Related papers: A computational reduction for many base cases in p…

We construct commutative algebra spectra that represent the operator $K$-theory of $C^*$-algebras, which are algebras over the commutative ring spectra that represent topological $K$-theory. The spectral multiplicative structure introduces…

We prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions that are only needed in the case l=2, we also prove various module Koszulity…

The goal of this note is to study the analog in unstable ${{\mathbb A}^1}$-homotopy theory of the unit map from the motivic sphere spectrum to the Hermitian K-theory spectrum, i.e., the degree map in Hermitian K-theory. We show that "Suslin…

We construct a family of moduli spaces, called generalized Lagrangian flag schemes, that are reducible Gorenstein (hence singular), and that admit well-behaved pushforward and pullback operations in Hermitian $K$-theory. These schemes arise…

Let $S$ be a K3 surface with primitive curve class $\beta$. We solve the relative Gromov-Witten theory of $S \times \mathbb{P}^1$ in classes $(\beta,1)$ and $(\beta,2)$. The generating series are quasi-Jacobi forms and equal to a…

We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop…

The mod p cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211--227] proved a conjecture due to N. Kuhn [On…

Let T be a compact torus and (M,\omega) a Hamiltonian T-space. In a previous paper, the authors showed that the T-equivariant K-theory of the manifold M surjects onto the ordinary integral K-theory of the symplectic quotient M \mod T of M…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes…

Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the…

We construct equivariant $KK$-theory with coefficients in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ as suitable inductive limits over ${\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties,…

We consider filtered or graded algebras $A$ over a field $K$. Assume that there is a discrete valuation $O_v$ of $K$ with $m_v$ its maximal ideal and $k_v:=O_v/m_v$ its residue field. Let $\Lambda$ be $O_v$-order such that $\Lambda K=A$ and…

Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gr\"obner basis theory. It is proved that there is a one-to-one…

We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30.…

This paper describes the $K$-theory structure for three algebra classes. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{Z}/p^s\mathbb{Z}$, we determine reduced $K_2$-structures via a common algebraic framework. For…

We conjecture that appropriate K-theoretic Gromov-Witten invariants of complex flag manifolds G/B are governed by finite-difference versions of Toda systems constructed in terms of the Langlands-dual quantized universal enveloping algebras…

Consider a complete discrete valuation ring $\mathcal{O}$ with quotient field $F$ and finite residue field. Then the inclusion map $\mathcal{O} \hookrightarrow F$ induces a map $\hat{\mathrm{K}}^\mathrm{M}_*\mathcal{O} \to…

In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand into the p-local complex connective K-theory spectrum, equipped with suitable log structures, is…

For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = \alpha u^n + \alpha_{n-1}u^{n-1} + \ldots + \alpha_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer,…