New class of maps restricts manifolds strongly in algebraic topology.
problem Restricting manifolds in algebraic topology.
method Proposed a class of generalized special generic maps.
result Extended fundamental results on structures and algebraic topological properties.
Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the D-topology. However, the D-topology has not yet been studied seriously in the existing literature. In this paper, we…
Topological normal generation proved for mapping class groups of certain surfaces.
problem Proving topological normal generation for mapping class groups of surfaces.
method Analyzing the end space of surfaces and using topological normal closure properties.
result Topological normal generation is equivalent to uniquely self-similar for surfaces with countable end space.
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.
We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…
Study one-dimensional topological theories with linear generating functions.
problem Understanding one-dimensional topological theories with defects.
method Construct bases of hom spaces for decorated unoriented one-dimensional cobordisms.
result Gram determinant and linear generating functions constructed.
Topology guidance controls generative model outputs by specifying topological features.
problem Lack of control in generative models for visual analysis.
method Coupling a coordinate-based neural network with a diffusion model to guide sampling.
result Generated fields adhere to specified topological features while staying within the generative distribution.
Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.
problem Minimal topological generating sets of mapping class groups consisting of torsion elements.
method Investigation of minimal topological generating sets for Map(S(n)) consisting entirely of torsion elements, with special attention to involutions. result Minimal topological generating sets for Map(S(n)) consisting of torsion elements are found for various n. Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
Generically, topological insulators have conical points leading to Dirac-like currents.
problem Understanding the conical structure of degeneracies in topological phases of matter.
method Analyzing Hermitian matrices with three parameters to show conical points.
result Adiabatic deformations of topological insulators result in Dirac-like currents whose total conductivity equals the chiral number of conical points.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
We consider principal bundles as generalized morphisms between topological groupoids. In the category of these generalized morphisms two topological groupoids are isomorphic if and only if they are Morita equivalent. We show that the fibers of a generalized morphism from H to G induce a singular foliation of the topolo…
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…
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then…
PhyloVAE learns tree topologies without supervision.
problem Learning accurate tree representations from evolutionary data.
method Unsupervised learning via variational autoencoders with efficient tree generation.
result PhyloVAE generates high-resolution tree topologies efficiently.
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.
Topological twists for 4d N=2 theories depend on spacetime type, gerbe connections, and generalized spin-c structures.
problem Defining topologically twisted partition functions for 4d N=2 theories.
method Topological twisting for general 4d N=2 theories, introducing generalized spin-c structures.
result Topological partition functions depend on spacetime type, gerbe connections, and generalized spin-c structures.
In this study, a novel topology optimization approach based on conditional Wasserstein generative adversarial networks (CWGAN) is developed to replicate the conventional topology optimization algorithms in an extremely computationally inexpensive way. CWGAN consists of a generator and a discriminator, both of which are…
Topological constraints improve neural network generalization.
problem Improving generalization in neural networks with limited data.
method Imposing topological constraints on internal representations of neural networks.
result Topological constraints lead to better mass concentration around training instances, improving generalization.
In this note we generalize a result by Alekseev and Strobl for the case of p-branes. We show that there is a relation between anomalous free current algebras and "isotropic" involutive subbundles of T⊕∧pT∗ with the Vinogradov bracket, that is a generalization of the Courant bracket. As an application …
Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
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
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
In this paper we show that certain generalizations of the Cr-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.
This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.
problem Analyzing the relationship between Seiberg-Witten curves and topological recursion.
method Analytical approach using Seiberg-Witten family of curves.
result A generalized formula relating Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.
Proves conjecture on graph configuration spaces' complexity.
problem Topological complexity of graph configuration spaces.
method Lower bound derived from insights into aspherical spaces.
result Proves Farber's conjecture on stable topological complexity.
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a …
Topology design optimization offers tremendous opportunity in design and manufacturing freedoms by designing and producing a part from the ground-up without a meaningful initial design as required by conventional shape design optimization approaches. Ideally, with adequate problem statements, to formulate and solve the…
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.
Involutions generate mapping class groups of infinite surfaces.
problem Generating involutions for mapping class groups of infinite surfaces.
method Analyzing infinite surfaces with n ends, showing involutions generate groups for n ≥ 6 and n ≥ 3.
result Involutions generate mapping class groups for n ≥ 6 and n ≥ 3.
Proposes TSBP for matching topological signal distributions.
problem Matching signal distributions on topological domains.
method Topological Schrödinger Bridge (TSBP) with linear topology-aware stochastic dynamics.
result Derives closed-form topological SB (TSB) for Gaussian boundary distributions.
This paper determines all possible topological symmetry groups of generalized Petersen graphs.
problem Identifying all topological symmetry groups of generalized Petersen graphs.
method Analyzing embeddings of generalized Petersen graphs in S3 and considering homeomorphisms. result All groups that can be topological symmetry groups of generalized Petersen graphs are identified.
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…
Method quantifies disentanglement of generative models using manifold topology.
problem Challenging and inconsistent measurement of disentanglement in generative models.
method Measures topological similarity of conditional submanifolds in learned representation.
result Method ranks models similarly to existing methods across multiple datasets.
This paper finds minimal sets of generators for mapping class groups of specific surfaces.
problem Finding minimal sets of generators for mapping class groups of infinite-type surfaces.
method Analyzing specific surfaces S(n) to determine minimal sets of generators. result Minimal sets of generators for Map(S(n)) are identified for n≥8 (3 elements), n≥3 (4 elements), and S(1) (2 elements). Generative Topological Networks (GTNs) simplify latent space generation.
problem Complex latent space generation in generative models.
method GTNs use topology theory for a simple, supervised learning approach.
result GTNs improve upon VAEs and converge quickly, generating realistic samples.
Proves planar graphs' configuration spaces have highest topological complexity.
problem Proving Farber's conjecture for planar graphs.
method Generic maximality argument for topological complexities.
result Generic maximality of topological complexities for planar graphs.
SOM-VQ tokenizes discrete models with semantic structure and navigable topology.
problem Lack of semantic structure in vector quantized representations limits interpretable human control.
method Combines vector quantization with Self-Organizing Maps to learn discrete codebooks with explicit topology.
result SOM-VQ produces more learnable token sequences and provides an explicit navigable geometry in code space.
Enhanced bounds on rho-invariants for 3-manifolds.
problem Establishing bounds on Cheeger-Gromov rho-invariants for 3-manifolds.
method Constructing chain null-homotopies with linear complexity.
result Linearly bounded complexity of constructed null-homotopies.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
The paper generalizes Yang-Mills theory to study four-manifold topology.
problem Understanding the topology of diffeomorphism groups of four-manifolds.
method Path integral formulation of supersymmetric Yang-Mills coupled to conformal supergravity.
result The invariants may contain nontrivial information about the topology of the diffeomorphism group.
Novel M-theory approach classifies topological phases of matter.
problem Classifying and understanding topological phases of matter.
method Establishing a correspondence between (2+1)d topological field theories and non-hyperbolic 3-manifolds, identifying topological phases from internal wrapped 3-manifolds.
result Paves a new route toward the classification of topological phases of matter, including fermionic and non-unitary phases.
It is known that the topological T-duality exchanges H and F-fluxes. In this paper, we reformulate the topological T-duality as an exchange of two Lie algebroids in the generalized tangent bundle. Then, we apply the same formulation to the Poisson-generalized geometry, which is introduced in arXiv:1408.2649 to defi…
Let G be a lattice in PSL(2,C). The pro-normal topology on G is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup H<G is closed in the pro-normal topology. As a corollary w…
In this paper we show that topological subgroupoids of Lie groupoids, under special circumstances are Lie subgroupoids. Giving an example, we indicate that having the same topological dimension is a necessary condition for topological subgroupoids to be Lie subgroupoids. Also, we provide some conditions for double subg…
BiHermitian geometry, discovered long ago by Gates, Hull and Roceck, is the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target spac…
Paper introduces a new topological loss for better convergence.
problem Optimizing topological losses for model's desired topological behavior.
method Introduces a new regularized topology-aware loss function.
result Guarantees efficient optimization of the new loss function.