Related papers: On the chromatic localization of the homotopy comp…
We show that the topological Hochschild homology THH(R of an E_n-ring spectrum R is an E_{n-1}-ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the the little n-cubes operad…
We examine the "homotopy coniveau tower" for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, the homotopy coniveau tower agrees with…
In this paper we develop the definition of a global orthogonal spectrum and its unitary version. It relates $G-$equivariant spectra by equivariant weak equivalence in a coherent way. This category of global spectra has a model structure…
We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G-homotopy theory is "pieced together" from the G/U-homotopy theories…
Let $E_n$ be Morava $E$-theory of height $n$. Let $R$ be a $p$-adically flat commutative ring spectrum. Then the Tate-valued Frobenius map endows $\pi_0 R$ with the structure of a $\delta$-ring. On the other hand, we may form the…
We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of…
We study the homogeneous coordinate rings of real multiplication noncommutative tori as defined by A. Polishchuk. Our aim is to understand how these rings give rise to an arithmetic structure on the noncommutative torus. We start by giving…
Given a noncommutative Hamiltonian space $A$, we prove that the conjecture ``{\it quantization commutes with reduction}'' holds for $A$. We further construct a semidirect product algebra $A \rtimes \mG^A$, and establish a correspondence…
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal…
We describe the construction of the noncommutative complex ball whose commutative analog is the Hermitian symmetric space $D=SU(m,1)/U(m)$, with the method of coherent state quantization. In the commutative limit we obtain the standard…
We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization…
We provide a homotopy theorist's point of view on $KK$- and $E$-theory for $C^{*}$-algebras. We construct stable $\infty$-categories representing these theories through a sequence of Dwyer-Kan localizations of the category of…
We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an…
We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data…
Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra ({\em full centre}) in the monoidal centre of the monoidal…
We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S-algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral…
In this paper we apply homotopical localization to the framework of differential graded algebras over an operad. We get plus construction by performing nullification with respect to an universal acyclic algebra. This plus construction for…
Let Map_T(K,X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space…
The topological Hochschild homology THH(R) of a commutative S-algebra (E_infty ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show,…
Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated…