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CATEGORIES:Lecture
DESCRIPTION:The Dehn complex: scissors congruence\, K-theory\, and regulato
rs\n\nAbstract: 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 \(Q = \bigcup_{i=1}^nQ_i\) such t
hat 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 c
onstructing a second scissors congruence invariant now called the "Dehn inv
ariant\," and showing that a cube and a regular tetrahedron never have equa
l Dehn invariants\, regardless of their volumes. We can then restate Hilbe
rt'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 ge
ometries. However\, the problem remains open past dimension 4. By iteratin
g Dehn invariants Goncharov constructed a chain complex\, and conjectured t
hat the homology of this chain complex is related to certain graded portion
s of the algebraic K-theory of the complex numbers\, with the volume appear
ing as a regulator. In joint work with Jonathan Campbell\, we have constru
cted a new analysis of this chain complex which illuminates the connection
between the Dehn complex and algebraic K-theory\, and which opens new route
s for extending Dehn's results to higher dimensions. In this talk we will d
iscuss this construction and its connections to both algebraic and Hermitia
n K-theory\, and discuss the new avenues of attack that this presents for t
he generalized Hilbert's third problem.
DTEND:20201030T003000Z
DTSTAMP:20201129T084314Z
DTSTART:20201029T234500Z
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SUMMARY:Math Colloquium: Inna Zakharevich\, Cornell University
UID:tag:localist.com\,2008:EventInstance_34352097995828
URL:https://events.reed.edu/event/virtual_math_colloquium_inna_zakharevich_
cornell_university
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