Many problems related to nonlinear analysis (qualitative investigation of differential equations with symmetries), topology (group actions on manifolds) and combinarotics (mass equipartitions, colored graphs) can be reduced to the following question raised by J. F. Adams: Let G be a compact Lie group, V and W orthogonal G-representations, and k a given integer. Does there exist a G-equivariant map f : S (V) → S (W) with deg(f, S(V)) = k (here "deg(f, S(V )") stands for the Brouwer degree of f and S(V) denotes the unit sphere in V)? Since 1963, this problem was intensively studied using different methods and techniques (equivariant obstruction theory, Borel spectral sequence method, fundamental domain method, equivariant K-theory, to mention a few). Our focus in this dissertation will be the case when G is a finite solvable group. The essential technical tools include geometric equivariant topology, representation theory, invariant theory and multilinear algebra.
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