Study on 3D surfaces and tangles formed by Poncelet triangles.
problem Geometric and topological properties of 3D surfaces and tangles.
method Exploration of Poncelet triangles and associated points.
result Properties of 3D surfaces and tangles formed by Poncelet triangles.
New framework for 3D spatial topology enumeration and identification.
problem Efficient navigation through complex engineering system topologies.
method Mathematical spatial graph theory to represent, enumerate, and identify unique topological classes.
result Identification of distinctive 3D topological classes for engineering systems.
Topology-enhanced loss improves 3D object reconstruction from 2D images.
problem Challenges in reconstructing 3D objects from 2D images, especially capturing shape information.
method Integrates multi-scale topological features into the reconstruction loss using cubical complexes and optimal transport distance.
result Topology-aware loss substantially improves 3D reconstruction quality.
Topology-GS improves 3D GS for better structural and feature integrity.
problem Compromised pixel-level and feature-level integrity in 3D GS.
method Incorporates Local Persistent Voronoi Interpolation (LPVI) and PersLoss based on persistent homology.
result Topology-GS outperforms existing methods in PSNR, SSIM, and LPIPS metrics.
EuLearn creates diverse 3D topological datasets for machine learning.
problem Training machine learning systems to discern topological features.
method Developed novel sampling and neural network architectures for graph and manifold data.
result Incorporating topological information improves deep learning performance on EuLearn datasets.
HLTF generates chemically valid 3D molecules with improved topology control.
problem Generating chemically valid 3D molecules is challenging due to bond topology errors.
method HLTF uses a latent multi-scale plan for global context and a constraint-aware sampler to suppress topology-driven failures.
result HLTF achieves high validity and uniqueness on QM9 and GEOM-DRUGS datasets.
Modeling 3D continua with singular points using Yin sets.
problem Treat singular points as central subjects in 3D continuum topology.
method Model 3D continua as Yin sets, regular open semianalytic sets with bounded boundary.
result Characterize local and global topology of Yin sets.
Defines a map connecting 3d-index and skein module.
problem Connecting mathematical physics predictions with topological quantum field theory.
method Defines a map from skein module to Laurent series ring.
result The map fulfills a supersymmetry prediction and is part of a conjectural topological quantum field theory.
Study 3d N=1 vacua from M-theory compactification on Spin(7) space.
problem Quantum corrections in 3d N=1 vacua from M-theory compactification.
method Use Higgs bundles to analyze 3d N=1 vacua and track corrections.
result Topological anomalies are robust and calculable in 3d effective field theory.
3D dual field theories for Virasoro minimal models constructed using Seifert fiber spaces.
problem Constructing 3D dual field theories for Virasoro minimal models.
method 3D-3D correspondence and Seifert fiber spaces.
result 3D dual field theories constructed for Virasoro minimal models.
Only vertical planes are asymptotic to other planes in 3D space.
problem Characterizing asymptotic planes in 3D space.
method Proof of uniqueness for complete translators with finite topology.
result Vertical planes are the only asymptotic planes in 3D space.
New method describes entanglement of straight lines in 3D space.
problem Tackles the geometry and topology of configurations of straight lines.
method Introduces direction matrices and a discrete motion principle.
result Shows n-crosses as links of pairwise connected unknots.
Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
problem Stability of the Schwarzschild 3-manifold in the context of the 3D Riemannian Penrose inequality.
method Pointed measured Gromov-Hausdorff topology, negligible domains and boundary area perturbations.
result Schwarzschild 3-manifold stability proven for 3D Penrose inequality.
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
Proves formula for 3D index change with Dehn filling.
problem Transforming 3D index under Dehn filling.
method Relative 3D index, gluing principle, inductive framework, q-hypergeometric functions.
result Rigorous proof of Gang-Yonekura formula.
Paper connects 3D gravity averages to 2D CFT correlators.
problem Understanding 3D gravity partition functions.
method 3D topological field theories and mapping class group averages.
result Established a correspondence between 3D gravity averages and 2D CFT correlators.
We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the spac…
3D spaces without boundaries are found that aren't manifolds.
problem Finding compact aspherical Alexandrov spaces without boundaries.
method Constructing specific examples of 3D spaces.
result Examples of 3D spaces without boundaries that are not topological manifolds.
Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in q1/2 with integer coefficients. Our goal is to explain how the 3D-index is a generating serie…
Study identifies topologies of 3D spaces with boundary.
problem Understanding the topologies of compact Alexandrov spaces with boundary.
method Continuation of previous work, determining topologies through analysis.
result Identified topologies of collapsing 3D Alexandrov spaces with boundary.
Decussation improves robustness of 3D neural networks.
problem Understanding the functional advantage of decussation in neural wiring.
method Examined constraints of topology on 3D wiring.
result Decussated arrangements are more robust against wiring errors than same-sided schemes.
Authors classify 3D locally standard T-pseudomanifolds under weaker conditions.
problem Classifying equivariant homeomorphism types of 3D locally standard T-pseudomanifolds.
method Introduced and classified by characteristic data under homotopy equivalence condition.
result Condition for classification can be removed when dimension is at most three.
Agent learns to navigate uncertain 3D maps using a hybrid planner.
problem Planning in 3D environments with uncertain topological maps.
method Hierarchical strategy combining graph planner and local policy, data-driven learning with neural network.
result Machine learning can overcome missing information in probabilistic topological maps.
3D topological order linked to Seifert manifolds and gauge groups.
problem Classifying 3D topological orders using Seifert manifolds and gauge groups.
method Correspondence between topological order, Seifert manifolds, and ADE gauge groups.
result Construction of modular fusion categories from Seifert manifolds and gauge groups.
Overview of 3D TQFTs and 3-manifold invariants.
problem Quantum invariants of 3-manifolds.
method Recall and review of TQFTs, fusion categories, and recent generalizations.
result Overview of various 3D TQFTs and their invariants.
Identifies 3D-index as invariant for cusped hyperbolic 3-manifolds.
problem Invariance of q-series for cusped hyperbolic 3-manifolds.
method Relates Frohman-Kania-Bartoszynska's q-series to Dimofte-Gaiotto-Gukov's 3D-index and tetrahedron index.
result Topological invariance of Frohman-Kania-Bartoszynska's q-series.
Topology optimization is computationally demanding that requires the assembly and solution to a finite element problem for each material distribution hypothesis. As a complementary alternative to the traditional physics-based topology optimization, we explore a data-driven approach that can quickly generate accurate so…
Explains research on 3D dynamics and manifold topology.
problem Understanding obstructions for Anosov flows on 3-manifolds.
method Expository note on partially hyperbolic diffeomorphisms and Anosov flows.
result Margulis and Plante-Thurston's topological obstructions for Anosov flows.
Characterizes Anosov flows in 3D using symplectic and contact geometry.
problem Understanding Anosov flows in 3D.
method Purely contact and symplectic geometric methods.
result Characterization of Anosov flows based on Reeb flows and underlying (bi)-contact structures.
There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimension…
We describe a family of 3d topological B-models whose target spaces are Hilbert schemes of points in C2. The interfaces separating theories with different numbers of points correspond to braid strands. The Hilbert space of the picture of a closed braid is the HOMFLY-PT homology of the corresponding link.
We show that 3D gravity, in its pure connection formulation, admits a natural 6D interpretation. The 3D field equations for the connection are equivalent to 6D Hitchin equations for the Chern-Simons 3-form in the total space of the principal bundle over the 3-dimensional base. Turning this construction around one gets …
Recent years have witnessed the emergence and increasing popularity of 3D medical imaging techniques with the development of 3D sensors and technology. However, achieving geometric invariance in the processing of 3D medical images is computationally expensive but nonetheless essential due to the presence of possible er…
The common assertion that the Ricci flows of Einstein spaces with cosmological constant can be modelled by certain classes of nonholonomic frame, metric and linear connection deformations resulting in nonhomogeneous Einstein spaces is examined in the light of the role played by topological three dimensional (3D) Taub-N…
The study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
New theorem on embedding Moebius bands in 3D space.
problem Proving the impossibility of placing uncountably many disjoint Moebius bands in 3D space.
method Generalization of Grushin and Palamodov's result to tame subsets in R^N and arbitrary topological embeddings in R^3.
result The impossibility of embedding uncountably many pairwise disjoint Moebius bands in 3D space, even for arbitrary topological embeddings.
Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.
problem Uniqueness of dynamical invariants for 3D volume-preserving diffeomorphisms.
method Examined failure of uniqueness on integral homology spheres and arbitrary three-manifolds using local and global invariants.
result Failure of uniqueness is severe, with continuous and non-constant invariants appearing in C1-open sets of nonvanishing exact fields of fixed helicity. Existing networks directly learn feature representations on 3D point clouds for shape analysis. We argue that 3D point clouds are highly redundant and hold irregular (permutation-invariant) structure, which makes it difficult to achieve inter-class discrimination efficiently. In this paper, we propose a two-faceted sol…
New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
Machine learning and topological data analysis identify unique geometric and topological features of human papillae.
problem Identifying unique features of human papillae across individuals.
method 3D microscopic scans, machine learning, discrete differential geometry, computational topology, persistent homology.
result Persistent homology features of papillae shape predict papillae type with high accuracy and can identify individuals with high accuracy.
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N=2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,Z) Weil representations, quantum mo…
Paper develops a new method to analyze 3D tree-like objects.
problem Analyzing complex geometrical and topological variations in 3D tree-like objects.
method Extended SRVF representation and new metric for tree-shaped 3D objects.
result Captures full elasticity and topological variations of branches.
Study of IR phases in 3D class R theories linked to non-hyperbolic 3-manifolds.
problem Understanding IR phases of 3D class R theories associated with non-hyperbolic 3-manifolds.
method Analysis of IR phenomena through `exceptional' Dehn fillings and gauging of flavor symmetries.
result 3D class R theories associated with certain atoroidal non-hyperbolic 3-manifolds exhibit supersymmetry enhancement at low energy.
New algorithm constructs characters of rational VOAs from knot complements.
problem Constructing characters of rational VOAs from knot complements.
method 3D N=2 gauge theories, Dimofte-Gaiotto-Gukov construction, 3D N=4 rank-0 SCFT, topological twist. result New Nahm-sum-like expressions for Virasoro minimal model characters.
Homology handles with trivial Alexander polynomial bound a 3D sphere.
problem Understanding when homology handles bound 3D spheres.
method Using Freedman and Quinn's result for Z-homology 3-spheres. result A distinguished homology handle with trivial Alexander polynomial bounds a homology S1imesD3. Improved 3D generative models for drug design reduce bias and enhance data efficiency.
problem Data sparsity and bias in 3D molecular design models.
method Multi-level contrastive learning protocol for bias control and data efficiency.
result Hierarchical generative models that are topologically unbiased and explainable.
A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, wh…
The study classifies flows of finite curvature in 3D space.
problem Classifying flows of finite curvature in 3D space.
method Partial classification of eternal mean convex flows.
result Topologically nonplanar flows must exit a catenoid.