We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups and discuss topological consequences. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. Of particular focus is the case of a toric arrangement: a finite collection of codimension-one subtori in a complex torus. If the intersection pattern of the subtori satisfies the combinatorial condition of supersolvability, the complement of the toric arrangement sits atop a tower of fiber bundles. This structure provides insight into topological invariants of these toric arrangement complements, including the homotopy groups, cohomology, and topological complexity. Based on joint work with Daniel C. Cohen and Emanuele Delucchi.
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Gromov-Wasserstein (GW) distances provide a method for comparing probability measures defined on different metric spaces, thereby giving an optimal transport-inspired variant of the well-known Gromov-Hausdorff distance. As GW distances admit computationally tractable approximations, they have become popular in machine learning applications where one wishes to learn trends in a dataset consisting of incomparable spaces, such as ensembles of graphs. In this talk, I will overview recent advances in the theory of GW distances. In particular, I will discuss a certain approximation technique which relies on comparing the distributions of pairwise distances between metric measure spaces. This approach naturally gives rise to fascinating questions about the geometrical and topological features that are encoded in this distributional information, and I will explain some partial answers to these questions.
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