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Many solutions to problems arising in discrete geometry have come from insights in equivariant topology. Configuration-Space/Test Map (CS/TM) type setups, pioneered by Zivaljevic, offer reductions of combinatorial or geometric facts to showing the nonexistence of certain $G$-equivariant maps $f:X \to V\setminus Z$. In particular, partitions of objects by arcs, planes, and convex sets, and Tverberg theorems have been particularly amenable to topological methods , since their solutions affect the global structure of the relevant topological objects. However, there have been limits to the method as demonstrated by a failure to solve of the celeberated and now settled Topological Tverberg conjecture and, more generally, difficulty in finding sharp bounds for various conjectures. Nonetheless, we seek to employ characteristic classes, a cohomological invariant common to Borsuk-Ulam type problems, since these allow us to use explicit polynomial calculations to sharpen results to related problems. While determining sharp topological results for equipartition problems is a hard problem, there has been recent success in finding precise solutions by adding geometric constraints to the problem of plane equipartitions. This suggests that the polynomial method still has its use in related problems, and we employ these methods to``cascading Makeev" type problems.
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Mejia, Andres N., "Applications of Equivariant Topology In Cascading Makeev Problems" (2018). Senior Projects Spring 2018. 162.