Proves mean convex neighborhood conjecture for ancient flows near singularities.
problem Proving mean convex neighborhood conjecture for mean curvature flow near singularities.
method General classification of ancient low entropy flows and mean curvature flow through singularities.
result Proves mean convex neighborhood conjecture for ancient flows near singularities.
Resolves conjecture on cylindrical mean curvature flows in all dimensions.
problem Mean Convex Neighborhood Conjecture for cylindrical singularities.
method Complete classification of ancient, asymptotically cylindrical flows; refined asymptotic analysis; leading mode condition; induction over thresholds.
result Establishes mean-convex neighborhood for cylindrical singularities; provides local models and canonical families.
We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isol…
We use Ilmanen's elliptic regularization to prove that for an initially smooth mean convex hypersurface in Euclidean n-space moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. Previously this was known only (i) for n < 7, and (ii) for arbitrary n up to the…
Study of singularities in mean curvature flow with focus on S3imesR.
problem Understanding singularities in mean curvature flow.
method Detailed analysis of singularities modeled on S3imesR, using normal form transformations and rescaled MCF. result Proves mean convexity and singularity isolation in a small neighborhood, conjectures singularity formation in entire neighborhood.
Study rigidity of minimal disks in specific 3-manifolds.
problem Rigidity of free boundary minimal disks in mean convex three-manifolds.
method Assuming strict stability, prove isometric neighborhoods using modified Hawking mass.
result Prove rigidity of minimal disks in specific 3-manifolds.
Study resolves flow through cylindrical singularities, proving nonfattening.
problem Analyzing free boundary flow through cylindrical singularities.
method Foundational results for free boundary Brakke flows and classification of ancient flows.
result Proves all cylindrical singularities have a mean-convex neighborhood, leading to well-posed flow.
The study examines mass drop and multiplicity in mean curvature flow.
problem Analyzing mass drop and multiplicity in mean curvature flow.
method Defined Brakke flow with variational inequality, proved mass drop conditions.
result Mass drop and multiplicity one conjecture are equivalent for Brakke flows.
New 1-parameter family of ovals identified in 4d Ricci flow classification.
problem Classifying κ-solutions in 4d Ricci flow. method Introducing conjectures and constructing new examples.
result Established canonical neighborhood theorem for 4d Ricci flow.
Fixed points of mean section operators found in convex bodies.
problem Characterizing fixed points of mean section operators in convex bodies.
method Characterization of rotation equivariant operators using spherical Laplacian mass distribution, and application of Minkowski valuations.
result Euclidean balls are the only fixed points of mean section operators in a C2 neighborhood of the unit ball. Almost all local minima in neural networks are strongly convex.
problem The prevalence of strongly convex neighborhoods around local minima in neural network optimization landscapes.
method Rigorous analysis of shallow neural networks with analytic activation functions, dividing parameter space into efficient and redundant domains.
result For shallow neural networks on the efficient domain, almost all local minima are strongly convex.
The study proves all limit flows are self-similar under specific conditions.
problem The selfsimilarity of limit flows in mean curvature flow.
method Proof of selfsimilarity conditions for mean convex surfaces and neck singularities.
result All limit flows are self-similar if and only if there are finitely many spherical singularities.
The study proves a neighborhood theorem for mean curvature flow in higher dimensions.
problem Proving a canonical neighborhood theorem for mean curvature flow in higher dimensions.
method Proved a canonical neighborhood theorem for mean curvature flow of compact submanifolds in RN with a pinching condition. result Proved a canonical neighborhood theorem for mean curvature flow in dimensions n≥5. This paper studies mean curvature flows near cylindrical singularities.
problem Understanding the behavior of mean curvature flows near cylindrical singularities.
method Proved the rescaled flow converges to a graph over a cylinder, defined nondegeneracy, and showed properties of nondegenerate singularities.
result Nondegenerate cylindrical singularities are isolated, have a mean convex neighborhood, and are type-I.
We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface Σ with nonpositive Yamabe invariant in a Riemannian n-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that Σ is locally volume-minimizing in a manif…
The paper proves a conjecture about mean curvature flows in higher dimensions.
problem Proving a conjecture about mean curvature flows in higher dimensions.
method Classifying ancient Brakke flows and introducing a novel moving plane method.
result Proves the mean-convex neighborhood conjecture for higher dimensional mean curvature flows.
We show that the torsion of any simple closed curve Γ in Euclidean 3-space changes sign at least 4 times provided that it is star-shaped and locally convex with respect to a point o in the interior of its convex hull. The latter condition means that through each point p of Γ there passes a plane H, not cont…
Study shows TAP free energy minimization provides better posterior inference in high-dimensional linear models.
problem Deviation from true posterior mean and underestimation of posterior uncertainty in variational inference.
method Minimization of TAP free energy in a high-dimensional asymptotic framework, showing geometric and statistical properties.
result Local minimizer of TAP free energy provides consistent estimate of posterior marginals and correctly calibrated posterior inference.
We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cu…
We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.
New algorithm tackles risk-aware learning problems efficiently.
problem Risk-aware learning with mean-semideviation objective.
method Zeroth-order compositional stochastic optimization algorithm.
result Algorithm converges to optimal solutions with explicit rates.
In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in n-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in cer…
We establish that over a C^{2,1} manifold the exponential map of any Lipschitz connection or spray determines a local Lipeomophism and that, furthermore, reversible convex normal neighborhoods do exist. To that end we use the method of Picard-Lindelof approximation to prove the strong differentiability of the exponenti…
In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space L3 and get the uniqueness of cr…
In empirical risk optimization, it has been observed that stochastic gradient implementations that rely on random reshuffling of the data achieve better performance than implementations that rely on sampling the data uniformly. Recent works have pursued justifications for this behavior by examining the convergence rate…
The paper bounds eigenvalues of hyperbolic manifolds with infinite volume.
problem Bounding eigenvalues of geometrically finite hyperbolic manifolds of infinite volume.
method Provided a lower bound on the kth eigenvalue of the Laplace-Beltrami operator by the kth eigenvalue of a neighborhood of the thick part of the convex core.
result Recovered a theorem bounding the bottom eigenvalue from below by a specific formula involving the volume of the 1-neighborhood of the convex core.
We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimens…
Classifies ancient ovals in higher dimensional mean curvature flow.
problem Classifying ancient ovals in higher dimensional mean curvature flow.
method Spectral parametrization to classify k-ovals.
result Classifies k-ovals in arbitrary dimensions.
Constructs 2-convex functions approximating distances in Alexandrov spaces.
problem Distance approximation in finite-dimensional Alexandrov spaces.
method Constructs 2-convex functions in Alexandrov spaces.
result Functions can be lifted to close Alexandrov spaces.
New study shows mean estimation algorithms can't beat sub-Gaussian rate in general.
problem Improving mean estimation beyond worst-case scenarios.
method Constructing counterexamples and introducing neighborhood optimality.
result No reasonable estimator can achieve better than sub-Gaussian error rate for any distribution.
Neighborhood sampling affects graph neural network training outcomes.
problem Understanding the impact of neighborhood sampling on graph neural network training.
method Theoretical analysis using neural tangent kernels and Gaussian processes.
result Posterior covariance differs for different neighborhood sampling approaches, indicating no dominant approach.
A novel distributed method tracks gradients for convex optimization over networks.
problem Distributed optimization of strongly-convex functions over a network.
method S-AB algorithm using auxiliary variables and row/column stochastic weights.
result Linear convergence to a neighborhood of the global minimizer.
WSFN overcomes saddle points for non-convex functionals in Wasserstein space.
problem Minimizing non-convex functionals over the Wasserstein space with saddle point avoidance.
method WSFN is a second-order method that preconditions the Wasserstein gradient to avoid saddle points.
result WSFN escapes saddle regions and reaches a global minimizer in polynomial time.
Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold. We make some remarks for the case when only timelike, null or spacelike geodesi…
We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…
Develops a new method for robust risk measurement by averaging nearby payoffs.
problem Measuring risk under uncertainty with a focus on robustness.
method Averaging nearby payoffs weighted by a chosen metric.
result The method leads to a convex risk measure and provides stability under large neighborhoods.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in the Heisenberg group.
method Proves surfaces in a neighborhood of non-umbilic points are solutions to a sinh-Gordon equation with a differential constraint.
result Surfaces in Heisenberg group with constant mean curvature described by solutions to sinh-Gordon equation.
Develops a tensor mixture model for high-dimensional data clustering.
problem Jointly modeling and clustering tensors in high dimensions.
method High-dimensional tensor mixture model with plausible dimension reduction assumptions. EHCMA algorithm for efficient estimation.
result The HECM algorithm converges geometrically to a neighborhood within statistical precision of the true parameter.
Foundations for free boundary Brakke flows established.
problem Analyzing free boundary flows through singularities
method Introducing unit-regular and cyclic free boundary flows, proving avoidance principle, and introducing inner and outer flows.
result General tools for analyzing free boundary flows through singularities.
The estimation of probabilities of network edges from the observed adjacency matrix has important applications to predicting missing links and network denoising. It has usually been addressed by estimating the graphon, a function that determines the matrix of edge probabilities, but this is ill-defined without strong a…
Study of bubble-sheet singularity in mean curvature flow.
problem Analyzing a specific type of singularity in mean curvature flow.
method Deriving an asymptotic profile for a neighborhood of the singularity.
result An asymptotic profile for a neighborhood of the singularity is derived.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core.
We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional directi…
Paper proposes an optimistic likelihood approximation for nonparametric likelihoods.
problem Computational intractability of evaluating likelihood functions in Bayesian statistics.
method Non-parametric approximation using distributionally robust optimization.
result Optimistic likelihood can be solved as a convex optimization problem with analytical expressions.
The paper tackles finding stationary points in stochastic convex optimization problems.
problem Finding stationary points for stochastic convex optimization problems.
method The approach relies on dimension theory to decompose the graph of the subdifferential of a convex function, showing how stochastic sampling preserves 'pieces' of these graphs, and allowing effective application of proximal-point-like methods.
result The paper provides convergence guarantees for finding stationary points in stochastic convex optimization problems.
This paper introduces a new and effective algorithm for learning kernels in a Multi-Task Learning (MTL) setting. Although, we consider a MTL scenario here, our approach can be easily applied to standard single task learning, as well. As shown by our empirical results, our algorithm consistently outperforms the traditio…
A real projective orbifold is an n-dimensional orbifold modeled on RPn with the group PGL(n+1,R). We concentrate on an orbifold that contains a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed (n−1)-dimensional orbifolds times …