Study end sum for surfaces and prove uniqueness results.
problem Uniqueness of end sum for surfaces and related manifolds.
method Analyzing end sum and adding a 1-handle at infinity for surfaces.
result The end sum of two surfaces with compact boundary is uniquely determined by the chosen ends.
Unique minimal translation surfaces found in Heisenberg group.
problem Classifying minimal translation surfaces in Heisenberg group.
method Complete classification through generating curves analysis.
result Uniqueness of minimal translation surfaces up to isometries.
Paper proves unique canonical form for certain highly twisted knots and links.
problem Classifying knots and links with specific plat projections.
method Analyzes 2m-plat projections with specific constraints on crossings and heights. result Unique canonical form for certain knots and links with plat projections.
Adversarial consistency depends on the uniqueness of adversarial Bayes classifiers.
problem Consistency of adversarial surrogate losses is not guaranteed.
method Connected consistency of adversarial surrogate losses to the uniqueness of adversarial Bayes classifiers.
result A convex surrogate loss is statistically consistent for adversarial learning if and only if the adversarial Bayes classifier is unique.
Unique asymptotics found for special geometric flows.
problem Classifying ancient ovals of Ricci flow.
method Analyzing invariant, compact, non-self-similar solutions.
result Uniqueness of the profile function G(z,t). Unique ancient convex flow in a ball with free boundary found.
problem Classifying convex ancient free boundary mean curvature flows in the ball.
method Proof of existence and uniqueness in every dimension.
result A unique (modulo rotations and translations) convex ancient mean curvature flow found.
Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.
problem Classifying ruled real hypersurfaces with constant norm.
method Analyzing nonflat complex space forms, proving existence and uniqueness.
result Existence of a unique inhomogeneous example in complex hyperbolic space.
New split rules improve subpopulation targeting in policy-making.
problem Improving binary classification for subpopulation targeting in policy-making.
method MDFS, PFS, wEFS for maximizing distance and penalizing final splits.
result Proposed methods target more vulnerable subpopulations than classic CART/KD-CART.
Unique Teichmüller curve found in complex geometry.
problem Classifying Teichmüller curves in complex geometry.
method Complete classification of algebraically primitive Teichmüller curves.
result Veech 14-gon generates unique algebraically primitive Teichmüller curve.
Proves uniqueness of translators in 3D space.
problem Uniqueness of pitchfork and helicoid translators in mean curvature flow.
method Arc-counting argument and rotational maximum principle.
result Proves conjecture on uniqueness of translators.
New uniqueness concept for adversarial Bayes classifier.
problem Understanding adversarial Bayes classifiers in binary classification.
method Developed a new notion of uniqueness and analyzed it for a family of one-dimensional data distributions.
result Improved regularity of adversarial Bayes classifiers as perturbation radius increases.
We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in CP2 and S2×S2. In particular, we show that a real Lagrangian in CP2 is unique up to Hamiltonian isotopy and that a real Lag…
Deep neural features identify unique vehicles from dash-cam feeds.
problem Identifying unique vehicles in dash-cam feeds for self-driving cars.
method Used pretrained YOLO network feature maps to create deep integrated feature signatures (DIFS) for 700 images of 35 vehicles and 340 images of 17 vehicles.
result Correctly identified unique vehicles at 96.7% for high resolution data and 86.8% for lower resolution data.
Simplified proof classifies surfaces using normal curves.
problem Classifying surfaces using algebraic topology invariants.
method Using normal curves and analogy to 3-manifolds classification.
result Simple proof without using algebraic topology invariants.
We prove uniqueness of the near-horizon geometries arising from degenerate Kerr black holes within the collection of nearby vacuum near-horizon geometries.
Higher-dimensional Ricci flows are shown to have unique and stable solutions.
problem Stability and uniqueness of Ricci flows in higher dimensions.
method Generalization of Bamler-Kleiner's proof to higher dimensions, use of Brendle's classification of κ-solutions, and maximum principle for linearized Ricci-DeTurck flow.
result Canonical evolution through singularities for manifolds with positive isotropic curvature.
Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are know…
We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CM…
Survey of machine learning methods for Windows malware classification.
problem Difficulties in malware classification through data collection, labeling, feature creation, and selection.
method Review of current methods and challenges in malware classification.
result Discussion of constraints and unaddressed problems for machine learning in cybersecurity.
Study on uniqueness of ad-invariant metrics in Lie algebras.
problem Uniqueness of ad-invariant metrics in Lie algebras up to automorphisms.
method Analysis of Lie algebras, cotangent Lie algebras, and specific conditions for uniqueness.
result Uniqueness of ad-invariant metric on T∗g implies solvability of g, but not conversely. Study higher-dimensional Ricci flow solutions, proving uniqueness.
problem Classifying ancient solutions to the Ricci flow on Sn. method Extending [13] to higher dimensions, proving uniqueness.
result Ancient solutions are either shrinking spheres or Type II solutions.
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…
Mathematical conditions and practical computations for adversarial robustness measures are established.
problem Existence, uniqueness, and scalability of adversarial robustness measures for AI classifiers.
method Formulated and proven mathematical conditions for existence, uniqueness, and explicit analytical computation of minimal adversarial paths and distances. Practical computation demonstrated on various AI tools and synthetic benchmarks.
result Explicit mathematical conditions and practical computations for adversarial robustness measures are established.
We classify static manifolds which admit more than one static decomposition whenever a condition on the curvature is fullfilled. For this, we take a standard static vector field and analyze its associated one parameter family of projections onto the base. We show that the base itself is a static manifold and the warpin…
Graph contrastive learning reveals unique inductive biases.
problem Understanding and optimizing graph contrastive learning methods.
method Systematic study of various GCL methods and their properties.
result GCL methods can work without positive or negative samples, and data augmentations have less impact.
Matrix SMD converges to unique solution minimizing Bregman divergence.
problem High-dimensional multi-output classification and matrix completion problems.
method Stochastic Mirror Descent with matrix parameters and matrix mirror functions.
result Matrix SMD converges exponentially to the unique solution minimizing Bregman divergence.
This paper classifies solutions to a specific hyperbolic geometry problem.
problem Classifying solutions to a specific hyperbolic geometry equation.
method Analytical and numerical methods to solve the equation.
result Classification of solutions for p≥−7, nonuniqueness for p<−7. Derives scalar reduction for generalized Kähler-Ricci solitons, proving uniqueness.
problem Uniqueness of generalized Kähler-Ricci solitons in complex dimension 2.
method Scalar reduction of soliton system, critical points of convex functional.
result Uniqueness of solitons on Hopf surfaces.
With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many exi…
Deep Neural Networks, and specifically fully-connected convolutional neural networks are achieving remarkable results across a wide variety of domains. They have been trained to achieve state-of-the-art performance when applied to problems such as speech recognition, image classification, natural language processing an…
New approach improves classification guarantees by focusing on direction rather than regression risk.
problem Improving classification guarantees in binary classification problems.
method Establishing a geometric distinction between classification and regression, leveraging scale invariance.
result Improved guarantees for classification risk compared to regression risk.
We establish a new uniqueness theorem for the three dimensional Schwarzschild-de Sitter metrics. For this some new or improved tools are developed. These include a reverse Lojasiewicz inequality, which holds in a neighborhood of the extremal points of any smooth function. We further prove smoothness of the set of maxim…
We characterize isometric actions on compact Kaehler manifolds admitting a Lagrangian orbit, describing under which condition the Lagrangian orbit is unique. We furthermore give the complete classification of simple groups acting on the complex projective space with a Lagrangian orbit, and we give the explicit list of …
Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.
problem Classifying SL(n) covariant matrix-valued valuations on Lp-spaces.
method Established a complete classification for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx), eliminating matrix symmetry assumption.
result Unique characterization of such valuations by the moment matrix in n>2, rotation matrix in 2D.
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number c grows exponentially with c, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings. Instead, we consider diagrams enumerated by bridge number, following the lead of S…
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.
When working with three-dimensional data, choice of representation is key. We explore voxel-based models, and present evidence for the viability of voxellated representations in applications including shape modeling and object classification. Our key contributions are methods for training voxel-based variational autoen…
New classification of complex hypersurfaces in 3D.
problem Classifying simply-transitive Levi non-degenerate hypersurfaces in C3. method Novel Lie algebraic approach, new coordinate-free formula for quartic tensor.
result Unique non-tubular model with geometric relations to planar equi-affine geometry.
The paper classifies metrics on a Heisenberg group's cotangent bundle.
problem Classifying Riemannian metrics on a Heisenberg group's cotangent bundle.
method Left-invariant Riemannian metrics classification and uniqueness proof.
result Complex structure is unique and pseudoKähler metrics are Ricci flat.
Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.
problem Identifying hyperbolic 3-manifolds with minimal volume and maximal cusps.
method Maximal cusp volume classification and analysis of low-volume manifolds.
result Figure-8 knot complement is unique in certain volume and filling categories.
Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ+δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight δ. Later, Si…
We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each H∈R, there exists a constant mean curvature H-sphere in the space that is unique up to an ambient isometry.
New metrics on C^3 defy uniqueness, differing even at infinity.
problem Non-uniqueness of Calabi-Yau metrics with maximal volume growth.
method Constructed a family of inequivalent metrics on C^3.
result First example of non-uniqueness in Calabi-Yau metrics asymptotic to a fixed cone.
We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.
We classify the normal CR structures on S3 and their automorphism groups. Together with [3], this closes the classification of normal CR structures on contact 3-manifolds. We give a criterion to compare 2 normal CR structures, and we show that the underlying contact structure is, up to homotopy, unique.
Study classifies translators for mean curvature flow in 3D.
problem Classifying semigraphical translators for mean curvature flow in R3. method Morse-Radó theory and angular maximum principle.
result No solution to the translator equation on the upper half-plane with alternating boundary values.
Active learning can't improve over passive in certain settings.
problem Active learning vs. passive learning in nonparametric settings.
method Analyzing margin conditions and their effects on active learning performance.
result Nuances in margin conditions determine whether active learning can outperform passive learning.
Graph neural networks struggle to distinguish certain graph structures.
problem Difficulty in distinguishing graphs with graph neural networks.
method Analysis of communication capacity in message-passing model of graph neural networks.
result Capacity of MPNN needs to grow linearly for trees and quadratically for general connected graphs.