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Organizers: Laura Anderson, Michael Dobbins, and Thomas Zaslavsky.
While equivariant methods have seen many fruitful applications in geometric combinatorics, their inability to answer the now settled Topological Tverberg Conjecture has made apparent the need to move beyond the use of Borsuk-Ulam-type theorems alone. This impression holds as well for one of the most famous problems in the field, dating back to at least 1960, which seeks the minimum dimension d:=Δ(m;k) such that any m mass distributions in Rd can be simultaneously equipartitioned by k hyperplanes. Precise values of Δ(m;k) have been obtained in only few cases, and the best-known general upper bound U(m;k) typically far exceeds the conjectured tight lower bound arising from degrees of freedom. By analogy with the "constraint method" of Blagojević, Frick, and Ziegler for Tverberg-type results, we show how the imposition of further conditions – on the hyperplane arrangements themselves (e.g., orthogonality, prescribed flat containment) or on the equipartition of additionally prescribed collections of masses by successively fewer hyperplanes ("cascades") – yields a variety of optimal results in dimension U(m;k), including in dimensions below Δ(m+1;k), which are still extractable via equivariance. Among these are families of exact values for full orthogonality, "maximal" cascades, and some strengthened equipartitions even in dimension Δ(m;k).