[Binghamton Geometry Topology Seminar] Geometry and Topology seminar: Cary Malkiewich

[Cary Malkiewich]

Hi everyone,

This week I'll be speaking in our geometry and topology seminar on scissors
congruence, title and abstract below. This will be an in person talk, on
Thursday at 2:50pm in WH 100E.

We will also have a lunch social, meet at 12pm just outside WH100E. See you
there!

Best,
Cary

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*Title:* Higher scissors congruence
*Abstract:* Hilbert's Third Problem asks for sufficient conditions that
determine when two polyhedra in three-dimensional Euclidean space are
scissors congruent. Classically, the attempts to solve this problem (in
this and other geometries) lead into group homology and algebraic K-theory,
in a somewhat ad-hoc way. In the last decade, Zakharevich has shown that
the presence of K-theory here is not ad-hoc, but is integral to the
definition of scissors congruence itself. This leads to a natural notion of
higher scissors congruence groups, in which the 0th group is the classical
one that determines the answer to Hilbert's Third Problem.

In this talk, I'll describe a surprising recent result that these higher
groups arise from a Thom spectrum. Its base space is the homotopy orbit
space of a Tits complex, and the vector bundle is the negative tangent
bundle of the underlying geometry. Using this result, we can explicitly
compute the higher scissors congruence groups for the one-dimensional
geometries, and give exact sequences that express them for the
two-dimensional geometries. Much of this is joint work with Anna-Marie
Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich.
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