In this paper we study functions on the interval that have the same persistent homology, which is what we mean by the fiber of the persistence map. By imposing an equivalence relation called graph-equivalence, the fiber of the persistence map becomes finite and a precise enumeration is given. Graph-equivalence classes are indexed by chiral merge trees, which are binary merge trees where a left-right ordering of the children of each vertex is given. Enumeration of merge trees and chiral merge trees with the same persistence makes essential use of the Elder Rule, which is given its first detailed proof in this paper. arXiv:1706.06059v2 [math.AT]

Abstract. Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.

Abstract. This paper affirms a conjecture of MacPherson: that the derived category of cellular sheaves is equivalent to the derived category of cellular cosheaves. We give a self-contained treatment of cellular sheaves and cosheaves and note that certain classical dualities give rise to an exchange of sheaves with cosheaves. Following a result of Pitts that states that cosheaves are cocontinuous functors on the category of sheaves, we use the derived equivalence provided here to gain a novel description of compactly supported sheaf cohomology.

In this paper we consider two topological transforms based on Euler calculus: the persistent homology transform (PHT) and the Euler characteristic transform (ECT). Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset M of R d , and associates to each direction v ∈ S d−1 a shape summary obtained by scanning M in the direction v. These shape summaries are either persistence diagrams or piecewise constant integer valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes-each shape has a unique transform. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result provides the first (to our knowledge) finite bound required to specify any shape in certain uncountable families of shapes, bounded below by curvature. This result is perhaps best appreciated in terms of shattering number or the perspective that any point in these particular moduli spaces of shapes is indexed using a tree of finite depth.

This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level set persistence is presented using sheaves and cosheaves.

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. The first important outcome of our study is a clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees. Generic BHV trees on n + 1 leaf nodes fall into (2n − 1)!! distinct strata, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n+1)!n!2 −n . The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data and opens the door to doing more precise science.

No abstract

No abstract