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

[Cary Malkiewich]

Hi everyone,

Unfortunately I'll have to postpone my seminar talk tomorrow, due to an
illness in my family. I'll speak at a later date, to be determined once
more of our invited speakers have picked dates for themselves.

Feel free to meet for topology lunch without me!

Next week we'll have Maxine Calle visiting from Penn. See you then!

Best,
Cary

On Mon, Sep 18, 2023 at 9:50 AM Cary Malkiewich <
malkiewich at math.binghamton.edu> wrote:

> 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
>
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> *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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