arXiv research

PERCEPT detects changes in high-dimensional data streams using topological data analysis.

Detects graph topology changes from noisy signals using prior spectral information.

Researchers prove a special case of Borde-Sorkin's conjecture about topology change in spacetimes.

We introduce a novel geometry-oriented methodology, based on the emerging tools of topological data analysis, into the change point detection framework. The key rationale is that change points are likely to be associated with changes in geometry behind the data generating process. While the applications of topological …

We study topology change in (2+1)D gravity coupling with non-Abelian SO(2,1) Higgs field from the point of view of Morse theory. It is shown that the Higgs potential can be identified as a Morse function. The critical points of the latter (i.e. loci of change of the spacetime topology) coincide with zeros of the Higgs …

TADA detects anomalies in time series using topological data analysis.

Neural networks simplify complex data topologies into simpler ones.

Lectures on mean curvature flow and its related equations.

Study shows non-orientable manifolds restrict signature-changing metrics globally.

QP perspective on Poisson-Lie T-duality topology changes.

Study nondegenerate singularities in mean curvature flow.

Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty, a]$ of a Morse function $f: M \to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a cri…

Paper shows Ricci flow with surgery converging to Taub-NUT metric.

New lattice path method for statistical inference of persistent diagrams.

We connect topological changes that can occur in $3$-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena, including relationships between quantum entanglement and wormhole formation. By considering the initial manifold as the $3$-dimensional spatial sectio…

Paper proposes a method to improve circular coordinate representation for detecting changes in high-dimensional datasets.

In this article we prove that any unitary, axiomatic topological quantum field theory in four-dimensions can not detect changes in the smooth structure of M, a simply connected, closed (compact without boundary), oriented smooth manifold. However, as Donaldson-Witten theory (a topological quantum field theory but not a…

The study explores spacetimes with changing spatial curvature, leading to topological transitions.

Paper tackles dynamic behavior of variable topology mechanisms, presenting new transition conditions.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics…

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics…

We show that many Lorentzian manifolds of dimension >2 do not admit a spacelike codimension-one foliation, and that almost every manifold of dimension >2 which admits a Lorentzian metric at all admits one which satisfies the dominant energy condition and the timelike convergence condition. These two seemingly unrelated…

In this paper, we extend the formal definition of topological surgery by introducing new notions in order to model natural phenomena exhibiting it. On the one hand, the common features of the presented natural processes are captured by our schematic models and, on the other hand, our new definitions provide the theoret…

Classical topological concepts are applied to understand high performance computing simulations of molecules writhing in three dimensional space. These simulations produce peta-bytes of floating point data, to describe 3 dimensional changes in molecular structure. A zero-th order analysis is achieved by viewing a compu…

This study examined how the correlation and network structure of 30 global indices and 145 local Korean indices belonging to the KOSPI 200 have changed during the 13-year period, 2000-2012. The correlations among the indices were calculated. The results showed that although the average correlations of the global indice…

We simplify neural networks to 3D to study their topological changes.

Framework for analyzing dynamic topological changes in point clouds using persistent homology and dynamic optimal transport.

We derive the topological obstruction to spin-Klein cobordism. This result has implications for signature change in general relativity, and for the $N=2$ superstring.

The study connects lamination and orbit closures in hyperbolic manifolds.

General equilibrium is the dominant theoretical framework for economic policy analysis at the level of the whole economy. In practice, general equilibrium treats economies as being always in equilibrium, albeit in a sequence of equilibria as driven by external changes in parameters. This view is sometimes defended on t…

We develop a topology data analysis-based method to detect early signs for critical transitions in financial data. From the time-series of multiple stock prices, we build time-dependent correlation networks, which exhibit topological structures. We compute the persistent homology associated to these structures in order…

Detects changes in brain signal topology to predict epileptic seizures.

Continuous analysis techniques for deforming domains in manifolds.

Paper classifies pillow box isometric deformations preserving crease patterns.

Locates entanglement in curves using knot intensity distribution.

Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensiona…

Scattering transforms are non-trainable deep convolutional architectures that exploit the multi-scale resolution of a wavelet filter bank to obtain an appropriate representation of data. More importantly, they are proven invariant to translations, and stable to perturbations that are close to translations. This stabili…

Study of disks in complex projective space with specific fundamental groups.

We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions,…

New method detects dynamical system changes in time series data.

The financial crisis clearly illustrated the importance of characterizing the level of 'systemic' risk associated with an entire credit network, rather than with single institutions. However, the interplay between financial distress and topological changes is still poorly understood. Here we analyze the quarterly inter…

Graph neural networks (GNNs) have emerged as a powerful tool for nonlinear processing of graph signals, exhibiting success in recommender systems, power outage prediction, and motion planning, among others. GNNs consists of a cascade of layers, each of which applies a graph convolution, followed by a pointwise nonlinea…

Study detects P-type bifurcations in single system realizations using unreliable kernel density estimates.

We study the physics of globally consistent four-dimensional $\mathcal{N}=1$ supersymmetric M-theory compactifications on $G_2$ manifolds constructed via twisted connected sum; there are now perhaps fifty million examples of these manifolds. We study a rich example that exhibits $U(1)^3$ gauge symmetry and a spectrum o…

Focusing on a small set of proteins that i) fold in a concerted, all-or-none fashion and ii) do not contain knots or slipknots, we show that the Gauss linking integral, the torsion and the number of sequence-distant contacts provide information regarding the folding rate. Our results suggest that the global topology/ge…

Graph neural networks (GNNs), consisting of a cascade of layers applying a graph convolution followed by a pointwise nonlinearity, have become a powerful architecture to process signals supported on graphs. Graph convolutions (and thus, GNNs), rely heavily on knowledge of the graph for operation. However, in many pract…

High dimensional piecewise stationary graphical models represent a versatile class for modelling time varying networks arising in diverse application areas, including biology, economics, and social sciences. There has been recent work in offline detection and estimation of regime changes in the topology of sparse graph…

The paper uses topological concepts to analyze neural networks, revealing complex structure and dynamics.