Simplified construction of contact triad connection for analysis.
problem Analyzing contact instanton equation on contact triads.
method Characterization via almost contact structure and construction using almost contact moving frame.
result Simpler and more canonical construction of contact triad connection.
We introduce a canonical affine connection on the contact manifold (Q,ξ), which is associated to each contact triad (Q,λ,J) where λ is a contact form and J:ξ→ξ is an endomorphism with J2=−id compatible to dλ. We call it the \emph{contact triad connection} of (Q,λ,J) and prove its existence and uniqu…
Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.
problem Analyzing nonlinear elliptic systems associated with contact Hamiltonian trajectories.
method Identifies correct action and energy functionals, develops elliptic regularity theory, and proves asymptotic convergence.
result Established C∞ convergence of perturbed contact instantons under finite energy hypothesis. The paper classifies symmetric triads with multiplicities and their applications.
problem Classifying symmetric triads with multiplicities and their applications.
method Developed the theory of symmetric triads with multiplicities, classified abstract triads, and determined corresponding triads for commutative compact triads.
result Classified symmetric triads with multiplicities and their applications.
Paper classifies compact symmetric triads using double Satake diagrams and canonical forms.
problem Classifying compact symmetric triads.
method Introducing double Satake diagrams and canonical forms, proving their existence and properties.
result Existence and properties of canonical forms for compact simple symmetric triads.
Homotopy theory for (2n+1)-dimensional manifold triads with fixed boundary.
problem Classifying stable moduli spaces of (2n+1)-dimensional manifold triads. method Homotopy-theoretic description of stable moduli spaces, stabilization by boundary connected sum with SnimesDn+1. result Established homology of stable moduli spaces for (2n+1)-dimensional manifold triads. The (variational) graph auto-encoder and its variants have been popularly used for representation learning on graph-structured data. While the encoder is often a powerful graph convolutional network, the decoder reconstructs the graph structure by only considering two nodes at a time, thus ignoring possible interaction…
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…
Survey article analyzes pseudoholomorphic curves on symplectization via contact instantons.
problem Analyzing pseudoholomorphic curves on symplectization.
method Using contact instantons and a coordinate-free covariant tensorial calculus.
result Stronger and more accurate a priori estimates for pseudoholomorphic curves.
Condition for intersection of real flag manifolds in complex flag manifold.
problem Intersection conditions of real flag manifolds in a complex flag manifold.
method Condition given in terms of symmetric triad, antipodal intersection proven.
result Intersection of real flag manifolds is antipodal.
TSSC images enhance chaotic signal classification using ConvNets.
problem Classifying chaotic signals accurately and robustly.
method Triad State Space Construction (TSSC) for image encoding, Convolutional Neural Network (ConvNet) for classification.
result TSSC-ConvNet achieves high accuracy and robustness in chaotic signal classification.
New universal automorphic functions capture monstrous moonshine.
problem Developing a universal framework for automorphic functions.
method Reformulating old results, constructing new coordinates, and defining central extensions.
result New invariant 1-forms and representations for universal Teichmüller space.
We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted…
Generates sutured manifolds from a surface using surgery.
problem Creating sutured manifolds from a given surface.
method Using surgery triads to generate sutured manifolds.
result The set of sutured manifolds is generated by a specific manifold.
Study shows spherical embedding and immersion components are related to homotopy groups.
problem Understanding the connected components of spherical embeddings and immersions.
method Analyzing the spaces of spherical embeddings and immersions modulo immersions, and relating them to homotopy groups.
result The set of connected components of spherical embeddings and immersions modulo immersions is isomorphic to π_{n+1}(SG,SG_q).
We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducibl…
Community detection is a fundamental task in social network analysis. In this paper, first we develop an endorsement filtered user connectivity network by utilizing Heider's structural balance theory and certain Twitter triad patterns. Next, we develop three Nonnegative Matrix Factorization frameworks to investigate th…
New methods stabilize Q-learning with linear approximations.
problem Stabilizing Q-learning with linear function approximation. method Target network and truncation.
result Provably stable Q-learning with linear function approximation. Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
The purpose of this article is to study co-dimension 2 iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold (M2n−1,ξM) iso-contact embeds in a contact manifold (N2n+1,ξN), provided M contact embeds in (N,ξN) with a trivial normal bundle and the contact s…
The paper defines and studies contact surgery numbers for contact 3-manifolds.
problem Understanding the minimal number of components of a surgery link describing a contact 3-manifold.
method Defined and studied various versions of contact surgery numbers, relating them to other invariants and computing specific cases.
result There exist infinitely many non-isotopic contact structures on certain manifolds that cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere.
Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.
problem Generalize K-contact and Sasakian manifolds criteria using weak quasi contact metric manifolds.
method Study weak quasi contact metric manifolds and generalize theorems for K-contact and Sasakian manifolds.
result Provide new criterions for K-contact and Sasakian manifolds in terms of curvature tensor and geometric objects.
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
We study the contact equivalence problem for toric contact structures on S3-bundles over S2. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To f…
Abstract: Survey on contact submanifolds.
problem Understanding contact submanifolds.
method None specified, survey of existing knowledge.
result Discussion of various contact submanifolds.
New contact structures on folded sums of contact mapping tori are tight under certain conditions.
problem Understanding tight contact structures on folded sums of contact mapping tori.
method Alternative bundle-theoretical construction and gluing process near the fold.
result Folded contact structures on folded sums of contact mapping tori are tight under specific conditions.
Study explores weak generalized K-contact structures in contact metric spaces.
problem Exploring weak generalized K-contact structures in contact metric spaces.
method Introducing a weak (κ,μ) condition and proving existence of K-contact and (κ,μ=2)-structures. result Existence of K-contact and (κ,μ=2)-structures under certain conditions on the Boeckx invariant. New contact structures detected by contact homology.
problem Detecting pseudo-Anosov flows in contact structures.
method Introducing pseudo-Anosov contact structures and using contact homology.
result Contact homology detects pseudo-Anosov flows and contact structures properties.
Local flatness theorem for paraquaternionic contact structures.
problem Local flatness of paraquaternionic contact manifolds.
method Defined paraquaternionic contact conformal curvature tensor and showed local flatness condition.
result Paraquaternionic contact conformal curvature vanishing implies local flatness.
New examples show contact invariants can be non-zero even with half Giroux torsion.
problem Understanding contact invariants and their obstructions in 3-manifolds.
method Use bordered contact invariants and innermost contact structures.
result Found closed contact 3-manifolds with non-vanishing contact invariants.
Extends Poincaré-Lefschetz duality to pairs of ∞-categories.
problem Generalizing Poincaré-Lefschetz duality to ∞-categories.
method Introduces Poincaré duality pairs of ∞-categories and uses them to study various diagrams of spaces.
result Unified treatment of Wall's Poincaré ads and iterated Poincaré cobordisms.
Survey of contact homology for contact manifolds.
problem Understanding contact manifolds through homology.
method Holomorphic curves invariants.
result Useful for students and young mathematicians.
We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact struc…
Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard con…
The paper connects complex contact structures to specific types of almost contact 3-structures.
problem Understanding the relationship between complex contact structures and almost contact 3-structures.
method Proving that every complex contact structure gives rise to a distinguished almost contact metric 3-structure.
result The paper provides new examples of manifolds with specific contact and almost contact structures.
The paper constructs contact-hyperbolic manifolds with large automorphism groups.
problem Finding contact-hyperbolic manifolds with large automorphism groups.
method Holomorphic contact structures, pseudometrics, and symplectic quotients.
result Explicit examples of contact-hyperbolic contact manifolds are constructed.
We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of …
Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.
problem Classifying contact structures with specific contact surgery numbers on Brieskorn spheres and lens spaces.
method Algorithm for computing Euler class from rational contact surgery, Legendrian knot classification, and analysis of contact structures.
result Infinitely many non-isotopic contact structures on Brieskorn spheres and lens spaces cannot be obtained by a single rational contact surgery.
Develops k-contact geometry theory for field theories.
problem Analyse field theories using k-contact geometry.
method Distributions maximally non-integrable with k commuting Lie symmetries.
result Established k-contact distributions and their relationships.
Contact round surgeries on (S3,ξst) help in constructing and understanding contact 3-manifolds.
problem Constructing contact 3-manifolds using Legendrian surgeries.
method Introducing contact round surgeries of indices 1 and 2, and associating them with surgery diagrams.
result Every closed connected contact 3-manifold can be obtained by a sequence of contact round surgeries on Legendrian knots in (S3,ξst). Contact surgeries yield algebraically overtwisted manifolds.
problem Understanding algebraically overtwisted contact manifolds through surgeries.
method Contact (+1)-surgeries on Legendrian spheres in flexibly fillable contact manifolds. result Yielding algebraically overtwisted manifolds when the Legendrian's homology class is not annihilated.
We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Matić in \cite{HKM08}. The Z2 SFH of these solid tori forms a "categorification of Pascal's triangle", and contact structur…
This paper presents and explores a theory of \emph{multiholomorphic maps}. This group of ideas generalizes the theory of pseudoholomorphic curves in a direction suggested by consideration of the kinds of compatible geometric structures that appear in the realm of special holonomy as well as some of the topological and …
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…
Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.
problem Tackles the need for a geometric tool in contact manifolds.
method Introduces a generalized Tulczyjew triple for contact manifolds.
result Contact Hamiltonians and Lagrangians as sections of line bundles determine dynamics on contact phase space.
The study finds tight contact structures without fillings in high dimensions.
problem Finding tight contact structures that cannot be filled by symplectic forms.
method Construction of specific contact structures on manifolds of various dimensions.
result Existence of tight contact structures without fillings in all dimensions n≥3 and for n=2 under certain conditions. We construct an open book decomposition compatible with a contact structure given by a rational contact surgery on a Legendrian link in the standard contact S3. As an application we show that some rational contact surgeries on certain Legendrian knots induce overtwisted contact structures.