Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when $M=\mathbb{R}$ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of $f$. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL is the new frontier for relational learning. TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

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.We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees.As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on n + 1 leaf nodes fall into (2n -1)!! distinct equivalence classes, 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 of our distribution in terms of Dirichlet convolution.This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the $p$-Wasserstein distance between the associated barcodes. For each $p\in[1,\infty]$, we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the $p=\infty$ case, this gives a novel proof of universality for the interleaving distance on merge trees.