From manifolds to data: towards an integrated mathematical lens for revealing shape and structure

Many ideas in modern mathematics—geometry, topology, spectral theory, and stochastic processes—are often presented as separate theories, even though they arise from the same underlying structure viewed through different mathematical lenses. In this talk, I will present several results that illustrate how concrete connections between these perspectives can be established. In particular, we will see:

(i) how curvature is used to detect equivalences in graphs and hypergraphs (joint with Nina Otter); (ii) how graphs can be used to characterize bipartiteness in simplicial complexes; and (iii) what geometric flows can tell us about the topology of manifolds and simplicial complexes (joint with Juergen Jost).

Several of the ideas and results presented here are joint work with, inspired by, or shaped through the support of Sayan, in whose memory this talk is given.