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We will explain a new setting of genuine equivariant homotopy theory for infinite groups (like the integers) inspired by work of Kaledin. This is different from the usual approach and allows to prove a comparison to a new relative version of cyclotomic spectra that we introduce (called polygonic spectra). We shall also explain the relation to TR and to Witt vectors with coefficients. This is joint work with A. Krause and J. McCandless.

13D03 ; 13F35 ; 19D55

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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.[-]
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 ...[+]

19D50 ; 19D55 ; 55P43 ; 55Q51

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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.[-]
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, ...[+]

19D55 ; 55N99 ; 19E99

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