We use sheaves of spectra to quantize a Hamiltonian $\coprod _n BO(n)$-action on $\varinjlim _{N}T^*\mathbf {R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed by Gaitsgory and Rozenblyum [A study in derived algebraic geometry, vol. I. Correspondences and duality, Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, 2017)] to give an enrichment of stratified Morse theory by the $J$-homomorphism. This provides a key step in the work of Jin [Microlocal sheaf categories and the $J$-homomorphism, Preprint (2020), arXiv:2004.14270v4] on the proof of a claim of Jin and Treumann [Brane structures in microlocal sheaf theory, J. Topol. 17 (2024), e12325]: the classifying map of the local system of brane structures on an (immersed) exact Lagrangian submanifold $L\subset T^*\mathbf {R}^N$ is given by the composition of the stable Gauss map $L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow B\mathrm {Pic}(\mathbf {S})$. We put special emphasis on the functoriality and (symmetric) monoidal structures of the categories involved and, as a byproduct, we produce several concrete constructions of (commutative) algebra/module objects and (right-lax) morphisms between them in the (symmetric) monoidal $(\infty, 2)$-category of correspondences, generalizing the construction out of Segal objects of Gaitsgory and Rozenblyum, which might be of independent interest.

We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology of $\pi $-finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$. We use this theory to generalize said duality in three different directions. First, we extend it from $\mathbb {Z}$-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of $E_n$. Second, we lift it to the telescopic setting by replacing $E_n$ with $T(n)$-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg–Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal $\infty $-categories of local systems of $K(n)$-local $E_n$-modules [-12pc] and relate it to (semiadditive) redshift phenomena.

The Great Wave off Kanagawa, Katsushika Hokusai.

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$- and $\mathrm {T}(n)$-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$, which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$, we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$-localized K-theory, showing that they coincide for any $p$-invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$.

We compute the connective spectra of maps from $\mathbb {Z}$ to the Picard spectra of the spherical Witt vectors associated with perfect rings of characteristic $p$. As an application, we determine the connective spectrum of maps from $\mathbb {Z}$ to the Picard spectrum of the sphere spectrum.

The algebraic K-theory of Lawvere theories is a conceptual device to elucidate the stable homology of the symmetry groups of algebraic structures such as the permutation groups and the automorphism groups of free groups. In this paper, we fully address the question of how Morita equivalence classes of Lawvere theories interact with algebraic K-theory. On the one hand, we show that the higher algebraic K-theory is invariant under passage to matrix theories. On the other hand, we show that the higher algebraic K-theory is not fully Morita invariant because of the behavior of idempotents in non-additive contexts: We compute the K-theory of all Lawvere theories Morita equivalent to the theory of Boolean algebras.

Loday’sassembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalisation that places both ingredients on the same footing. Building on Elmendorf–Mandell’s multiplicativity results and our earlier work, we show that the K-theory of Lawvere theories is lax monoidal. This result makes it possible to present our theory in a user-friendly way without using higher-categorical language. It also allows us to extend the idea to new contexts and set up a nonabelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

For a weight structure w on a triangulated category $\underline {C}$ we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case $w=w^{sph}$ (the spherical weight structure on $SH$), we deduce the following converse to the stable Hurewicz theorem: $H^{sing}_{i}(M)=\{0\}$ for all $i<0$ if and only if $M\in SH$ is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement.

The main idea is to study M that has no weights $m,\dots ,n$ (‘in the middle’). For $w=w^{sph}$, this is the case if there exists a distinguished triangle $LM\to M\to RM$, where $RM$ is an n-connected spectrum and $LM$ is an $m-1$-skeleton (of M) in the sense of Margolis’s definition; this happens whenever $H^{sing}_i(M)=\{0\}$ for $m\le i\le n$ and $H^{sing}_{m-1}(M)$ is a free abelian group. We also consider morphisms that kill weights $m,\dots ,n$; those ‘send n-w-skeleta into $m-1$-w-skeleta’.

For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.

We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.

We define stable homotopy refinements of Khovanov’s arc algebras and tangle invariants.

We discuss some general properties of $\mathrm {TR}$ and its $K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm {TR}$ of $H\mathbb {Z}$-algebras is a truncating invariant in the Land–Tamme sense, and deduce $h$-descent results. We show that for regular rings in mixed characteristic, $\mathrm {TR}$ is asymptotically $K(1)$-local, extending results of Hesselholt and Madsen. As an application of these methods and recent advances in the theory of cyclotomic spectra, we construct an analog of Thomason's spectral sequence relating $K(1)$-local $K$-theory and étale cohomology for $K(1)$-local $\mathrm {TR}$.

We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.

We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study ‘functors with reality’ such as the Real classifying space functor, . The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of . We further give model categorical considerations, producing a zigzag of Quillen equivalences between spectra with an action of and a model structure on the category of input functors which captures the homotopy theory of the n-th layer of the Taylor tower.

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set-level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding $\infty$-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted $K$-theory.

This note is on spherical classes in $H_*(QS^0;k)$ when $k=\mathbb{Z}, \mathbb{Z}/p$, with a special focus on the case of p=2 related to the Curtis conjecture. We apply Freudenthal's theorem to prove a vanishing result for the unstable Hurewicz image of elements in ${\pi _*^s}$ that factor through certain finite spectra. After either p-localization or p-completion, this immediately implies that elements of well-known infinite families in ${_p\pi _*^s}$, such as Mahowaldean families, map trivially under the unstable Hurewicz homomorphism ${_p\pi _*^s}\simeq {_p\pi _*}QS^0\to H_*(QS^0;\mathbb{Z} /p)$. We also observe that the image of the submodule of decomposable elements under the integral unstable Hurewicz homomorphism $\pi _*^s\simeq \pi _*QS^0\to H_*(QS^0;\mathbb{Z} )$ is given by $\mathbb{Z} \{h(\eta ^2),h(\nu ^2),h(\sigma ^2)\}$. We apply the latter to completely determine spherical classes in $H_*(\Omega ^dS^{n+d};\mathbb{Z} /2)$ for certain values of n>0 and d>0; this verifies Eccles' conjecture on spherical classes in $H_*QS^n$, n>0, on finite loop spaces associated with spheres.

We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.

This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

We develop a theory of $R$-module Thom spectra for a commutative symmetric ring spectrum $R$ and we analyze their multiplicative properties. As an interesting source of examples, we show that $R$-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on $R$. We apply the general theory to obtain a description of the $R$-based topological Hochschild homology associated to an $R$-algebra Thom spectrum.

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p\,>\,0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order $2$ or $3$. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd’s generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps that mimic ghosts in high degrees.

We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, . This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕn. If the characteristic of k does not divide any of the ai we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k = ℤ.

To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.