Hilbert's third problem asks: do there exist two polyhedra

with the same volume which are not scissors congruent? In other

words, if $P$ and $Q$ are polyhedra with the same volume, is it always

possible to write $P = \bigcup_{i=1}^n P_i$ and $Q = \bigcup_{i=1}^n

Q_i$ such that the $P$'s and $Q$'s intersect only on the boundaries

and such that $P_i \cong Q_i$? In 1901 Dehn answered this question in

the negative by constructing a second scissors congruence invariant

now called the "Dehn invariant," and showing that a cube and a regular

tetrahedron never have equal Dehn invariants, regardless of their

volumes. We can then restate Hilbert's third problem: do the volume

and Dehn invariant separate the scissors congruence classes? In 1965

Sydler showed that the answer is yes; in 1968 Jessen showed that this

result extends to dimension 4, and in 1982 Dupont and Sah constructed

analogs of such results in spherical and hyperbolic geometries.

However, the problem remains open past dimension 4. By iterating Dehn

invariants Goncharov constructed a chain complex, and conjectured that

the homology of this chain complex is related to certain graded

portions of the algebraic K-theory of the complex numbers, with the

volume appearing as a regulator. In joint work with Jonathan

Campbell, we have constructed a new analysis of this chain complex

which illuminates the connection between the Dehn complex and

algebraic K-theory, and which opens new routes for extending Dehn's

results to higher dimensions. In this talk we will discuss this

construction and its connections to both algebraic and Hermitian

K-theory, and discuss the new avenues of attack that this presents for

the generalized Hilbert's third problem.

with the same volume which are not scissors congruent? In other

words, if $P$ and $Q$ are polyhedra with the same volume, is it always

possible to write $P = \bigcup_{i=1}^n P_i$ and $Q = \bigcup_{i=1}^n

Q_i$ such that the $P$'s and $Q$'s intersect only on the boundaries

and such that $P_i \cong Q_i$? In 1901 Dehn answered this question in

the negative by constructing a second scissors congruence invariant

now called the "Dehn invariant," and showing that a cube and a regular

tetrahedron never have equal Dehn invariants, regardless of their

volumes. We can then restate Hilbert's third problem: do the volume

and Dehn invariant separate the scissors congruence classes? In 1965

Sydler showed that the answer is yes; in 1968 Jessen showed that this

result extends to dimension 4, and in 1982 Dupont and Sah constructed

analogs of such results in spherical and hyperbolic geometries.

However, the problem remains open past dimension 4. By iterating Dehn

invariants Goncharov constructed a chain complex, and conjectured that

the homology of this chain complex is related to certain graded

portions of the algebraic K-theory of the complex numbers, with the

volume appearing as a regulator. In joint work with Jonathan

Campbell, we have constructed a new analysis of this chain complex

which illuminates the connection between the Dehn complex and

algebraic K-theory, and which opens new routes for extending Dehn's

results to higher dimensions. In this talk we will discuss this

construction and its connections to both algebraic and Hermitian

K-theory, and discuss the new avenues of attack that this presents for

the generalized Hilbert's third problem.