We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc.

We describe all Witt invariants and mod 2 cohomological invariants of the functor $I^n$ as combinations of fundamental invariants; this is related to the study of operations on mod 2 Milnor K-theory. We also study behaviour of these invariants with respect to products, restrictions, similitudes and ramification.

We construct a groupoid equivariant Kasparov class for transversely oriented foliations in all codimensions. In codimension 1 we show that the Chern character of an associated semifinite spectral triple recovers the Connes–Moscovici cyclic cocycle for the Godbillon–Vey secondary characteristic class.

We give a complete $K$-theoretical description of when an essential extension of two simple graph $C^{\ast }$-algebras is again a graph $C^{\ast }$-algebra.

We give the first example of a derived scheme $X$ and a line bundle $\mathcal{ℒ}$ on the truncation $tX$ so that $\mathcal{ℒ}$ does not extend to the original derived scheme $X$. In other words the pullback map $Pic\left(X\right)\to Pic\left(tX\right)$, and hence also the pullback map $K^0\left(X\right)\to K^0\left(tX\right)$, is not surjective. The derived schemes we construct have the further property that while their truncations are projective hypersurfaces, they fail to have any nontrivial line bundles, and therefore they are not quasiprojective.

We provide an axiomatic framework that characterizes the stable $\infty$-categories that are module categories over a motivic spectrum. This is done by invoking Lurie’s $\infty$-categorical version of the Barr–Beck theorem. As an application, this gives an alternative approach to Röndigs and Østvær’s theorem relating Voevodsky’s motives with modules over motivic cohomology and to Garkusha’s extension of Röndigs and Østvær’s result to general correspondence categories, including the category of Milnor–Witt correspondences in the sense of Calmès and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and Déglise.

The 8-periodic theory that comes from the $\mathit{KO}$-theory of the mod 2 Moore space is the same as the real first Morava $K$-theory obtained from the homotopy fixed points of the $\mathbb{Z}∕\left(2\right)$ action on the first Morava $K$-theory. The first Morava $K$-theory, $K\left(1\right)$, is just mod 2 $\mathit{KU}$-theory. We compute the homology Hopf algebras for the spaces in this Omega spectrum.