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Bradley Hicks – Hilbert’s Third Problem and the Dehn Invariant – GMA Seminar
September 30, 2019 @ 4:00 pm - 5:00 pm
Title: Hilbert’s Third Problem and the Dehn Invariant
Abstract: Of Hilbert’s 23 problems posed in 1900 the third was resolved first a year later by Max Dehn. It was settled by Dehn’s construction of an algebraic invariant that detects when two polyhedra are not scissors-congruent to each other. This was just one early example of new algebraic approaches to attack geometric problems. While Dehn’s invariant applied only to polyhedra in 3 dimensional Euclidean space it has been generalized to higher dimensions and to polyhedra in different geometries (Spherical and Hyperbolic), and has been connected to ideas in K Theory. For the majority of this talk I will give a laid back presentation of the original question, its motivation, and solution by Dehn, which can be laid out very basically only requiring a familiarity with tensor products of Z-modules. Time permitting I will also show how the problem has been modernized and how the Dehn Invariant is connected to more modern notions.