Related papers: Complex orientations for THH of some perfectoid fi…
This paper provides a systematic treatment of Gorenstein homological aspects for cleft extensions of rings. In particular, we investigate Goresnteinness, Gorenstein projective modules and singularity categories in the context of cleft…
The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of K\"ahler differentials. Here we generalize it for simplicial, graded and anticommutative…
We give a new formula for real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for topological cyclic homology to one involving genuine $C_2$-spectra. To accomplish this, we give a new…
We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from…
We use geometric fixed points to describe the homotopy theory of genuine equivariant commutative ring spectra after inverting the group order. The main innovation is the use of the extra structure provided by the Hill-Hopkins-Ravenel norms…
Matthew Ando produced power operations in the Lubin-Tate cohomology theories and was able to classify which complex orientations were compatible with these operations. The methods used by Ando, Hopkins and Rezk to classify orientations of…
For a scheme X, we construct a sheaf C of complexes on X such that for every quasi-compact open subset U of X, C(U) is quasi-isomorphic to the Hochschild complex of the scheme U. Since C is moreover acyclic for taking sections on…
We provide a new description of logarithmic topological Andr\'e-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and…
In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper…
An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove…
We define an $\infty$-category $\mathrm{CycSyn}$ of $p$-typical cyclotomic synthetic spectra and prove that the motivic filtration on $\mathrm{THH}(R;\mathbf{Z}_p)$, defined by Bhatt, Morrow, and Scholze when $R$ is quasisyntomic and by…
$CPT$ groups of higher spin fields are defined in the framework of automorphism groups of Clifford algebras associated with the complex representations of the proper orthochronous Lorentz group. Higher spin fields are understood as the…
The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space $X$ and the space of maximal ideals of the ring of real…
We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as…
For simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex whose vertices are the graph homomorphisms $G\to H$ and whose edges connect the pairs of homomorphisms which differ in a single vertex of $G$. Hom…
We introduce the notion of a polygonic spectrum which is designed to axiomatize the structure on topological Hochschild homology $\mathrm{THH}(R,M)$ of an $\mathbb{E}_1$-ring $R$ with coefficients in an $R$-bimodule $M$. For every polygonic…
A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence constructed by Cohen, Jones and Yan for the…
J McClure's Dyer-Lashof operation in $p$-adic $K$-theory defines, in particular, a prismatic structure on the complex representation ring of the circle group. Work of Ando, Rezk, Stapleton, and others generalizes this to define a canonical…
We construct for any algebra over an operad an Hochschild chain complex. In the case of the singular cochain complex of a topological space, considered as a commutative algebra up to homotopy, we show that this complex computes the singular…
Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule…