Related papers: Persistent Homology analysis of Phase Transitions
The topological theory of phase transitions has its strong point in two theorems proving that, for a wide class of physical systems, phase transitions necessarily stem from topological changes of some submanifolds of configuration space. It…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
Topological data analysis, as a tool for extracting topological features and characterizing geometric shapes, has experienced significant development across diverse fields. Its key mathematical techniques include persistent homology and the…
We develop a method for analyzing spatial and spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), which allows one to algorithmically detect geometric voids in a…
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…
We critically analyze the possibility of finding signatures of a phase transition by looking exclusively at static quantities of statistical systems, like e.g., the topology of potential energy sub-manifolds (PES). This topological…
In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the…
Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often…
Higher-form symmetries are a valuable tool for classifying topological phases of matter. However, emergent higher-form symmetries in interacting many-body quantum systems are not typically exact due to the presence of topological defects.…
We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric…
Understanding the magnetic fields of the Sun is essential for unraveling the underlying mechanisms driving solar activity. Integrating topological data analysis techniques into these investigations can provide valuable insights into the…
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally,…
The dynamics of phase transitions plays a crucial r\^ole in the so-called interface between high energy particle physics and cosmology. Many of the interesting results generated during the last fifteen years or so rely on simplified…
Within the context of topological data analysis, the problems of identifying topological significance and matching signals across datasets are important and useful inferential tasks in many applications. The limitation of existing solutions…
We report upon the numerical computation of the Euler characteristic \chi (a topologic invariant) of the equipotential hypersurfaces \Sigma_v of the configuration space of the two-dimensional lattice $\phi^4$ model. The pattern…
Understanding and predicting thermal transport in disordered materials remains a significant challenge due to the absence of periodicity and the complex nature of medium-range structural motifs. In this work, we investigate amorphous…
Persistent homology theory is a relatively new but powerful method in data analysis. Using simplicial complexes, classical persistent homology is able to reveal high dimensional geometric structures of datasets, and represent them as…
The topological phase transition in the Qi-Wu-Zhang model is studied using a real-space approach. An effective Hamiltonian for the topologically protected edge-modes in a finite-size system is developed. The topological phase transition is…