A Legendrian or transverse knot in an overtwisted contact 3-manifold is non-loose if its complement is tight and loose if its complement is overtwisted. We define three measures of the extent of non-looseness of a non-loose knot and show they are distinct.
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New twist classes help classify contact structures and looseness.
problem Classifying contact structures and looseness in open books.
method Introducing twist left-veering mapping classes and proving their properties.
result Twist left-veering open books support overtwisted contact structures and determine looseness conditions.
Paper classifies non-loose knots and discusses conditions for their existence.
problem Existence and classification of non-loose knots.
method Analyzes necessary and sufficient conditions for non-loose knots in specific contact structures.
result Classifies all non-loose rational unknots in lens spaces.
On infinite order corksmath.GT
Infinite order loose corks created.
problem Creating loose corks of infinite order.
method Constructing an infinite order loose cork.
result Demonstrated construction of an infinite order loose cork.
Proves h-principle for loose Legendrian embeddings in contact topology.
problem Existence of non-loose Legendrian embeddings.
method h-principle from microlocal sheaf theory.
result Existence of non-loose Legendrian embeddings.
Loose Legendrian knots in rational homology spheres are Legendrian isotopic if they have the same classical invariants.
problem Classifying loose Legendrian knots in rational homology spheres.
method Combining Dymara's result on loose Legendrian knots and Eliashberg's classification of overtwisted contact structures.
result Loose Legendrian knots in rational homology spheres are Legendrian isotopic if and only if they have the same classical invariants.
New property ensures non-looseness of ribbon boundaries.
problem Non-looseness of ribbon boundaries for Legendrian graphs.
method Define and prove the Tight Reattachment Property.
result Ribbon boundaries of Legendrian graphs with the Tight Reattachment Property are non-loose.
Research shows that certain metric spaces cannot contain rigid structures and provides evidence for loose embeddings into Euclidean spaces.
problem The inability of certain metric spaces to contain rigid structures like regular simplices or equidistant sequences.
method Proof of non-embeddability of certain metric spaces into finite-dimensional Euclidean spaces and a local-to-global principle for loose embeddability.
result Compact Riemannian manifolds cannot contain arbitrarily large regular simplices or long equidistant sequences, suggesting loose embeddings into Euclidean spaces.
Quantum isometry groups extend to all countable metric spaces, and loose embeddings help understand metric space relationships.
problem Understanding the quantum isometry groups of all countable metric spaces.
method Defining and studying loose embeddability, showing that 0-dimensional compact metric spaces are generically loosely embeddable into the real line.
result 0-dimensional compact metric spaces are generically loosely embeddable into the real line.
New concept of quasi-right-veering for braids helps classify non-loose links.
problem Classifying non-loose transverse links in contact 3-manifolds.
method Introducing quasi-right-veering for braids and proving equivalence of different definitions.
result Non-loose transverse links are characterized by quasi-right-veering braids.
Disproves the Smale Conjecture for S^4 by showing Diff(S^4) is not SO(5).
problem Disproving the Smale Conjecture for S^4.
method Directly showing π₀Diff(S^4) ≠ 0 by proving a loose-cork cannot be a loose-cork.
result Diff(S^4) ≠ SO(5).
Classifies Legendrian Hopf links in lens spaces.
problem Classifying Legendrian Hopf links in lens spaces.
method Complete classification using contact surgery diagrams.
result Explicit algorithm for all Legendrian representatives.
Homotopy 4-spheres created by iterated Gluck twists of S^4.
problem Understanding the creation and properties of homotopy 4-spheres.
method Using iterated Gluck twists and handle cancellations to show homotopy equivalence to S^4.
result Homotopy spheres Σ_n formed by doubling an infinite order loose-cork are homotopy equivalent to S^4.
Non-loose torus knots in math.GT
Classifies knots in a special 3D space.
problem Classifying knots in a specific geometric space.
method Complete coarse classification of non-loose torus knots.
result Complete classification of knots in .
Study on Legendrian and transverse realizations of negative torus knots.
problem Classification of transverse and Legendrian realizations of negative torus knots.
method Analysis of contact structures, knot Floer homology, Legendrian surgeries.
result Classification of strongly non-loose transverse and Legendrian realizations.
On a homotopy -spheremath.GT
Standard proved to be diffeomorphic to a curious homotopy sphere.
problem Determining the diffeomorphism of a curious homotopy sphere to the standard .
method Proof based on properties of homotopy spheres and loose corks.
result The curious homotopy sphere is diffeomorphic to the standard .
Concrete description of infinite order cork automorphism.
problem Describing the infinite order loose-cork automorphism.
method Concatenating the defining ribbon disk by an infinite order isotopy.
result Concrete description of the infinite order cork automorphism.
We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n+1>3. More precisely, we prove that every Legendrian knot whose complement contains a "nice" plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certai…
Loose Engel structuresmath.SG
The article classifies Engel structures up to homotopy.
problem Classifying Engel structures up to homotopy.
method Introduced the notion of a loose family of Engel structures and showed homotopy equivalence conditions.
result Two Engel structures are homotopic if and only if they are formally homotopic.
Loose bounds found for least-norm interpolant in over-parameterized settings.
problem Failures of model-dependent generalization bounds for least-norm interpolation.
method Analysis of generalization performance of least-norm linear regressor in over-parameterized regime.
result Generalization bounds for least-norm interpolant can be very loose, even when true excess risk goes to zero.
New algorithm learns coordinated decisions in loosely-coupled multi-agent systems.
problem Learning coordinated decisions in multi-agent systems with sparse interactions.
method Multi-Agent Thompson Sampling (MATS) for multi-agent multi-armed bandits.
result MATS achieves sublinear regret and outperforms MAUCE on synthetic and real benchmarks.
We consider the structure learning problem for graphical models that we call loosely connected Markov random fields, in which the number of short paths between any pair of nodes is small, and present a new conditional independence test based algorithm for learning the underlying graph structure. The novel maximization …
Classifies Legendrian unknots in overtwisted contact structures.
problem Classifying Legendrian unknots in overtwisted contact structures on .
method Contact isotopy, Thurston-Bennequin invariant, rotation number.
result Exactly two non-loose Legendrian unknots with specific invariants exist for each pair .
This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. …
LC-FL uses generative models to reduce communication costs in federated learning.
problem High communication costs and strict model homogeneity in federated learning.
method LC-FL employs generative models to transmit data and aggregate models.
result LC-FL reduces communication costs and supports heterogeneous models.
We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.
Non-loose torus knotsmath.GT
Classifies Legendrian and transverse torus knots in .
problem Classifying Legendrian and transverse torus knots in .
method Complete coarse classification using Legendrian and transverse properties.
result Complete classification of torus knots in contact structures on .
The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice. While these relaxations can be solved globally optimally, they are often too loos…
Given two maps f1 and f2 from the sphere Sm to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On the one hand the resulting bordism class of coincidence data and the corresponding Niels…
The study explores Legendrian invariants and half Giroux torsion in contact structures.
problem Understanding Legendrian invariants and their behavior with half Giroux torsion.
method Analysis of Legendrian links with non-vanishing contact invariants and the study of half Giroux torsion.
result Null-homologous links with irreducible complements have non-loose Legendrian realizations with non-zero invariants.
In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.
InfoNCE objective is equivalent to ELBO in RPM, linking to self-supervised learning.
problem Improving self-supervised learning methods by connecting them to variational inference.
method Recognizing RPM and showing InfoNCE as a simplified lower bound on MI, equal to ELBO in infinite sample limit.
result The actual InfoNCE objective is equal to the ELBO (up to a constant) in the infinite sample limit.
Paper tightens optimization bounds using conformal prediction.
problem Lack of practical informiveness in dual bounds from optimization solvers.
method Introduces conformal prediction framework to tighten loose primal and dual bounds.
result Proposed method produces tighter, more informative prediction intervals.
Gossip-based actor-learner architectures improve deep reinforcement learning efficiency and scalability.
problem Stabilizing and scaling deep reinforcement learning through multi-simulator training.
method Gossip-based communication among actor-learners in a peer-to-peer topology, reducing synchronization.
result Gossip-based architectures outperform A2C in terms of robustness and sample efficiency.
We set forth a definition of hyperfinite knots. Loosely speaking, these are limits of certain sequences of knots with increasing crossing number. These limits exist in appropriate closures of quotient spaces of knots. We give examples of hyperfinite knots. These examples stem from an application of the Thermodynamic Li…
We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a -manifold that are transverse to a nowhere-zero vector field up to the corresponding isotopy relation. Such knots are called …
This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to filli…
Study fractional twist behavior in branched covers, with applications to 3-manifolds.
problem Behavior of fractional Dehn twist coefficients in branched coverings.
method Analyzes fully ramified branched coverings and their effects on 3-manifolds.
result Non-right-veering braids represent virtually loose transverse links.
We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. More…
Improves GBDT accuracy with differential privacy.
problem Balancing privacy and accuracy in GBDT models.
method Adaptive gradient control and novel boosting framework for privacy budget allocation.
result Achieves better model accuracy with differential privacy.
We derive a novel norm that corresponds to the tightest convex relaxation of sparsity combined with an penalty. We show that this new {\em -support norm} provides a tighter relaxation than the elastic net and is thus a good replacement for the Lasso or the elastic net in sparse prediction problems. Through …
We provide sharp lower bounds for the simplicial volume of compact -manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of -manifolds, including handlebodies and products of surfaces with the interval. Our results provide the firs…
This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold . A superpath in is, loosely speaking, a path in together with an odd vector field in along the path. We also develop a notion of parall…
Improves probabilistic inference with new weighting method.
problem Improving variational inference for probabilistic models.
method Importance Weighted Variational Inference (IWVI) using augmented variational inference.
result IWVI is a practical technique for probabilistic inference.
We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We pr…
Minimal hypersurfaces with bounded index behave like stable ones, up to controlled errors.
problem Understanding the behavior of minimal hypersurfaces with bounded index.
method Proved a structural theorem for sequences of closed embedded minimal hypersurfaces in Riemannian manifolds.
result Embedded minimal hypersurfaces with bounded index behave qualitatively like stable minimal hypersurfaces, up to controlled errors.
Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knotmath.GT
Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the …
We prove that two Legendrian knots in a contact structure which is trivializable as a plane bundle are Legendrian isotopic provided that (1) they are isotopic as framed knots, (2) they have the same rotation number with respect to some parallelization of the contact structure, and (3) there is an overtwisted disk disjo…