Workshop

I will review the theory of topological group actions on complex-linear categories and their algebraic calculus of characters, with (old and new) applications to Gromov-Witten theory and dualities in mathematical physics.

I will describe a Linearization Conjecture that identifies the spectral dualizing module of a p-adic Lie group in terms of a representation sphere built from the Lie algebra. We can prove this when the action is restricted to certain small finite subgroups. These results are enough to determine Spanier-Whitehead duals of some chromatically interesting spectra. This is joint work in progress with Beaudry, Goerss, and Hopkins.

In this talk I will review spectral Lie algebras, an adaptation of the theory of Lie algebras to the category of spectra. These generalize differential graded Lie algebras over the rational numbers and admit similar applications in derived algebraic geometry and homotopy theory. As an example of this, I will discuss their role as algebraic models for spaces, generalizing Quillen's rational homotopy theory.

I will discuss some recent results on the p-adic algebraic K-theory of p-adic rings, obtained using the cyclotomic trace from K-theory to topological cyclic homology. Parts of this are joint with Bhargav Bhatt, Dustin Clausen, and Matthew Morrow.

I will report on joint work with Markus Land. To any pullback diagram of ring spectra we associate a new square of ring spectra which becomes cartesian upon applying K-theory, or in fact any localizing invariant. The new square canonically maps to the original one, and this map is an equivalence in three corners. In the fourth corner, this map is generally not an equivalence. However, we understand this map well enough to deduce simple proofs of various excision results, most of which were previously proven by different methods in work of Suslin-Wodzicki, Cuntz-Quillen, Cortiñas, and Geisser-Hesselholt.

The Chern character is a central construction which appears in topology, representation theory and algebraic geometry. In algebraic topology it is for instance used to probe algebraic K-theory which is notoriously hard to compute, in representation theory it takes the form of classical character theory. Recently, Toen and Vezzosi suggested a construction, using derived algebraic geometry, which allows to unify the various Chern characters. We will categorify this Chern character. In the categorified picture algebraic K-theory is replaced by the category of non-commutative motives. It turns out that the categorified Chern character has many interesting applications. For instance we show that the DeRham realisation functor is of non-commutative origin.

This is joint work with Michael Larsen and Ayelet Lindenstrauss. Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers $\ell \neq p$ and integers $j \geq 1$, there exists a canonical "reduced norm" isomorphism of $\ell$-adic $K$-groups $\operatorname{Nrd}_{D/K} \colon K_j(D,\mathbb{Z}_{\ell}) \to K_j(K,\mathbb{Z}_{\ell})$ such that $d \cdot \operatorname{Nrd}_{D/K}$ is equal to the norm homomorphism $N_{D/K}$. We prove the analogous statement for the $p$-adic $K$-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence $\operatorname{Trd}_{A/S} \colon \operatorname{THH}(A\,|\,D,\mathbb{Z}_p) \to \operatorname{THH}(S\,|\,K,\mathbb{Z}_p)$ between two $p$-complete cyclotomic spectra associated with $D$ and $K$, respectively, from which the statement for the $p$-adic $K$-groups ensues. Interestingly, we show that if $p$ divides $d$, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, $d \cdot \operatorname{Trd}_{A/S}$ agrees with the trace $\operatorname{Tr}_{A/S}$, although this is possible as maps of spectra with $\mathbb{T}$-action.

One of the miracles of Topological Hochschild Homology is that THH(F_p) is very simple: it is just the algebra of polynomials in one variable of degree 2. However, the existing proofs of this fact are not simple at all. I will present yet another proof that uses quite a few additional general structures THH is known to have (multiplication, a trace functor structure), but almost nothing specific to F_p.