Rade Zivaljevic (6/27/17) Bedlewo: Topological methods in discrete geometry; new developments

Some new applications of the configurations space/test map scheme can be
found in Chapter 21 of the latest (third) edition of the Handbook of Discrete
and Computational Geometry [2]. In this lecture we focus on some of the
new developments which, due to the limitations of space, may have been
included in the Chapter 21 only partially (or were not mentioned at all).
Among the promising new research directions is the ‘algebraic topology
of cooperative games’. Recall that in cooperative games the ‘players’ create
coalitions in order to achieve some common goal (voting games, profit
games, ‘simple games’ of von Neumann and Morgenstern, etc.).
One of the central new ideas is to study and compare configuration
spaces (usually simplicial complexes) arising in cooperative game theory
(threshold complexes, simple games) with complexes arising as obstructors
for embedding (mapping) spaces into higher dimensional euclidean spaces
without double (multiple) points (Kuratowski graphs, Tverberg–Van
Kampen–Flores obstructions, r-unavoidable complexes (Gromov–Blagojević
–Frick–Ziegler reduction), etc.)
As an illustration we outline a new proof (based on symmetrized deleted
joins and discrete Morse theory) of the very general ‘balanced Van
Kampren–Flores theorem’ ([2, Theorem 1.2]), which confirmed a conjecture
of Blagojević, Frick, and Ziegler [1].