Related papers: Computing Generalized Rank Invariant for 2-Paramet…
Symmetry arises often when learning from high dimensional data. For example, data sets consisting of point clouds, graphs, and unordered sets appear routinely in contemporary applications, and exhibit rich underlying symmetries.…
In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules can be viewed as representations for the poset…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a continuous vector-valued function. In this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in…
We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group…
We describe a class (called regular) of invariant generalized complex structures on a real semisimple Lie group G. The problem reduces to the description of admissible pairs (\gk, \omega), where \gk is an appropriate regular subalgebra of…
We dualize previous work on generalized persistence diagrams for filtrations to cofiltrations. When the underlying space is a manifold, we express this duality as a Poincar\'e duality between their generalized persistence diagrams. A heavy…
We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…
$1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation…
In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the…
In the present paper, we give an upper bound for the generic degree of the generalized Verschiebung between the moduli spaces of rank two stable bundles with trivial determinant.
In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution…
For a unital ring R, a Sylvester rank function is a numerical invariant which can be described in 3 equivalent ways: on finitely presented left R-modules, or on rectangular matrices over R, or on maps between finitely generated projective…
Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features.…
The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous…
The paper mostly collects material on generic rank of $A$--modules with respect to differential geometric applications. Our research was motivated by geometry of $A$--structures. In particular, we discuss the case where $A$ is an unitary…
Suppose $\mathbb{K}$ is a large enough field and $\mathcal{P} \subset \mathbb{K}^2$ is a fixed, generic set of points which is available for precomputation. We introduce a technique called \emph{reshaping} which allows us to design…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher…
Persistent homology is a popular technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a…