## Talbot Workshop: Scissors Congruence in Algebraic K-Theory

- Start Date
- 2022-06-12
- End Date
- 2022-06-18
- Institution
- MIT
- City
- New York
- Country
- USA
- Meeting Type
- workshop
- Homepage
- https://math.mit.edu/events/talbot/index.php?year=2022
- Contact Name
- Adam Holeman, Liam Keenan, Lucy Yang

## Description

The application for this year's Talbot workshop is now live.

**2022 TALBOT WORKSHOP: SCISSORS CONGRUENCE AND ALGEBRAIC K-THEORY**

Mentored by Inna Zakharevich and Jonathan Campbell

June 12 - 18, 2022

New York (to be confirmed)

**What:** The Talbot Workshop is a one week learning workshop for roughly 35 graduate students and a few postdocs. Most of the talks will be given by participants, and will be expository in nature. Due to the pandemic, the workshop will most likely utilize a virtual format, with half of the participants invited to join us in person.

Topic description: Hilbert’s third problem asked to produce two polyhedra of equal volume which are not scissors congruent, i.e. which cannot be decomposed into the same (finite) set of isomorphic sub-polygons. In 1901 Dehn showed that a second invariant, now called the Dehn invariant, was preserved under such decompositions, and that this invariant is zero for the cube but nonzero for the regular tetrahedron, thus providing the example Hilbert requested. In 1965 Sydler showed that no other invariants are necessary: if two polyhedra have the same volume and the same Dehn invariant, then they are scissors congruent. Analogs of this problem, in other geometries and in other dimensions, are still open.

By defining the sum of polytopes to be their disjoint union we can define a group structure on scissors congruence classes. Hilbert’s third problem can then be reinterpreted as the question of analyzing these scissors congruence groups. The first part of this course will focus on learning about computations and connections between scissors congruence and algebraic K-theory. The second part will discuss how to construct scissors congruence invariants using K-theory and various implications of these constructions. The third part dives deeper into the technical underpinnings of K-theory to illustrate how the algebraic approaches of classical K-theory can be expanded to include more combinatorial descriptions. More details can be found here.

Applications close on Thursday, March 10 at 11:59PM Eastern Time. You can apply online here: http://math.mit.edu/events/talbot/index.php?pageID=application

Talbot is meant to encourage collaboration among young researchers, with an emphasis on graduate students. We also aim to gather participants with a diverse array of knowledge and interests, so applicants need not be an expert in the field -- in particular, students at all levels of graduate education are encouraged to apply. As we are committed to promoting diversity in mathematics, we also especially encourage women and minorities to apply.

We will cover all local expenses including lodging and food. We also offer partial funding for participants' travel costs.

If you have any questions, please do not hesitate to e-mail the organizers at talbotworkshop (at) gmail.com.

Organizers:

Adam Holeman

Liam Keenan

Lucy Yang

## Problems?

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