Topology Of Manifolds Research Articles

Encoding manifolds constructed from grating responses organize responses to natural scenes in cortical visual areas.

We have created "encoding manifolds" to reveal the overall responses of a brain area to a variety of stimuli. Encoding manifolds organize response properties globally: each point on an encoding manifold is a neuron, and nearby neurons respond similarly to the stimulus ensemble in time. We previously found, using a large stimulus ensemble including optic flows, that encoding manifolds for the retina were highly clustered, with each cluster corresponding to a different ganglion cell type. In contrast, the topology of the V1 manifold was continuous. Now, using responses of individual neurons from the Allen Institute Visual Coding-Neuropixels dataset in the mouse, we infer encoding manifolds for V1 and for five higher cortical visual areas (VISam, VISal, VISpm, VISlm, and VISrl). We show here that the encoding manifold topology computed only from responses to various grating stimuli is also continuous, not only for V1 but also for the higher visual areas, with smooth coordinates spanning it that include, among others, orientation selectivity and firing-rate magnitude. Surprisingly, the encoding manifold for gratings also provides information about natural scene responses. To investigate whether neurons respond more strongly to gratings or natural scenes, we plot the log ratio of natural scene responses to grating responses (mean firing rates) on the encoding manifold. This reveals a global coordinate axis organizing neurons' preferences between these two stimuli. This coordinate is orthogonal (i.e., uncorrelated) to that organizing firing rate magnitudes in VISp. Analyzing layer responses, a preference for gratings is concentrated in layer 6, whereas preference for natural scenes tends to be higher in layers 2/3 and 4. We also find that preference for natural scenes dominates the responses of neurons that prefer low (0.02 cpd) and high (0.32 cpd) spatial frequencies, rather than intermediate ones (0.04 to 0.16 cpd). Conclusion: while gratings seem limited and natural scenes unconstrained, machine learning algorithms can reveal subtle relationships between them beyond linear techniques.

Causal completion of globally hyperbolic conformally flat spacetimes

In their paper Geroch, Kronheimer, and Penrose, “Ideal points in space-time,” [Proc. R. Soc. London, Ser. A 327, 545–567 (1972)] introduced a way to attach ideal points to a spacetime M, defining the causal completion of M. They established that this is a topological space which is Hausdorff when M is globally hyperbolic. In this paper, we prove that if, in addition, M is simply-connected and conformally flat, its causal completion is a topological manifold with boundary homeomorphic to S × [0, 1] where S is a Cauchy hypersurface of M. We also introduce three remarkable families of globally hyperbolic conformally flat spacetimes and provide a description of their causal completions.

Fast and Efficient Drone Path Planning Using Riemannian Manifold in Indoor Environment

This paper introduces an innovative dual-path planning algorithm rooted in a topological three-dimensional Riemannian manifold (T3DRM) to optimize drone navigation in complex environments. It seamlessly integrates strategies for both discrete and continuous obstacles, employing spherical navigation for the former and hyperbolic paths for the latter. Serving as a transformative tool, the T3DRM facilitates efficient path planning by transitioning between discrete and continuous domains. In uncertain environments with unpredictable obstacle positions, our methodology categorizes these positions as discrete or continuous based on their distribution patterns. Discrete obstacles exhibit random distributions, while continuous obstacles display symmetrical patterns with continuity. Leveraging topological metrics, the T3DRM efficiently classifies these patterns for effective path planning. The findings of this research demonstrate the efficiency of path planning based on classified obstacle positions, enabling swift and efficient drone navigation. This research introduces a pioneering application of a T3DRM, accelerating drone navigation in uncertain environments through a dual approach that simultaneously transforms navigation in primal and dual domains. By enabling spherical and hyperbolic navigation concurrently, the T3DRM offers a comprehensive solution to discrete and continuous path planning challenges. The proposed approach can be used for various indoor applications, especially for warehouse management, surveillance and security, navigation in complex structures, indoor farming, site inspection, healthcare facilities, etc.

Erratum: "Topology of homology manifolds"

Erratum: "Topology of homology manifolds"

A study of topology of the flip Stiefel manifolds

A study of topology of the flip Stiefel manifolds

1D CNNs and face-based random walks: A powerful combination to enhance mesh understanding and 3D semantic segmentation

In this paper, we present a novel face-based random walk method aimed at addressing the 3D semantic segmentation issue. Our method utilizes a one-dimensional convolutional neural network for detailed feature extraction from sequences of triangular faces and employs a stacked gated recurrent unit to gather information along the sequence during training. This approach allows us to effectively handle irregular meshes and utilize the inherent feature extraction potential present in mesh geometry. Our study's results show that the proposed method achieves competitive results compared to the state-of-the-art methods in mesh segmentation. Importantly, it requires fewer training iterations and demonstrates versatility by applying to a wide range of objects without the need for the mesh to adhere to manifold or watertight topology requirements.

Attributed graph subspace clustering with residual compensation guided by adaptive dual manifold regularization

As an important technology for recommendation system, attributed graph clustering has received extensive attention recently. Attributed graph clustering methods based on graph neural networks are the mainstream methods . However, their assumption that the attributes of adjacent nodes are similar is often not satisfied in the real world, which influences the clustering performance. To this end, this paper proposes an attributed graph subspace clustering algorithm with residual compensation guided by adaptive dual manifold regularization (ADMRGC). On the basis of the low rank representation subspace clustering (LRR) model, ADMRGC introduces attributed manifold regularization, topological manifold regularization and residual compensation, which make ADMRGC possible to utilize attributed similarity and topological similarity at the same time, thus solving the problem that traditional subspace clustering only considers attributed information. In addition, ADMRGC balances the contribution of node attributes and topology by adaptively weighting the dual manifold regularization, and uses the residual representation matrix to describe the difference between node attributed similarity and topological neighbor relationship. Therefore, ADMRGC can avoid the limitation of graph neural network models assuming that the attributes of adjacent nodes are similar. Experimental results on 7 public graph datasets show that ADMRGC can achieve the best clustering performance on both high-homophily and low-homophily datasets. Especially, for datasets with low homophily, ADMRGC as a shallow model can also achieve better clustering performance than state-of-the-art deep neural network models.

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We consider G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{G_2}\\,}}$$\\end{document} manifolds with a cohomogeneity two T2×SU(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\mathbb {T}}\\,}}^2\ imes {{\\,\ extrm{SU}\\,}}(2)$$\\end{document} symmetry group. We give a local characterization of these manifolds and we describe the geometry, including regularity and singularity analysis, of cohomogeneity one calibrated submanifolds in them. We apply these results to the manifolds recently constructed by Foscolo–Haskins–Nordström and to the Bryant–Salamon manifold of topology /S(S3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$/\\!\\!\\!S(S^3)$$\\end{document}. In particular, we describe new large families of complete T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\mathbb {T}}\\,}}^2$$\\end{document}-invariant associative submanifolds in them.

On the shape description of general solids using Morse theory

The automatic shape description of solids is a problem of interest in manufacturing engineering, amongst other related areas. This description can be either geometrical or topological in nature and can be applied to either surfaces or solids (embedded manifolds). Topological descriptions are specially interesting for the problem of shape comparison and retrieval, where one wants to know if a given shape resembles some other known shape. Some popular topological descriptions use Morse theory to study the topology of manifolds and encode their shape characteristics. A Morse function f is defined on the manifold and the manifold’s shape is indirectly studied by studying the behavior of the critical points of f. This family of methods is well defined for surfaces but does not consider the case of solids. In this paper we address the topological description of solids using Morse theory. Our methodology considers three cases: solids without internal boundaries, solids with internal boundaries and thin-walled solids. We present an algorithm to identify topological changes on these solids using the principle of shape decomposition by Morse handles. The presented algorithm deals with Morse functions that produce parallel planar level sets. Future endeavors should consider other candidate functions.

Weyl Points on Nonorientable Manifolds.

Weyl fermions are hypothetical chiral particles that can also manifest as excitations near three-dimensional band crossing points in lattice systems. These quasiparticles are subject to the Nielsen-Ninomiya "no-go" theorem when placed on a lattice, requiring the total chirality across the Brillouin zone to vanish. This constraint results from the topology of the (orientable) manifold on which they exist. Here, we ask to what extent the concepts of topology and chirality of Weyl points remain well defined when the underlying manifold is nonorientable. We show that the usual notion of chirality becomes ambiguous in this setting, allowing for systems with a nonzero total chirality. This circumvention of the Nielsen-Ninomiya theorem stems from a generic discontinuity of the vector field whose zeros are Weyl points. Furthermore, we discover that Weyl points on nonorientable manifolds carry an additional Z_{2} topological invariant which satisfies a different no-go theorem. We implement such Weyl points by imposing a nonsymmorphic symmetry in the momentum space of lattice models. Finally, we experimentally realize all aspects of their phenomenology in a photonic platform with synthetic momenta. Our work highlights the subtle but crucial interplay between the topology of quasiparticles and of their underlying manifold.

The Morse index theorem for mechanical systems with reflections

We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.

Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms ω with r=0,1,⋯,n. Because d∘d is always zero where d is the exterior differential, it follows that every exact r-form (i.e., ω=dλ where λ is an r−1-form) is closed (i.e., dω=0) but not every closed r-form is exact. This implies the existence of a third type of differential r-form that is closed but not exact. Such forms are called harmonic forms. Every smooth n-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham’s theory of differential forms relates the presence of r-dimensional holes in the underlying topology of a smooth n-manifold M to the presence of harmonic r-form fields on the smooth manifold. A large amount of theory is required to understand de Rham’s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham’s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains.

Brillouin Klein space and half-turn space in three-dimensional acoustic crystals

The Bloch band theory and Brillouin zone (BZ) that characterize wave-like behaviors in periodic mediums are two cornerstones of contemporary physics, ranging from condensed matter to topological physics. Recent theoretical breakthrough revealed that, under the projective symmetry algebra enforced by artificial gauge fields, the usual two-dimensional (2D) BZ (orientable Brillouin two-torus) can be fundamentally modified to a non-orientable Brillouin Klein bottle with radically distinct manifold topology. However, the physical consequence of artificial gauge fields on the more general three-dimensional (3D) BZ (orientable Brillouin three-torus) was so far missing. Here, we theoretically discovered and experimentally observed that the fundamental domain and topology of the usual 3D BZ can be reduced to a non-orientable Brillouin Klein space or an orientable Brillouin half-turn space in a 3D acoustic crystal with artificial gauge fields. We experimentally identify peculiar 3D momentum-space non-symmorphic screw rotation and glide reflection symmetries in the measured band structures. Moreover, we experimentally demonstrate a novel stacked weak Klein bottle insulator featuring a nonzero Z2 topological invariant and self-collimated topological surface states at two opposite surfaces related by a nonlocal twist, radically distinct from all previous 3D topological insulators. Our discovery not only fundamentally modifies the fundamental domain and topology of 3D BZ, but also opens the door towards a wealth of previously overlooked momentum-space multidimensional manifold topologies and novel gauge-symmetry-enriched topological physics and robust acoustic wave manipulations beyond the existing paradigms.

Actions of large finite groups on manifolds

In this paper, we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group [Formula: see text] on a manifold [Formula: see text], these results provide information on the restriction of the action to a subgroup of [Formula: see text] of index bounded above by a number depending only on [Formula: see text]. Some of these results refer to the algebraic structure of the group, such as being abelian or nilpotent or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups or to the action on cohomology.

Refinement and Computation Method for Line/Body Topological Relationships

Three-dimensional topological relationships serve as a theoretical foundation for quality control, update processing, and spatial analysis of three-dimensional spatial data in real-world three-dimensional GIS. The existing 3D topological relationship models are all basic relationship models that cannot distinguish the refined topological relationship between the line and the body with multiple intersections. In this study, we develop a 3D refined topological relationship description framework that draws from the two-dimensional refined topological relationship model, defines the unit intersection between the line and the body based on manifold topology, and proposes a method for describing the unit intersections between the line and the body considering Euler numbers and adjacency types. In total, 23 basic types between the line and the body are deduced. An example is provided to illustrate the distinguished refined topological relationship between the line and the body with multiple intersections. Subsequently, an algorithm for determining the basic type of line/body is developed. Finally, a line/body refined topological relationship computation prototype system is developed using the Nef polyhedron model, C++ language, and an open-source geometric algorithm library, and the effectiveness of our method is verified using actual building and pedestrian data.

Multi-task collaborative method based on manifold optimization for automated test case generation based on path coverage

Automated test case generation based on path coverage is not only a large-scale black-box optimization problem, but also a key scientific problem in software automatic testing technology. Evolutionary algorithms and other search-based algorithms are representative methods for this problem. However, existing research mainly focuses on the multi-function case, where generating test cases for multiple functions is difficult due to the combinatorial explosion of the path number. In this paper, we propose a multi-task collaborative method based on manifold optimization by considering the topological manifold relationship between the test case space (decision space) and the program path space (target space). This method achieves the goal of collaborative optimization for different function coverage tasks by coordinating the allocation of computing resources and knowledge transfer mechanisms among them. To verify the effectiveness of the proposed method, we compare it with general solution methods for single-function automated test case generation based on path coverage, such as the manifold-inspired search-based algorithm. The experimental results show that the proposed method outperforms the compared single-function optimization algorithms especially on programs with strong coding similarities. This study verifies the feasibility of collaborative optimization methods for solving large-scale black-box optimization problems, such as the automated test case generation based on path coverage, and expands their application scenarios.

The proper homotopy of Alexandroff spaces

This paper introduces the study of the infinity in the class of Alexandroff spaces by establishing the proper category of locally finite Alexandroff spaces. We observe that the McCord and Clader theorems are available in this setting. The paper can be considered as the starting point of a seemingly new research: relating the combinatorics and topology of the proper category of Alexandroff spaces to the proper invariants used in the geometric topology of non-compact polyhedra and manifolds.

Generalized soap bubbles and the topology of manifolds with positive scalar curvature

Generalized soap bubbles and the topology of manifolds with positive scalar curvature

Homology Growth, Hyperbolization, and Fibering

We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and—for almost all primes p—\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{F} _{p}$\\end{document}-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic 7-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.

The Properties of Topological Manifolds of Simplicial Polynomials

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition.