Loen, August 2nd-8th 2009

Algebraic K-theory of the fraction field of topological K-theory

In this talk, I report on joint work with John Rognes. We compute (in V(1)-homotopy) the algebraic K-theory of mod p complex K-theory l/p. Using iterated localization cofibre sequences, this allows us to evaluate the V(1)-homotopy of a spectrum interpreted as the algebraic K-theory K(F) of the "fraction field" F of l. The answer fits well with a generalization of the Lichtenbaum-Quillen Conjecture to ring spectra.

Introducing a suitable definition of F and K(F) and extending the trace methods to cover such "fields" is one motivation for studying logarithmic structures in the context of ring spectra.

Equivariant derived algebraic geometry and crystalline realizations of K-theory, I and II"

I introduce equivariant derived algebraic geometry. I describe algebraic K-theory as a "derived equivariant scheme," first for Galois groups, and then for crystalline fundamental groups. I then describe recent and ongoing work, in which the deformation theory of derived equivariant schemes is used to verify conjectural computations of certain K-theory groups.

Syntomic regulators and special values of p-adic L-function

Syntomic cohomology is the p-adic analogue of Deligne cohomology and syntomic regulators, from algebraic K-theory into syntomic cohomology, are the analogues of Beilinson regulators into Deligne cohomology. At the same time they are also connected, via the Bloch-Kato exponential map, with the regulator into continuous étale cohomology.

Like the Beilinson conjectures relating Beilinson regulators with special values of L-functions, there are p-adic Beilinson conjectures, which relate syntomic regulators with special values of p-adic L-functions.

In the talk I will first recall the motivation and definitions of syntomic cohomology. I will then describe the p-adic Beilinson conjecture and discuss the known evidence for it. I will also say a few words on the proposed conjecture of syntomic regulators with the regulators defined by Karoubi.

p-adic cohomologies and comparison theorems

In this survey, I'll describe how crystalline and log-schemes techniques can be used to add more structures on the algebraic de Rham cohomology of proper and smooth algebraic varieties over a p-adic field. I'll explain how one can recover the p-adic étale cohomology from these structures and conversely, with a special emphasis on the syntomic methods.

Fixed points of THH and the de Rham-Witt complex

There are trace maps relating algebraic K-theory to fixed point spectra of topological Hochschild homology (THH). As topological Hochschild homology is comparatively easy to compute, these maps provide an approach to previously inaccessible computations in algebraic K-theory. Many of these computations are facilitated by exploiting the rigid algebraic relationships between different fixed point spectra of THH---altogether the fixed point spectra form a Witt complex, and the initial Witt complex, called the de Rham-Witt complex, can sometimes be constructed explicitly. This talk will introduce THH and describe various operators on and relations among its fixed point spectra.

On the slice filtration and the Kervaire invariant one problem

Review of de Rham-Witt theory

I will recall constructions and properties of de Rham-Witt complexes, both in the non log and the log contexts, with a view toward the p-adic comparison theorems.

Rigid cohomology and its coefficients

Rigid cohomology was introduced by Berthelot to serve as a unified p-adic cohomology theory with field coefficients, applicable to any scheme of finite type over a field of characteristic p. It reproduces rational crystalline cohomology of smooth proper schemes as well as Monsky-Washnitzer cohomology of smooth affine schemes. I'll describe the basic construction of rigid cohomology, some categories of coefficients analogous to lisse and constructible l-adic étale sheaves, and survey the (recently much improved) state of knowledge about these coefficients.

Topological Automorphic Forms

Lurie has announced a generalization of the Hopkins-Miller theorem that allows construction of strictly commutative ring spectra from purely algebraic data, a 1-dimensional p-divisible group. We will discuss computational progress (joint work with Behrens and Hill) in understanding ring spectra arising from moduli of abelian varieties.

A theory of base motives

When A is a commutative local ring with residue field k, the functor
- ⊗^{L}_{A} k : D(A) ---> D(k)
lifts to a functor taking values in a category of modules over the
`Tate cohomology' RHom_{A}(k,k) (cf. eg Cartan-Eilenberg Ch X), which
is the universal enveloping algebra of a certain Lie algebra; k ⊗^{L}_{A}
k is the dual Hopf algebra. Under reasonable conditions
this lift yields a descent spectral sequence of Adams (or Bockstein)
type.

In a suitable category of ring-spectra, replacing A --> k by A(*) --> S^{0}
or TC(S^{0}) --> S^{0} in this construction yields prounipotent Lie algebras,
free after tensoring with Q, analogous to the Lie algebras of motivic
groups studied recently by Deligne, Connes and Marcolli, and others.

Logarithmic K-theory

Having a log-scheme one can consider some natural definitions of its K-theory using the log-etale and log-syntomic topologies. Using the category of locally free sheaves in these topologies we obtain the algebraic K-theory; using fibrant models -- we obtain topological K-theory. I will survey what we know about such K-theories and their relations to log-etale and log-syntomic cohomology.

p-adic comparison morphism via K-theory

I will present a construction of the p-adic comparison morphism for proper log-smooth varieties using Chern classes from higher algebraic K-theory to etale and log-syntomic cohomology.

Logarithmic Geometry

Logarithmic geometry brings together many classical themes, ranging from toric geometry to differential equations, and thus provides a framework for the systematic study of problems involving compactification and degeneration in a wide array of arithmetic and geometric settings. I will try to describe the main ideas of the theory, concentrating on the classical case of log schemes over the complex numbers, their associated topological spaces, and their Betti, de Rham, and crystalline realizations.

Logarithmic THH and TC

I will discuss how to define topological Hochschild homology (THH) and topological cyclic homology (TC) for symmetric ring spectra with log structure, giving topological analogues of the log de Rham and log de Rham-Witt complexes. These constructions provide insight into the algebraic K-theory of suitable symmetric ring spectra.

Graded log symmetric ring spectra

I will review John Rognes' notion of symmetric ring spectra with log structures and explain how these objects can be treated conveniently with the language of Quillen model categories. Based on joint work with Christian Schlichtkrull, I will illustrate how Rognes' definition can be modified to give symmetric ring spectra with `graded' log structures. This `graded' version is better suited for the study of non-connective ring spectra like the periodic complex K-theory spectrum KU.

Symmetric spectra and diagram spaces

We give an introduction to the theory of symmetric spectra and the diagram space approach to infinite loop space theory. The diagram space part of the talk represents joint work with Steffen Sagave.

Brave new rings from K3 surfaces

While elliptic curves give rise to formal groups via their formal Picard group, K3 surfaces give rise to formal groups via their formal Brauer group. This makes them interesting for the homotopy theorist. In this talk, I will adress the problem to realise the corresponding Landweber exact homology theories in the world of brave new rings.

John Rognes / July 26th 2009