Study on LS-category and topological complexity of connected sum manifolds.
problem Behavior of LS-category and topological complexity under connected sum operation.
method Analyzes the invariants LS-category and topological complexity for connected sum of manifolds.
result For orientable manifolds, $\cat(M\# N)=\max\{\cat M,\cat N\}$; for simply connected manifolds, $\TC (M\# N)\ge\max\{\TC M,\TC N\}$.
Develops topological concepts for Morrey-Sobolev bundles in high dimensions.
problem Lack of continuity in transition maps for Morrey-Sobolev bundles.
method Introduces topological isomorphism classes and uses connection-oriented approach.
result Derives approximability results for bundles and connections in Morrey-Sobolev setting.
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.
Topology connects to black holes and wormholes.
problem Understanding topological changes in 3D space.
method Direct connection via surgery.
result New insights into black hole and wormhole formation.
Research connects topological changes to black hole formation and wormholes.
problem Understanding topological changes in mathematical three-space.
method Direct connection via surgery to black hole and wormhole formation.
result New platform for exploring geometrical physics.
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.
New technique connects graph matching complexes to Morse theory for better topology understanding.
problem Understanding the topology of matching complexes of complete graphs.
method Developed discrete Morse theory technique to analyze Mn. result Showed Mn is geometrically (νn−1)-connected, improving on previous homotopical results. GANs can analyze graph topology, ranking edge sets by their importance.
problem Capturing topological features of graphs using GANs.
method Leveraging hierarchical connectivity structure of graphs, GANs rank edge sets by their contribution to topology reconstruction.
result GANs can preserve important topological features in graphs.
There is a concept in digital topology of a shy map. We define an analogous concept for topological spaces: We say a function is shy if it is continuous and the inverse image of every path-connected subset of its image is path-connected. Some basic properties of such maps are presented. For example, every shy map onto …
The paper improves bounds on topological complexity for certain manifolds.
problem Determining bounds on topological complexity for specific manifolds.
method Analyzing cohomology classes and their pullbacks, using Gromov norm.
result Improved bounds on topological complexity for connected sums.
Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomi…
The boundary of hyperbolic groups is locally simply connected.
problem Topology of hyperbolic group boundaries
method Proving local simple connectedness in terms of global topology
result Boundary is locally simply connected if and only if complement of any point is simply connected
In this paper we try to establish a connection between a three-dimensional Lotka--Volterra dynamical system and two-dimensional topological surgery. There are many physical phenomena exhibiting two-dimensional topological surgery through a `hole drilling' process. By our connection, such phenomena may be modelled mathe…
We propose Sparse Neural Network architectures that are based on random or structured bipartite graph topologies. Sparse architectures provide compression of the models learned and speed-ups of computations, they can also surpass their unstructured or fully connected counterparts. As we show, even more compact topologi…
Space of hyperbolic surfaces is path-connected.
problem Topology of hyperbolic surfaces and their subspaces.
method Constructing paths using Fenchel-Nielsen coordinates and shrinking curves.
result Path-connectivity of the space of hyperbolic surfaces.
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
Connected sums defined for codimension two locally flat submanifolds in higher dimensions.
problem Defining connected sums for codimension two locally flat submanifolds in various dimensions.
method Using results from higher dimensional topological manifolds and four-manifolds, defining connected sums for codimension two locally flat submanifolds.
result A well-defined connected sum exists up to orientation preserving homeomorphism.
Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.
problem Proving topological and smooth pseudo-isotopies of simply connected 4-manifolds.
method Provided different arguments that bypass the replacement criterion, thus completing Quinn's proofs.
result Corrected and completed Quinn's proofs of both topological and stable smooth pseudo-isotopy theorems.
The study examines whether certain 4-manifolds can have flat embedded spheres.
problem Whether specific 4-manifolds can have smooth or topologically flat embedded 2-spheres.
method Analyzes the second homology of specific simply connected closed 4-manifolds.
result Determines conditions under which certain 4-manifolds can have flat embedded spheres.
New communication topologies improve deep reinforcement learning efficiency.
problem Optimizing communication topology for faster and more robust learning in deep reinforcement learning.
method Introduced alternative network topologies (Erdos-Renyi random graphs) and compared their performance with fully-connected and star topologies.
result Erdos-Renyi random graphs outperform fully-connected networks in deep reinforcement learning tasks.
Dunkl connections on complex plane don't preserve metrics.
problem Preserving metrics with Dunkl connections on \(\mathbb{C}^2\).
method Analysis of the topology of spherical tori with conical points.
result General Dunkl connections on \(\mathbb{C}^2\) do not preserve non-zero Hermitian forms.
Classifies 1-connected 8-manifolds with specific homology.
problem Classifying 1-connected 8-manifolds with the same homology as S3imesS5. method Topological and smooth manifold classification.
result Classification of 1-connected 8-manifolds with the same homology as S3imesS5. Topological autoencoders preserve complex structures in latent spaces.
problem Preserving topological structures in latent representations of autoencoders.
method Using persistent homology, a topological data analysis technique, to calculate topological signatures of input and latent spaces and derive a differentiable topological loss term.
result Our approach preserves multi-scale connectivity information in latent representations, leading to favorable latent representations on synthetic and real-world data.
The purpose of this paper is to discuss how topology and geometry provide, in many instances, the connective tissue that enables logical comprehension. We illustrate this theme with many examples including Venn diagrams, knot diagrams, knot-logical diagrams and an arrow of reference that elucidates self-reference and G…
Kernel for multi-parameter persistent homology connects TDA with ML.
problem Connecting persistent homology with machine learning for multivariate data.
method Integrating a one-parameter kernel weighted along straight lines.
result Stable and efficiently computable kernel for multi-parameter persistence.
Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a vol…
New method associates topological classes to Sobolev bundles in critical dimensions.
problem No standard topology in critical dimensions for Sobolev bundles.
method Coulomb gauges and strong approximations of smooth connections.
result Topology stabilizes for sequences of Sobolev bundles with bounded Yang-Mills energy.
Study classifies moduli spaces of spin connections on 3D homogeneous spaces.
problem Classifying moduli spaces of spin connections on 3D homogeneous spaces.
method Analysis of the topology of moduli spaces, focusing on finite-dimensional topological manifolds with trivial homotopy groups.
result Moduli spaces are finite-dimensional topological manifolds with trivial homotopy groups, essential for consistent cosmological models.
It is shown that the hyperspace of all nonempty closed subsets $\Cld_{AW}(X)$ of a separable metric space X endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of X is proper, locally connected and contains no bounded connected component, X is topologi…
In this paper we give a characterization of 2-dimensional topological field theories over a space X as Frobenius bundles with connections over LX, the free loop space of X. This is a generalization of the folk theorem stating that 2-dimensional topological field theories (over a point) are described by finite-dim…
Connects 4-manifold topology to topological modular forms.
problem Understanding the topology of 4-manifolds.
method Topologically twisted compactification of 6d (1,0) theories on 4-manifolds.
result New invariant of 4-manifolds using equivariant weakly holomorphic modular forms.
Topological surgery occurs in natural phenomena where two points are selected and attracting or repelling forces are applied. The two points are connected via an invisible `thread'. In order to model topologically such phenomena we introduce dynamics in 1-, 2- and 3-dimensional topological surgery, by means of attracti…
We present the alternative topological twisting of N=4 Yang-Mills, in which the path integral is dominated not by instantons, but by flat connections of the COMPLEXIFIED gauge group. The theory is nontrivial on compact orientable four-manifolds with nonpositive Euler number, which are necessarily not simply connected. …
Survey on topological properties of discrete groups.
problem Understanding asymptotic topological properties of discrete groups.
method Exploring quasi-simple filtration, geometric simple connectivity, and topological inverse-representations.
result Discussing the conditions under which all finitely presented groups satisfy these properties.
For each composite number n=2k, there does not exist a single connected closed (n+1)-manifold such that any smooth, simply-connected, closed n-manifold can be topologically flat embedded into it. There is a single connected closed 5-manifold W such that any simply-connected, 4-manifold M can be topologica…
Formulates a new connection between topological and geometric categories.
problem No specific problem stated; focuses on category formulation.
method Formulates a connection between a topological and geometric category.
result Provides a more precise and improved version of a previous proposal.
Explains how Reeb dynamics connects to topology.
problem Understanding the relationship between Reeb dynamics and topology.
method Contact cuts, lifting group actions, and other construction methods.
result Methods for constructing Reeb flows with finite periodic orbits.
Study local limits of subgroups in SL3(R).
problem Understanding subgroups of SL3(R) in a specific topology. method Conjugation and Chabauty topology analysis.
result Described local limits of all connected subgroups.
Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
New communication topologies improve deep reinforcement learning performance.
problem Improving performance of learning agents in distributed reinforcement learning.
method Examined four graph families for communication topologies and found Erdos-Renyi random graphs to outperform fully connected topologies.
result Erdos-Renyi random graphs can improve performance of distributed learning agents.
Study topological components of surface group representations into SL(2,R) and PSL(2,R).
problem Understanding the structure of representation spaces of surface groups.
method Utilized the signature formula to determine connected components of representation spaces.
result Determined the number of connected components for different types of holonomies.
The paper studies spectral convergence of connections on vector and principal bundles.
problem Continuity of eigenvalues of connection Laplacians on vector bundles.
method Introducing a new topology on metric measure spaces with isometric G-actions to analyze convergence of G-connections. result Established spectral convergence of connections on vector and principal bundles.
Minimal hypersurfaces can't always be connected by mean curvature flow.
problem Existence of connecting mean curvature flows for minimal hypersurfaces.
method Minimal hypersurface analogue of gradient flow trajectories between critical points.
result Additional topological and variational obstructions to connecting mean curvature flows.
The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…
The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
The wall-crossing formula for Donaldson invariants of smooth, simply connected four manifolds with b+=1 is shown to be a topological invariant of the manifold for reducible connections with two or fewer singular points. The explicit formulas derived agree with those of Ellingsrud and Gottische and Friedman and Qin f…
Study distance functions on manifolds linking geometry to topology.
problem Understanding the relationship between curvature and topology on manifolds.
method Analyzing distance functions and their connection to manifold geometry.
result Alternative proofs of theorems linking curvature and topology.
The study characterizes neural network capacity using algebraic topology.
problem Characterizing the capacity of neural networks based on data complexity.
method Reframing architecture selection as data complexity understanding, using algebraic topology.
result Neural networks exhibit topological phase transitions at different levels of dataset complexity.