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.

Accurate segmentation of organoids in bright-field microscopy is essential for drug screening and personalized medicine, yet separating touching instances remains challenging. We present a training-free method that combines phase congruency and persistent homology to delineate touching instances without shape priors or learned representations. By utilizing maximally persistent H₁ cycles with their birth and death simplices, our method remains robust to common brightfield imaging artifacts while producing interpretable separation of contours that align with true organoid boundaries.

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.