The fifth edition of the Atlantic Meeting on Topology, Representation theory, and K-theory will be held on Friday, April 17 at the University of Pennsylvania. The conference is themed around Scissors Congruence K-Theory and features talks by Maru Sarazola and Cary Malkiewich.
The conference is organized by Riley Shahar (Penn) and Nooria Ahmed (Hopkins). Past organizers include Maxine Calle, Anish Chedalavada, and Ben Spitz. See previous editions of the conference on Maxine’s website.
| Time | Event | Location |
|---|---|---|
| 9–10 | Breakfast | DRL A6 |
| 10–11 | Pretalk 1: Fangji Liu | DRL A6 |
| 11–11:30 | Break | — |
| 11:30–12:30 | Pretalk 2: Mattie Ji | LRSM 112B |
| 12:30–2 | Lunch | — |
| 2–3 | Talk 1: Maru Sarazola | DRL A4 |
| 3–3:45 | Department tea | — |
| 4–5 | Talk 2: Cary Malkiewich | DRL A4 |
(DRL is David Rittenhouse Laboratory; LRSM is the Laboratory for Research on the Structure of Matter, both at Penn.)
The classical problem of scissors congruence for polytopes admits a natural translation to the world of manifolds: given two manifolds M and N, one can study whether it’s possible to cut M into pieces and reassemble them to obtain N. Recent advances have made it possible to interpret this problem within the framework of algebraic K-theory, providing an exciting new perspective on this problem that upgrades this classical algebraic invariant to the homotopical realm.
While the Grothendieck group of this K-theory spectrum is well-understood, not much else is known in terms of its higher homotopical information. In this talk, I will describe joint work with Maxine Calle where we provide a new simplicial model for cut-and-paste K-theory in the spirit of Waldhausen’s S-dot construction. We will discuss our progress in computing K_1 of this spectrum, and how it encodes information about the cut-and-paste automorphisms of manifolds.
Scissors congruence is the study of polytopes, up to the relation of cutting into finitely many pieces and rearranging the pieces. In the 2010s Zakharevich defined a “higher” version of scissors congruence via K-theory, and in the early 2020s my collaborators and I showed that Zakharevich’s scissors congruence K-theory is a Thom spectrum, allowing us to make many new computations of the higher K-groups.
It’s rare for K-theory to have such a nice description. Why does it happen in this case? In this talk I’ll discuss the proof, and a couple of ways that our understanding of the proof has changed in the last few years. In particular, I’ll describe a generalization that restricts to polytopes whose facets lie along any prescribed set of hyperplanes. This makes more computations possible, including a recent computation by Holley for all polygons in the Euclidean plane whose edges have rational slope.