We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we our method four two-dimensional lattice spin models: Ising, square ice, XY, fully-frustrated XY models. In particular, use persistent homology, which computes births deaths individual topological features as a coarse-graining scale or sublevel threshold is incr...

Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD)....

Topological descriptors such as the generators of homology groups are very useful in the analysis of complex data sets. It is often desired to find the smallest such generators to help localize the interesting features. One interpretation of localization utilizes a covering of the underlying space and computes generators contained within these covers. A similar construction was later used to co...

In this work, we develop a pipeline that associates Persistence Diagrams to digital data via the most appropriate filtration for type of considered. Using grid search approach, determines optimal representation methods and parameters. The development such topological Machine Learning involves two crucial steps strongly affect its performance: firstly, must be represented as an algebraic object ...

4 We design and implement a framework for parallel computation of homology of cellular spaces over 5 field coefficients. We begin by cutting a space into local pieces. We then compute the homology of 6 each piece in parallel using the persistence algorithm. Finally we glue the pieces together by construct7 ing the Mayer-Vietoris blowup complex and computing its homology. Theoretically, we show ...

We prove that the existence of a $1$-Lipschitz retraction (a contraction) from space $X$ onto its subspace $A$ implies persistence diagram embeds into $X$. As tool we introduce tight injections modules as maps inducing said embeddings. show contractions always exist shortest loops in metric graphs and conjecture on planar all homology basis. Of primary interest are geodesic spaces. These act id...

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data characterization, identification and analysis (CIA). Indeed, persistent homology ha...

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed f...

We investigate the use of persistent homology, a tool from topological data analysis, as means to detect and quantitatively describe center vortices in $\mathrm{SU}(2)$ lattice gauge theory gauge-invariant manner. provide evidence for sensitivity our method by detecting vortex explicitly inserted using twisted boundary conditions deconfined phase. This inspires definition new phase indicator de...

We develop a unifying framework for the treatment of various persistent homology architectures using notion correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to mappings. one-dimensional case, among other things, allows us to: (i) treat persistence modules and zigzag algebraic objects same type; (ii) give categorical f...