Related papers: Scissors congruence of the line and the regulator

In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the…

Scissors congruence groups have traditionally been expressed algebraically in terms of group homology. We give an alternate construction of these groups by producing them as the $0$-level in the algebraic $K$-theory of a Waldhausen…

We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents…

We solve the higher version of Hilbert's Third Problem for one-dimensional geometries, and in higher dimensions we reduce the problem to a computation in group homology. Our central result concerns the scissors congruence $K$-theory…

Cut-and-paste $K$-theory has recently emerged as an important variant of higher algebraic $K$-theory. However, many of the powerful tools used to study classical higher algebraic $K$-theory do not yet have analogues in the cut-and-paste…

Let $\Gamma$ be a dense countable subgroup of $\mathbb{R}$. Then, consider $IE(\Gamma)$; the group of piecewise linear bijections of $[0,1]$ with finitely many angles, all in $\Gamma$. We introduce and systematically study a family of…

In the two parts of this paper we solve a problem of De Rham, proving that Reidemeister torsion invariants determine topological equivalence of linear G-representations, for G a finite cyclic group. Methods in controlled K-theory and…

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as…

We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the…

In Grayson's combinatorial description of higher K-groups, the generators are bounded acyclic binary multi-complexes of arbitrary size. Generalising work by Kasprowski, Winges and the author, we show in this paper that multi-complexes of…

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop the theory of covering spaces for k-graphs, obtaining a…

We introduce a scissors congruence $K$-theory spectrum which lifts the equivariant scissors congruence groups for compact $G$-manifolds with boundary, and we show that on $\pi_0$ this is the source of a spectrum level lift of the Burnside…

A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross…

We prove a Schoenberg-type correspondence for non-unital semigroups which generalizes an analogous result for unital semigroup proved by Michael Sch\"urmann. It characterizes the generators of semigroups of linear maps on $M_n(C)$ which are…

We construct a version of Beilinson's regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Bloch's…

We show how to use equidimensional algebraic correspondences between complex algebraic varieties to construct pull-backs and transforms of certain classes of geometric currents. Using this construction we produce explicit formulas at the…

We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group…

In this thesis we generalise the six-term exact sequence in graded $KK$-theory obtained in a paper of Kumjian, Pask and Sims (2017) to allow correspondences with non-compact left action. In particular, this allows us to compute the graded…

We provide a Cuntz-Pimsner model for graph of groups $C^*$-algebras. This allows us to compute the $K$-theory of a range of examples and show that graph of groups $C^*$-algebras can be realised as Exel-Pardo algebras. We also make a…

Let X be a smooth complex algebraic variety. In this paper, we associate, to each exact n-cube of hermitian vector bundles over X, a differential form, called higher Bott Chern form, which generalizes the Bott Chern forms associated to an…