Proves planarity and convexity for ancient solutions of mean curvature flow.
problem Ancient solutions of mean curvature flow in higher codimension.
method Parabolically scale-invariant variation of planarity estimate, convexity proof for pinched solutions.
result Characterizes certain pinched complete ancient solutions and shrinkers in higher codimension.
Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.
problem Proving Hamilton's pinching conjecture for three-manifolds.
method Ricci flow with scale-invariant curvature decay and pinching preservation.
result Hamilton's pinching conjecture is proven without additional hypotheses.
We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension n≥3 has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists …
The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.
problem Proving a sharp mean value inequality for non-negative superharmonic functions.
method Develops a new sharp mean value inequality and an explicit formula for weighted scalar curvature.
result The new inequality removes the radius restriction of Schoen-Yau's result and provides an explicit formula for integral of weighted scalar curvature.
Scales attention for long contexts in LLMs.
problem Development of attention mechanisms for long context inference.
method Scale-invariant total attention and sparsity conditions, with a position-dependent transformation of logits.
result Scale-invariant attention scheme improves validation loss and long-context retrieval.
Three training regimes found for scale-invariant neural networks on the sphere.
problem Training scale-invariant neural networks on the sphere with varying effective learning rate.
method Investigated three regimes of training: convergence, chaotic equilibrium, and divergence.
result Discovered three distinct training regimes with unique characteristics.
In theoretical analysis of deep learning, discovering which features of deep learning lead to good performance is an important task. In this paper, using the framework for analyzing the generalization error developed in Suzuki (2018), we derive a fast learning rate for deep neural networks with more general activation …
Proves uniqueness of Ricci flow with scaling invariant estimates.
problem Proving uniqueness of Ricci flow with scaling invariant curvature bound.
method Solving Ricci-harmonic map heat flow in unbounded curvature background.
result Complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique in dimension three.
Critical points of scale-invariant curvature energies in 4D are analytic.
problem Analyzing critical points of curvature energies in 4D manifolds.
method Applying Noether's theorem to identify conservation laws and lower order elliptic system of PDEs, then using integrability by compensation and interpolation theory.
result Critical points of scale-invariant curvature energies in 4D are analytic.
New bounds on self-normalized martingales improve online linear regression performance.
problem Improving regret bounds in online linear regression.
method Characterizing scale-invariant bounds on self-normalized martingales.
result For d=1, O(logT) doubly-uniform regret is possible; for d>1, sublinear doubly-uniform regret is impossible. We consider a variant of online convex optimization in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of th…
AdamP optimizes momentum-based optimizers for scale-invariant weights, improving model performance.
problem Premature decay of effective step sizes in momentum-based optimizers for scale-invariant weights.
method Proposes SGDP and AdamP to eliminate the radial component at each optimizer step, preserving convergence properties.
result Uniform gains across multiple benchmarks, improving model performance.
Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
problem Conditions for Willmore surfaces to have finite ends or finite total curvature.
method Analyzes scale-invariant second fundamental form near infinity.
result Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
Study pinched self-dual Weyl curvature in compact 4-manifolds.
problem Analyzing compact 4-manifolds with specific curvature properties.
method Examining harmonic self-dual Weyl curvature under pinching conditions.
result Characterized compact 4-manifolds with pinched self-dual Weyl curvature.
We refine a metric bunching estimate for pinched manifolds.
problem Improving an unstable bunching estimate for pinched metrics.
method Compact Riemannian manifolds with pointwise negatively pinched curvature tensor.
result Improved unstable bunching estimate.
Paper proves Hamilton's pinching theorem using mean curvature flow.
problem Hamilton's pinching theorem in extrinsic geometry.
method Mean curvature flow approach.
result Proof of Hamilton's pinching theorem.
Power iteration has been generalized to solve many interesting problems in machine learning and statistics. Despite its striking success, theoretical understanding of when and how such an algorithm enjoys good convergence property is limited. In this work, we introduce a new class of optimization problems called scale …
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
The paper pinches curvature in expanding Ricci solitons.
problem Curvature pinching in expanding Ricci solitons.
method Hamilton-Ivey type curvature pinching estimates.
result Three-dimensional Hamilton-Ivey type curvature pinching theorem.
This paper extends 3D results to higher dimensions, proving compactness for PIC1 pinched manifolds.
problem Proving compactness for higher-dimensional manifolds with specific curvature conditions.
method Constructing Ricci flows for non-compact PIC1 pinched manifolds to prove compactness.
result Proves that PIC1 pinched manifolds of non-negative complex sectional curvature must be flat or compact.
Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when w…
Paper proves pinching theorem for minimal surfaces in spheres.
problem Pinching rigidity of minimal surfaces in spheres.
method Simon conjecture and Simons-type integral inequalities.
result New proof of pinching theorem for minimal surfaces in spheres.
Alternative proof of flatness for Ricci-pinched 3-manifolds.
problem Hamilton's pinching conjecture for 3-manifolds.
method Nonlinear potential theory with superquadratic volume growth.
result Flatness of Ricci-pinched 3-manifolds with superquadratic volume growth.
Study shows how near crushing singularities, Kasner-like regions can exist.
problem Understanding spatial volume densities near crushing singularities.
method Relates existence of Kasner-like regions to asymptotics of spatial volume densities under scale-invariant curvature bounds.
result Kasner-like regions can exist near crushing singularities under certain curvature conditions.
New learning dynamics achieve fast convergence in games without needing to know utility scales.
problem Fast convergence guarantees in learning games require prior knowledge of utility scales.
method Developed scale-free and scale-invariant learning dynamics using optimistic follow-the-regularized-leader with adaptive learning rates and clipping techniques.
result Achieved fast convergence rates to Nash and correlated equilibria without prior utility scale knowledge.
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
The study pinches the rigidity of self-shrinking surfaces in mean curvature flow.
problem Rigidity of self-shrinking hypersurfaces in mean curvature flow.
method Spectral upper-pinching theorem and weighted Poincaré estimate.
result Self-shrinking hypersurfaces are restricted to specific forms under certain conditions.
Adam performs better with equal momentum parameters, revealing a gradient scale invariance principle.
problem Why Adam performs better with β1=β2. method Formalized gradient scale invariance and proved it for Adam with equal β1 and β2. result Adam becomes gradient scale invariant of first order if and only if β1=β2. In this article, we generalize the classical Bochner-Weitzenböck theorem for manifolds satisfying an integral pinching on the curvature. We obtain the vanishing of Betti numbers under integral pinching assumptions on the curvature, and characterize the equality case. In particular, we reprove and extend to higher degre…
Study pinched submanifolds in space forms, proving rigidity results.
problem Pinching condition on submanifolds in space forms.
method Analyzing geometry and topology under pinching conditions.
result Pinching condition forces homology to vanish or determines submanifolds up to congruence.
Harmonic maps between pinched Hadamard surfaces are quasi-conformal.
problem Characterizing harmonic maps between Hadamard surfaces.
method Proving harmonic quasi-isometries are quasi-conformal diffeomorphisms.
result Harmonic quasi-isometries of pinched Hadamard surfaces are injective.
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
Study on surfaces pinched by curvature in space forms converging under specific conditions.
problem Investigating convergence of surfaces pinched by curvature in space forms.
method Proving convergence theorems for surfaces pinched by normal curvature in 4-dimensional space forms.
result Generalizes Baker-Nguyen's convergence theorem for surfaces pinched by curvature.
Study pinched submanifolds, proving homology vanishing results.
problem Understanding the geometry and topology of pinched submanifolds.
method Investigates submanifolds with a pinching condition on extrinsic invariants.
result Homology vanishing theorems for pinched submanifolds.
Paper finds critical metrics with pinched curvature are geodesic balls.
problem Identifying critical metrics with specific curvature constraints.
method Proved isometry to geodesic balls in S^n and provided conditions for the gradient of the potential function.
result Critical metrics with pinched curvature are isometric to geodesic balls in S^n.
Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.
problem Understanding Weyl curvature pinching on 4-manifolds.
method Analyzing harmonic and pinched self-dual Weyl curvature, proving anti-self-duality.
result Proves anti-self-duality for compact 4-manifolds with pinched self-dual Weyl curvature.
New restrictions found on 4-manifolds with pinched curvature.
problem Restrictions on Euler characteristic and signature of 4-manifolds with pinched curvature.
method Proved new restrictions on Euler characteristic and signature of oriented 4-manifolds with pinched sectional curvature.
result Simply connected 4-manifolds with δ≤sec≤1 are homeomorphic to S4 or CP2. SAM improves deep learning tasks by promoting balancedness, reducing outlier impact.
problem Improving generalization in deep learning tasks, especially with scale-invariant problems.
method Introduces balancedness as a new concept to depict global behaviors of SAM, focusing on the difference between squared norms of two variables.
result SAM promotes balancedness and is data-responsive, outperforming SGD in outlier scenarios.
In this paper, the pinching problems of complete λ-hypersurfaces in a Euclidean space Rn+1 are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete λ-hypersurfaces in a Euclidean space Rn+1.
Classifies self-shrinkers in arbitrary dimensions under specific curvature conditions.
problem Classifying self-shrinkers with quadratic pinching conditions.
method Purely elliptic approach using weighted parabolicity, tailored to self-shrinkers.
result Generalized self-shrinking cylinders as solutions under quadratic pinching.
We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. We show that if the pinching condion is strong enough with a dependance on the norm of the second foundamental form, then the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.
The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.
problem Quantifying rigidity in manifolds with nonnegative Ricci curvature.
method Investigates pinching of Colding's monotone functionals and constructs k-splitting functions. result Quantitative control of splitting functions by pinching at independent points controls the distance to the nearest cone.
We develop a scale-invariant truncated Lévy (STL) process to describe physical systems characterized by correlated stochastic variables. The STL process exhibits Lévy stability for the probability density, and hence shows scaling properties (as observed in empirical data); it has the advantage that all moments are fini…
The paper constructs a new metric on Kähler manifolds.
problem Finding metrics with specific curvature properties on Kähler manifolds.
method Constructing an almost negatively 1/4-pinched Riemannian metric.
result First known examples of not locally symmetric Kähler manifolds with this metric.
Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.