Related papers: Examples of chromatic redshift in algebraic $K$-th…

Let A and B be $C^*$-algebras, A separable, and B $\sigma$-unital and stable. It is shown that there are natural isomorphisms $E(A,B)=KK(SA,Q(B))=[SA,Q(B)\otimes K]$, where $SA=C_0(0,1)\otimes A$, $[\cdot,\cdot]$ denotes the set of homotopy…

This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a…

We compute certain strict Picard spectra of $K(1)$-local $K$-theory spectra of schemes in terms of Brauer groups, using the map that takes an Azumaya algebra to its $K(1)$-local $K$-theory and proving a K\"unneth formula in that setting.…

Nilpotence in the homotopy of $\mathbb{E}_\infty$-ring spectra is detected by the classical $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in…

Let $K$ be a finite extension of $\mathbb{Q}_p$, and choose a uniformizer $\pi\in K$, and put $K_\infty:=K(\sqrt[p^\infty]{\pi})$. We introduce a new technique using restriction to $\Gal(\ol K/K_\infty)$ to study flat deformation rings. We…

We use equivariant K-theory to classify charges of new (possibly non-supersymmetric) states localized on various orientifolds in Type II string theory. We also comment on the stringy construction of new D-branes and demonstrate the discrete…

The primary goal of this paper is to study Spanier-Whitehead duality in the $K(n)$-local category. One of the key players in the $K(n)$-local category is the Lubin-Tate spectrum $E_n$, whose homotopy groups classify deformations of a formal…

In this paper we introduce exotic twisted $\mathbb T$-equivariant K-theory of loop space $LZ$ depending on the (typically non-flat) holonomy line bundle ${\mathcal L}^B$ on $LZ$ induced from a gerbe with connection $B$ on $Z$. We also…

We use Drinfeld style generators and relations to define an algebra $\mathfrak{U}_n$ which is a ``$q=0$'' version of the affine quantum group of $\mathfrak{gl}_n.$ We then use the convolution product on the equivariant $K$-theory of…

We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and…

In this paper, we introduce Euclidean Gallai-Ramsey theory, by combining Euclidean Ramsey theory and Gallai-Ramsey theory on graphs. More precisely, we consider the following problem: For an integer $r$ and configurations $K$ and $K'$, does…

We compute the value of finitary localizing invariants, including algebraic $K$-theory, on categories of sheaves over stably locally compact spaces $X$. Our formula simultaneously generalizes the cases of locally compact Hausdorff and…

We introduce the notion of an E_k-ring with prelogarithmic structure, define logarithmic topological Hochschild homology and logarithmic topological cyclic homology in this context, and establish localization sequences for these theories.…

By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic K-theory. We give a new and direct proof of Suslin's result based on an exact sequence of categories of perfect modules. In fact, we prove a…

We consider the space of $n$-tuples of pairwise commuting elements in the Lie algebra of $U(m)$. We relate its one-point compactification to the subquotients of certain rank filtrations of connective complex $K$-theory. We also describe the…

We introduce a variant of homotopy K-theory for Tate rings, which we call analytic K-theory. It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain…

For an additive Waldhausen category linear over a ring $k$, the corresponding $K$-theory spectrum is a module spectrum over the $K$-theory spectrum of $k$. Thus if $k$ is a finite field of characteristic $p$, then after localization at $p$,…

For a reductive group scheme over a regular semi-local ring, we prove an equivarinat version of the Gersten conjecture. We draw some interesting consequences for the representation rings of such reductive group schemes. We also prove the…

We construct Galois extensions of the T(n)-local sphere, lifting all finite abelian Galois extensions of the K(n)-local sphere. This is achieved by realizing them as higher semiadditive analogues of cyclotomic extensions. Combining this…

Recently, we have witnessed tremendous applications of algebraic intersection theory to branches of mathematics, that previously seemed very distant. In this article we review some of them. Our aim is to provide a unified approach to the…