Pure Maths Colloquium: Hilbert's Third Problem and Scissors Congruence Groups
19 May 2014 16:00 in CM221
Hilbert's third problem was to find two polyhedra of equal volume
neither of which can be subdivided into
finitely many pieces and re-assembled to equal the other (we say they
are `scissors-congruent' if this can be done).
It was solved in 1900 by Max Dehn, who introduced a new invariant of
(scissors-congruence classes of) polyhedra for
the purpose. Much later, in 1965, J. P. Sydler showed that volume and
Dehn invariant are a complete set of invariants for
classes of polyhedra in 3-dimensional Euclidean space.
However, the corresponding problems for hyperbolic and spherical space
have been much studied in the last thirty years
because of their connections with K-theory, motivic cohomology,
regulators and polylogarithms,
homology of Linear groups and several other topics of current interest.
I will give an overview of the history of these questions and discuss
some recent related developments.
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