Workshop

I will report on joint work with Tamme. I will recall that algebraic K-theory does not satisfy excision and describe an effective way of studying the failure for excision in any localizing invariant. As applications we find a short proof of Suslin's excision theorem, that what we call truncating invariants satisfy excision, and new ways of calculating algebraic K-groups of certain rings. Examples of truncating invariants include periodic cyclic homology over the rationals and the fibre of the cyclotomic trace. If time permits I will also indicate joint work with Tamme and Meier, where we study chromatic localizations of algebraic K-theory. As applications of our approach we show that K(1)-localized algebraic K-theory satisfies excision (first proven by Bhatt--Clausen--Mathew) and that the K(n)-localized K-theory of any bounded above connective ring spectrum vanishes for n at least 2.

Hochschild homology of a ring has a topological analogue for ring spectra, topological Hochschild homology (THH), which plays an essential role in the trace method approach to algebraic K-theory. Using norms in equivariant stable homotopy theory, one can define a twisted version of THH, which takes as input an equivariant ring spectrum. In this talk I will discuss categorical approaches to studying twisted THH, as well as computational tools. In particular, I will discuss the twisted THH of Thom spectra and the Real bordism spectrum. This talk includes joint work with Angeltveit, Blumberg, Hill, Lawson, and Mandell, as well as joint work with Adamyk, Hess, Klang, and Kong.

If R is an associative ring satisfying some mild technical conditions, Beilinson constructs a fiber sequence of spectra identifying the fiber of the map \lim_n K(R/p^n)\rightarrow K(R/p) with a suspension of the ordinary cyclic homology of R, all up to p-completion followed by inverting p. Joint work with Akhil Mathew, Matthew Morrow, and Thomas Nikolaus provides a new proof of this fact using recent advances in the theory of cyclotomic spectra due to Nikolaus and Scholze. I will explain the motivation for this fiber sequence, which has to do with the infinitesimal part of the p-adic variational Hodge conjecture, and then I will give a construction of the fiber sequence and of a more general fiber square.

Algebraic K-theory and topological cyclic homology are invariants defined very generally (e.g., for stable \infty-categories). When applied to a commutative ring, they inherit an additional important (and slightly mysterious) structure: a motivic filtration. For topological cyclic homology, the motivic filtration has been constructed in the recent work of Bhatt--Morrow--Scholze using a very direct approach; the graded pieces are a type of filtered Frobenius eigenspace on prismatic cohomology. I will describe the BMS construction and some new structural features of the motivic filtration on TC, in particular, an identification of the graded pieces, either in low weights or with denominators, with syntomic cohomology -- a construction which relies only on more classical techniques, and which is easier to compute in practice. Joint work with Benjamin Antieau, Matthew Morrow, and Thomas Nikolaus.

In collaboration with Martina Rovelli.

We will present how, by means of a suitable nerve construction, the established homotopy theory of strict 2-categories is fully recovered in the model of (\infty,2)-categories given by 2-complicial sets.

In the first talk, we will review the basics of 2-complicial sets, and we will introduce the nerve construction. We will then discuss some of its formal properties, and explain that it induces an embedding of the Lack's homotopy theory of 2-categories into the homotopy theory of Verity's 2-complicial sets.

In the second talk, we will give an explicit combinatorial description of the nerve which we can use to study many relevant homotopical properties. For instance, we will show that the nerve commutes up to equivalence with many relevant constructions, such as certain pushouts or suspensions.

Commutative semirings can be described in terms of bispans of finite sets, meaning spans with an extra forward leg; if we instead take bispans in finite G-sets we get Tambara functors, which are the structure on $\pi_0$ of $G$-equivariant commutative ring spectra. Motivated by applications of the $\infty$-categorical upgrade of such descriptions to motivic and equivariant ring spectra, I will discuss the universal property of $(\infty,2)$-categories of bispans. I will focus on the simplest case of bispans in finite sets, where this gives a new construction of the semiring structure on a symmetric monoidal $\infty$-category whose tensor product commutes with coproducts. This is joint work with Elden Elmanto.

For an exact Lagrangian $L$ in a cotangent bundle, one can define a sheaf of stable-infinity categories on it, called the Kashiwara--Schapira stack. Assuming the Lagrangian is smooth, the sheaf of categories is a local system with fiber equivalent to Mod(k), where k is the coefficient ring spectrum (at least E_2). I will show that the classifying map for the local system of categories factors through the stable Gauss map L--->U/O and the delooping of the J-homomorphism U/O-->BPic(S), where S is the sphere spectrum. Part of the proof employs the (\infty,2)-category of correspondences developed by Gaitsgory--Rozenblyum. If time permits, I will also talk about some applications of this result.

I'll talk about a new way to encode paracyclic structures (a combinatorial structure in Waldahausen's s-dot construction). This new way is through sheaves on broken cycles. This stack also parametrizes families of certain symplectic manifolds, so this new way allows us to see why the data for constructing Fukaya categories for exact 2-dimensional manifolds is equivalent to the data of a 2-Segal object (in the sense of Dyckerhoff-Kaparanov). A bonus is that we can prove theorems like: The K theory of the integers is equivalent to a Floer theory of Lagrangian cobordisms.