligns a kernel in the path-lifting space with a chosen reference.
Ensemble decoders to capture latent space topology in deep generative models.
problem Topological mismatch between latent space geometry and data manifolds.
method Using ensembles of decoders to compute geodesics on the expected manifold.
result Ensemble approach provides a simple and reliable way to capture model uncertainty in latent space.
This work addresses the convergence of SGD's final iterate without restrictive assumptions.
problem Prove optimal convergence rate of SGD's final iterate without compact domains or bounded noise.
method Unified proof for general domains, composite objectives, non-Euclidean norms, etc.
result First unified convergence rates in expectation and high probability.