Visiting addressNiels Henrik Abels hus Moltke Moes vei 35 (map)
A continuation of the previous talk.
In 1980 R. W. Thomason published a proof that CAT, the category of small categories, is a proper closed model category that is Quillen equivalent to SSet, the category of simplicial sets, with the standard model structure defined by Quillen. D-C Cisinski has since corrected the proof of left properness by replacing the central term of Dwyer morphism - a class of morphisms that Thomason believed to be the cofibrations - with a rough analogue in CAT of the NDR-pairs. The cofibrations, then, which are all retracts of Dwyer morphisms, are really the NDR-pair analogues. I will go through the main parts of Thomason's argument, incorporating Cisinski's adjustment, point out Thomason's mistake and here and there use more recent terminology from M. Hovey's book Model Categories. Towards the end I'll compare Thomason's method with modern, standardized ways of confirming a cofibrantly generated (closed) model structure, like the necessary and sufficient conditions listed in Hovey's Model Categories (thm. 2.1.19) and transferring a model structure across an adjunction by using Kan's lemma on transfer and similar results
Abstract: We will begin by reviewing and constructing power operations in the familiar setting of chain complexes. In stable homotopy, these operations help distinguish different geometric objects. These operations are also the residue of a rich homotopical structure. We will also define such structure and explain its role in stable homotopy theory. Specifically, we will consider what structure on a filtration might give rise to power operations in the associated spectral sequence, if time allows. This first talk will be accessible to graduate students. Such power operations also act on the homotopy of highly structured ring spectra. We will compute these operations on relative smash products using the Kunneth spectral sequence. We will interpret the homotopy of these relative smash products and the algebra of operations in terms of different realizations of highly structured DGAs. We will also discuss the relation to the relevant notion of cotangent complexes.
We explain how motivic categories with reasonable properties for arbitrary schemes can be constructed. A crucial property used for the construction is base change for a motivic Eilenberg-MacLane spectrum over Dedekind rings.
I'll review some basic ideas about topological Andre-Quillen theory and how it relates to E-infinity cell structures. As applications I'll discuss a new approach to calculating TAQ for HF_p and HZ, and various other recent results. These make heavy use of Dyer-Lashof operations and the coaction of the dual Steenrod algebra.
Algebraic cobordism MGL was introduced by Voevodsky as an algebro-geometric analogue of complex cobordism MU: it is the universal oriented cohomology theory for smooth schemes. A fundamental result in homotopy theory is Quillen's identification of the homotopy groups of MU with the Lazard ring. Voevodsky conjectured an analogous result for MGL, and his conjecture was recently proved for regular schemes of characteristic zero and up to p-torsion for regular schemes of charateristic p>0. I will explain Voevodsky's conjecture and sketch the proof in these cases.
Abstract: We will give a brief introduction to motivic homotopy theory followed by a discussion on how a theorem of Gabber may be used to avoid assuming that resolution of singularities holds in positive characteristic. The first half will be aimed at a general audience of topologists. The second will feature more algebraic geometry, however we will still try and keep it accessible to topologists.
Abstract: In groundbreaking work Thomason establishes a fundamental comparison between Bott-inverted algebraic K-theory and étale K-theory with finite coefficients. Over the complex numbers, Walker has shown how to deduce Thomason's theorem using a semi-topological K-homology theory. In joint work with J. Hornbostel we establish an equivariant generalization of Walker's Fundamental Comparison Theorem and use it to deduce the equivariant version of Thomason's theorem for complex varieties with action by a finite group.
Abstract: We introduce the notion of Arakelov motivic cohomology, and discuss the beautiful reformulation (due to Jakob Scholbach) of the Beilinson conjectures on special values of L-functions.
Abstract: In topology, there is a correspondence between generalized cohomology theories (in the sense of the Eilenberg-Steenrod axioms) on one hand and spectra on the other hand, the latter being objects in the stable homotopy category SH. In algebraic geometry and motivic homotopy theory, the situation is much more complicated in several ways. Firstly, there are many stable homotopy categories, one for each scheme, and various functors between them. Secondly, there are many sets of axioms for what a cohomology theory should be (Weil cohomology, Bloch-Ogus cohomology, oriented cohomology, ...) and a huge zoo of cohomology theories. The aim of the talk will be to give an overview of all generalized cohomology theories in algebraic geometry, using the language of motivic stable homotopy theory.
We study a modified version of Rognes' logarithmic structures on structured ring spectra. In our setup, we obtain canonical logarithmic structures on connective K-theory spectra which approximate the respective periodic spectra. The inclusion of the p-complete Adams summand into the p-complete connective complex K-theory spectrum is compatible with these logarithmic structures. The vanishing of appropriate logarithmic topological Andre-Quillen homology groups confirms that the inclusion of the Adams summand should be viewed as a tamely ramified extension of ring spectra.
I extend my 2005 AG&T paper with Bruner from the circle case to more general Lie groups. There are new results about infinite cycles for actions by the torus T2 or the rotation group SO(3).
I will go through the simplest case of my 2005 AG&T paper with Bruner, showing that certain classes, in the homological homotopy fixed point spectral sequence for a circle action on a commutative ring spectrum, are infinite cycles. The idea of using an universal example may lead to generalizations for actions by tori or other Lie groups.
We show that the hermitian K-theory of regular schemes (with 2 a unit in the ring of regular functions) is represented in the A^1-homotopy category of Morel-Voevodsky by the ind-scheme of non-degenerate Grassmanians.