Abstract
We will show that the famous, intractible 1959 Kadison-Singer
problem in $C^*$-algebras is equivalent to fundamental unsolved problems in a
dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as
well as explaining why each of these areas has volumes of literature on their
respective problems without a satisfactory resolution. In each of these areas
we will reduce the problem to the minimum which needs to be proved to solve
their version of Kadison-Singer. In some areas we will prove what we believe
will be the strongest results ever available in the case that Kadison-Singer
fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer.
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