We introduce Orthogonal Möbius Inversion, a concept analogous to Möbius inversion on finite posets, applicable to order-preserving functions from a finite poset to the Grassmannian $\mathrm{Gr}(V)$ of an inner product space $V$. This notion relies critically on the inner product structure on $V$, enabling it to capture finer information than standard integer-valued persistence diagrams. Orthogonal inversion is a special case of the broader concept of orthomodular inversion, in which the target is an arbitrary orthomodular lattice (which we also identify). We apply orthogonal inversion to construct a “nonnegative” persistence diagram for any given multiparameter filtration $F$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$, by applying it to the birth–death spaces of $F$. Analogously to classical one-parameter persistence diagrams, these multiparameter Grassmannian persistence diagrams admit a straightforward interpretation. Specifically, for each segment $(b,d)\in \mathrm{Seg}(P)$: the Grassmannian persistence diagram canonically assigns a vector subspace of degree-$*$ cycles in $K$ that are born at $b$ and become boundaries at $d$, and this assignment is exhaustive at the homology level. This is joint work with Aziz Gülen and Zhengchao Wan.

Organoids are miniaturized representations of human organs and play an important role in precision medicine. They are widely used in applications such as drug discovery and personalized treatment development, making accurate instance segmentation essential. In this work, we present a method for organoid instance separation in bright-field images using phase congruency and persistent homology. Phase congruency provides illumination-invariant edge responses, which define a filtration over which persistent homology is computed. Focusing on H1(loop) features, maximally persistent cycles correspond to closed contours that align with organoid boundaries. These cycles are mapped back to the image domain to separate instance contours, enabling training-free delineation of touching organoids without reliance on shape priors or supervised learning. This approach demonstrates how persistent cycles can serve as a robust structural signal for resolving ambiguous boundaries under varying imaging conditions.

In this talk, for a finite group G, we consider G-metric spaces: metric spaces equipped with an isometric G-action. We introduce a G-equivariant Gromov–Hausdorff distance for compact G-metric spaces and derive lower bounds using equivariant persistent invariants and related constructions in equivariant topology. To analyze and compare these bounds, we further develop two complementary G-equivariant distances—the homotopy-type and interleaving distances—and establish stability relations linking them to the G-Gromov–Hausdorff distance. As applications: (1) we analyze how the G-actions descend to and enrich persistence modules and obtain lower bounds via the G-interleaving distance, comparing these to those induced by equivariant topology; (2) we prove equivariant rigidity and finiteness theorems; (3) we obtain sharp bounds on the Gromov–Hausdorff distance between spheres; and (4) we obtain a G-equivariant quantitative Borsuk–Ulam theorem. This is joint work with Sunhyuk Lim.