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

[Cary Malkiewich]

A quick reminder about topology seminar today (in 5 minutes), sorry for not
sending it out this morning!

Cary

On Tue, Oct 3, 2023 at 8:14 AM Cary Malkiewich <
malkiewich at math.binghamton.edu> wrote:

> Hi everyone,
>
> OK, let's try this again! 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
>
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> *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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