Survey on hyperplane arrangements and their topology.
problem Topology of hyperplane arrangements.
method Focus on the relationship between topology and real structure.
result Relationship between topology and real structure of hyperplane arrangements.
New algebraic structures for topological pairs.
problem No specific problem stated; generalization of knot quandles.
method Introducing multi-quandles for topological pairs.
result New algebraic structures for topological pairs.
A new approach uses circuit topology to study complex polymer interactions.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
Novel theory combines combinatorial and topological elements.
problem Understanding combinatorial phenomena at the intersection of topology.
method Synthesizes combinatorial and topological approaches with a new framing concept.
result Framed combinatorial spaces exhibit better behavior than classical spaces.
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
We analyze oversquashing in topological message-passing using relational structures.
problem Oversquashing in topological message-passing remains understudied.
method A unifying axiomatic framework that bridges graph and topological message-passing.
result Potential to advance topological deep learning.
Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of existing Bayesian optimization algorithms can only handle vector data and canno…
New topology defined from spacetime paths, reconstructing spacetime structure.
problem Reconstructing spacetime structure from path homotopy classes.
method Defining a topology on spacetime based on timelike and causal homotopy classes.
result The topology on spacetime is reconstructed from the space of homotopy classes.
Study the structure of equidistant decompositions in manifolds.
problem Geometric and topological structure of equidistant decompositions.
method Investigation of Riemannian manifolds.
result Detailed understanding of equidistant decompositions.
It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend o…
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
Regularization plays a crucial role in supervised learning. Most existing methods enforce a global regularization in a structure agnostic manner. In this paper, we initiate a new direction and propose to enforce the structural simplicity of the classification boundary by regularizing over its topological complexity. In…
New method estimates causal effects in complex spaces using topological structures.
problem Challenges in estimating causal effects in non-Euclidean spaces.
method Developed a topological causal inference framework using power-weighted silhouette functions of persistence diagrams.
result Successfully quantifies topological treatment effects across various complex outcomes.
Paper proposes learnable topological features for efficient phylogenetic inference.
problem Finding appropriate topological structures for phylogenetic inference tasks requires significant design effort and domain expertise.
method Combines raw node features with graph neural networks to automatically adapt to different tasks.
result Demonstrates effectiveness and efficiency on simulated and real data phylogenetic inference tasks.
Topological Flow Matching: A Generative Modeling Framework for Structured Spaces
problem Handling structured spaces in generative modeling
method Introducing topological flow matching
result Captures the structure of the underlying domain while preserving desirable properties
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.
Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
We define a moduli space of translation structures on the open topological disk with a basepoint and endow it with a locally-compact metrizable topology. We call this the immersive topology, because it is defined using the concept of immersions: continuous maps between subsets of translation surfaces that respect the b…
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
New EZ-structure maps mapping class group actions.
problem Mapping class group actions on surfaces.
method Constructing a boundary with minimal, strongly proximal, and topologically free action.
result Boundary acts on mapping class group with desired properties.
A new method clusters complex networks using topological and geometric structure.
problem Clustering complex networks with intricate topology.
method Centroid-based clustering strategy using Wasserstein distance and barycenter for persistence barcodes.
result Demonstrated effectiveness on simulated and real-world networks.
The paper studies Liouville structures for taut foliations and Anosov flows, proving their topological invariance.
problem Understanding the topological invariance of Liouville structures for taut foliations and Anosov flows.
method Combining smoothing schemes for topological conjugacies and a refinement of Vogel's uniqueness result.
result Liouville structures are topological invariants of taut foliations and orbit equivalent Anosov flows.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
A brief survey of real algebraic structures on topological spaces is given. This article is written for the Gokova Gemetry/Topology Conference proceedings.
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
Let M be an orientable topological manifold of dimension m, m greater or equal to 5, with fundamental group Γ. Let S(M) be the topological structure set, endowed with the group structure induced by its identification with Ranicki's algebraic structure set. We prove that the (rationalized) rho map $ρ_Γ: S(M)\rightarro…
This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition.
New compact manifolds with closed G2 structures found.
problem Existence of torsion-free G2 structures on compact manifolds.
method Equivariant resolution procedure for G2 orbifolds with Z2 isotropy.
result Construction of new compact manifolds with closed G2 structures.
Galois action on manifold structures of complex varieties is abelian.
problem Understanding the Galois action on topological manifold structures of complex varieties.
method Definition of profinite normal structure set and Galois action analysis.
result Galois action on manifold structures of simply-connected varieties is abelian.
Let X be a closed oriented connected topological manifold of dimension n≥5. The structure group of X is the abelian group of equivalence classes of all pairs (f,M) such that M is a closed oriented manifold and f:M→X is an orientation-preserving homotopy equivalence. The main purpose of this a…
Paper encodes textile structures and classifies them up to complexity five.
problem Topological classification of textile structures.
method Extends Gauss code to textile structures, introduces a linear time algorithm.
result First classification of all oriented textile structures up to complexity five.
Persistent homology reveals a topological signature of grokking in neural networks.
problem Understanding how neural networks learn and generalize from modular arithmetic tasks.
method Persistent homology on point clouds derived from embedding matrices of models trained on modular arithmetic.
result A sharp increase in first homology persistence indicates grokking, with a dominant long-lived topological feature and structured secondary features.
A faster, more stable method for optimizing topological functions.
problem Optimizing topological functions is computationally expensive and unstable.
method Introduces a novel backpropagation scheme for faster and more robust optimization.
result Produces more robust optima and stable visualizations.
We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete group…
Topological data analysis quantifies structural dynamics using persistent homology.
problem Analyzing the shape and topology of structural dynamics data.
method Topological Data Analysis (TDA) with persistent homology to quantify shape over scales.
result Persistent homology reveals significant changes in manifold shape due to damage, not temperature.
Tropical geometry aids in computing topological quantum field theories.
problem Computing Gromov-Witten invariants using tropical geometry.
method Using mathematical techniques of tropical geometry to compute topological quantum field theories of pseudoholomorphic maps.
result Identifies the tropicalization of localization equations and studies the geometry and symmetries of the theory.
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with Z2-twists and higher cohomology. We consider the topological and geometric structures associated with cohomological and homological objects in M-theory. For the latter, we have M2-branes and M5-branes, the analysis of which requires the underlying spacetime to admit a String structure and a Fivebrane structure, respectively. For the former, we study h…
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable …
Introduces topological deep learning for neural network classification problems.
problem Classifying neural networks using minimal topological structures.
method Formalizes classification problems in a topological setting.
result Demonstrates conditions for the feasibility of classification problems in neural networks.
We describe off-shell N=1 M-theory compactifications down to four dimensions in terms of eight-dimensional manifolds equipped with a topological Spin(7)-structure. Motivated by the exceptionally generalized geometry formulation of M-theory compactifications, we consider an eight-dimensional manifold $\mat…
TADA detects anomalies in time series using topological data analysis.
problem Detecting global changes in dependency structure between channels in multivariate time series.
method Topological Data Analysis for detecting anomalies in multivariate time series.
result The approach is more suitable for detecting global changes of correlation structures than existing methods.
A new model learns graph structures from data.
problem Learning graph topologies from data.
method Proposes a learning to optimise (L2O) approach to learn graph structures from node data.
result The proposed model learns graph structures more efficiently than classic iterative algorithms.
New topological methods for hypergraph data improve community detection and pattern recognition.
problem Community detection and pattern recognition in hypergraph data.
method Introducing a new topological space structure of hypergraph data, proposing modified nearest neighbors methods.
result Improved methods for community detection and pattern recognition in hypergraph data.
Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on RP2. result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on RP2. Study on symplectic structures and their deformations.
problem Preservation of complex symplectic structures under deformations.
method Analyzes various cohomologies and conditions for deformations.
result Obtains topological obstructions for compact complex symplectic manifolds.
Study on stable Hamiltonian topology finds non-density of certain structures.
problem Non-density of stable hypersurfaces and Hamiltonian structures.
method Proving non-density results for stable hypersurfaces and Hamiltonian structures in various dimensions.
result Non-density of stable hypersurfaces and Hamiltonian structures in specific isotopy and homotopy classes.