Develops slope detection for 3-manifolds with torus boundaries.
problem Determining slopes on the boundary of 3-manifolds with torus boundaries.
method Introduces order-detection and representation-detection of slopes, proving their equivalence.
result Shows how slopes' behavior changes with cabling, improving previous results.
Geometric approach improves functional outlier detection.
problem Detecting outliers in functional data sets.
method Developed a geometric perspective on functional manifold for outlier detection.
result Improved understanding and differentiation of outliers.
Study slopes on knot manifolds to understand their fundamental groups.
problem Characterize slopes on knot manifolds to determine fundamental group properties.
method Develops new order-detection notions, parallels existing slope detection methods, and uses dynamics of 3-manifold group actions.
result Conjectured structure theorems connecting Heegaard-Floer homology and foliation dynamics to left-orderability.
Geometric framework detects outliers in high-dimensional data.
problem Detecting outliers in high-dimensional data.
method Geometric framework exploiting manifold structure.
result Significant improvement in outlier detection in high-dimensional data.
New method detects anomalies without labeled data.
problem Detecting anomalies in unsupervised settings.
method Nonlinear manifold learning using LMGP or AE.
result Superior performance compared to existing methods.
Constructs Gabor frames for curved manifolds to detect boundaries.
problem Signal analysis on curved manifolds with boundaries.
method Higher-dimensional Gabor frames for local linearizations.
result Detection of higher-dimensional boundaries in curved manifolds.
Torus decomposition shows foliation detected slopes for glued knot manifolds.
problem Detecting foliation detected slopes in glued knot manifolds.
method Torus decomposition and foliation analysis.
result Gluing knot manifolds identifies rational boundary slopes.
We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H where G is a metrizable topological group and H is a closed balanced subgroup of G.
Paper tackles L-space conjecture for knot manifolds, proving equivalence for some properties.
problem Tackles L-space conjecture for knot manifolds, proving equivalence for some properties. method Introduces relative L-space conjecture, characterizes slope detection, uses Heegaard Floer homology, left-orders, and foliations. result Confirms equivalence of CTF and NLS for slope detected knots, identifies exceptional slopes. A new framework detects anomalies in structured data.
problem Detecting anomalies in samples not conforming to low-dimensional manifolds.
method Preference Isolation Forest (PIF) framework combining adaptive isolation methods and preference embedding.
result Anomalies identified as isolated points in a high-dimensional preference space.
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect fa…
Homological stability fails for 4-manifold moduli spaces, detected by new class.
problem Homological instability for moduli spaces of 4-manifolds.
method Used Seiberg-Witten equations to construct a new characteristic class.
result Homological discrepancy between smooth and topological 4-manifold moduli spaces.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
Proposes manifold-based unsupervised anomaly detection for visual data.
problem Rare anomalies in unlabeled data.
method Constant curvature manifolds, hyperspherical Variational Auto-Encoders (VAE) with gyroplane layer.
result State-of-the-art results on visual anomaly benchmarks and histopathology.
CHAODA detects anomalies in high-dimensional data.
problem Anomaly detection in high-dimensional spaces.
method Hierarchical clustering, manifold mapping, transfer learning.
result CHAODA outperforms other algorithms on 16 out of 18 datasets.
New unsupervised methods for anomaly detection and clustering in structured and streaming data.
problem Anomaly detection and clustering in structured and streaming data.
method Preference Isolation Forest (PIF), Sliding-PIF, MultiLink, Online-iForest, MaxLogit.
result Methods outperform existing techniques on synthetic and real datasets.
Essential tori in certain 3-manifolds are missed by ideal points in character varieties.
problem Essential tori in 3-manifolds are not detected by ideal points in character varieties.
method Infinite families of 3-manifolds are constructed to show the existence of essential tori not detected by ideal points in character varieties over any algebraically closed field.
result Essential tori in 3-manifolds are missed by ideal points in character varieties over any algebraically closed field.
Proposes a boundary detection method inspired by LLE for high-dimensional data.
problem Identifying boundary points from data on an embedded manifold.
method Inspired by locally linear embedding, uses nearest neighbor search schemes and spectral properties of local covariance matrix.
result Enhanced boundary detection in noisy data.
For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.
Diverging Flows detects extrapolations in flow models, ensuring reliable predictions.
problem Flow models extrapolate into invalid data, leading to silent failures.
method Structurally enforce inefficient transport for off-manifold inputs.
result Effective detection of extrapolations without compromising predictive fidelity or inference latency.
PAGER detects failures in deep regression models using a new framework.
problem Detecting failures in deep regression models.
method PAGER uses a combination of epistemic uncertainty and manifold non-conformity scores.
result PAGER accurately characterizes and detects failures in deep regressors.
We extend Culler and Shalen's construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the SL2(C) character variety, and following Boyer and Zhang in the PSL2(C) character variety as well. We show that any slope detected on a canoni…
If a given behavior of a multi-agent system restricts the phase variable to a invariant manifold, then we define a phase transition as change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct…
Two methods monitor high-dimensional processes via manifold fitting or learning.
problem Monitoring high-dimensional, dynamic industrial processes.
method Manifold fitting and learning approaches for online SPC.
result Manifold-fitting approach achieves performance competitive with classical methods.
We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …
Detects knots in thickened surfaces using instanton homology.
problem Detecting knots in thickened surfaces using homology.
method Uses Asaeda-Przytycki-Sikora (APS) homology and sutured instanton homology.
result Detects the unknot in (−1,1)imesΣ and characterizes minimal sutured instanton homology. We discuss corks, and introduce new objects which we call plugs. Though plugs are fundamentally different objects, they also detect exotic smooth structures in 4-manifolds like corks. We discuss relation between corks, plugs and rational blow-downs. We show how to detect corks and plugs inside of some exotic manifolds.…
The paper improves boundary detection and density estimation on noisy data.
problem Detecting boundary points and estimating density on noisy data from compact manifolds.
method Doubly stochastic scaling of the Gaussian heat kernel via Sinkhorn iterations.
result The new estimates of boundary points and density outperform standard methods, especially under noise.
Detects singularities in complex data to improve machine learning models.
problem Real-world data often contains non-manifold structures (singularities) that can mislead machine learning models.
method Develops a topological framework to quantify local intrinsic dimension and Euclidicity score for multiple scales.
result Identifies singularities and captures local geometric complexity in image data.
Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data representation possibly characterised by convenient geometric properties. Euclidean spaces…
We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floe…
The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields…
This paper describes a novel approach to change-point detection when the observed high-dimensional data may have missing elements. The performance of classical methods for change-point detection typically scales poorly with the dimensionality of the data, so that a large number of observations are collected after the t…
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
New method detects symmetries beyond affine transformations.
problem Current methods limit symmetry detection to affine transformations.
method Framework for discovering continuous symmetry beyond affine transformations.
result Method is competitive for large sample sizes and superior for small sample sizes.
New foliations show knot meridians are detectable.
problem Detecting knots in 3-manifolds.
method Constructing co-oriented taut foliations intersecting knot meridians.
result Evidence supports conjecture related to L-space conjecture.
Develops L∞ spaces over dg manifolds and establishes an equivalence with L∞ algebroids.
problem Defining and comparing L∞ spaces and algebroids over dg manifolds. method Establishes an equivalence between categories of L∞ algebroids and L∞ spaces, constructs a faithful functor. result Detects weak equivalences between L∞ algebroids and L∞ spaces. Paper finds a counterexample showing rectangle condition doesn't detect strong irreducibility.
problem Detecting strong irreducibility from the rectangle condition.
method Constructing a double branched cover of a knot in S^3.
result Found a genus 2 Heegaard splitting that is strongly irreducible but fails rectangle condition.
Study potential theory to detect completeness of Finsler manifolds.
problem Detecting completeness of Finsler manifolds via potential theory.
method Potential theoretic aspects of eikonal and infinity Laplace operator, Liouville properties, maximum principles at infinity, viscosity solutions.
result Forward completeness of Finsler manifolds can be detected using Liouville properties and maximum principles at infinity.
We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to def…
Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a 3-manifold from an ideal point of a curve in the SLn-character variety. There exists an essential surface in some 3-manifold known to be not detected in the classical $\o…
We show that there are algorithms to determine if a 3-manifold contains an essential lamination or a Reebless foliation.
Hashing detects anomalies in structured data efficiently.
problem Identifying non-conforming samples on low-dimensional manifolds.
method Locality Sensitive Hashing in Preference Space.
result State-of-the-art performance at lower computational cost.
HADES detects data singularities quickly and accurately.
problem Detecting singularities in data efficiently.
method Kernel goodness-of-fit test based on differential geometry and optimal transport theory.
result Correctly detects singularities with high probability.
Closed essential surfaces in a three-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro's module-theoretic interpretation of Culler-Shalen th…
Improves change-point detection for high-dimensional time-series.
problem Uncertainty in latent variable estimation affects change-point detection.
method Proposes multinomial sampling to improve detection rate and reduce delay.
result Results outperform baseline method in experiments.
The space of graphs is often characterised by a non-trivial geometry, which complicates learning and inference in practical applications. A common approach is to use embedding techniques to represent graphs as points in a conventional Euclidean space, but non-Euclidean spaces have often been shown to be better suited f…
Detects adversarial examples using autoencoders at hidden layers.
problem Identifying adversarial examples in deep neural networks.
method Trains autoencoders on intermediate layers to detect deviations from true data.
result Outperforms state-of-the-art methods in both supervised and unsupervised settings.