In joint work with Gabriel Angelini-Knoll, Christian Ausoni, Dominic Leon Culver and Eva Höning, we calculate the $\bmod \left(p, v_1, v_2\right)$ homotopy $V(2)_* T C(B P\langle 2\rangle)$ of the topological cyclic homology of the truncated Brown-Peterson spectrum $B P\langle 2\rangle$, at all primes $p \geq 7$, and show that it is a finitely generated and free $\mathbb{F}_p\left[v_3\right]$-module on $12 p+4$ generators in explicit degrees within the range $-1 \leq * \leq 2 p^3+2 p^2+2 p-3$. Our computation is the first that exhibits chromatic redshift from pure $v_2$-periodicity to pure $v_3$-periodicity in a precise quantitative manner.[-]

For a geometry $X$ (such as Euclidean, spherical, or hyperbolic) with isometry group $G$ the scissors congruence group $\mathcal{P}(X, G)$ is defined to be the free abelian group generated by polytopes in $X$, modulo the relation that for polytopes $P$ and $Q$ that intersect only on the boundary, $[P \cup Q]=[P]+[Q]$, and for $g \in G,[P]=[g \cdot P]$. This group classifies polytopes up to 'scissors congruence', i.e. cutting up into pieces, rearranging the pieces, and gluing them back together. With some basic group homology one can see that $\mathcal{P}(X, G) \cong H_0(G, \mathcal{P}(X, 1))$. Using combinatorial $K$-theory $\mathcal{P}(X, G)$ can be expressed as the $K_0$ of a spectrum $K(X, G)$. In this talk we will generalize this formula to show that, in fact, $K(X, G) \simeq K(X, 1)_{h G}$, and in fact more generally that this is true for any assembler with a $G$-action.This is joint work with Anna Marie Bohmann, Teena Gerhardt, Cary Malkiewich, and Mona Merling.[-]