Rade Zivaljevic: The method of configuration spaces in topological combinatorics

The method of configuration spaces, known also as the configuration space/test map scheme, has been for decades one of very useful tools for applying topological methods in discrete geometry and combinatorics. We review some of more recent advances, comparing new and old ideas and old and new configuration spaces (chessboard complexes and their generalizations).
The first circle of applications include extensions and refinements of N. Alon’s necklace-splitting theorem (almost equicardinal, binary, envy free necklace-splitting). The second application is an Optimal colored Tverberg theorem for multisets of colored points, which extends to multisets known (Type A and Type C) optimal colored Tverberg theorems. The third application is to mathematical economics where we propose a new approach to the problem of envy-free division (extensions of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the „cake“).
This is joint work with Duško Jojić (University of Banja Luka) and Gaiane Panina (St. Petersburg State University and Steklov Mathematical Institute).