Seminarer - Side 4

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.   

Bjørn Jahren fortsetter seminaret om 4 mangfoldigheter.

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.

Bjørn Jahren fortsetter seminaret om 4 mangfoldigheter.

Bjørn Jahren fortsetter seminaret om 4-mangfoldigheter og vil nå snakke om Clifford-algebraer, Spin-bunter og Spin^c-strukturer på 4-mangfoldigheter.

Geir Ellingsrud fortsetter seminaret om 4-mangfoldigheter.

Geir Ellingsrud fortsetter seminaret om 4-mangfoldigheter og  om elliptiske fibrasjoner.

Geir Ellingsrud fortsetter seminaret om 4-mangfoldigheter og denne gangen snakker han om elliptiske fibrasjoner.

Geir Ellingsrud fortsetter seminaret om 4-mangfoldigheter

Vi vil gi en kortfattet intoduksjon til  algebraiske flater og en like kortfattet oversikt over Enriques-Kodaira klassifikasjonen.

Bjørn Jahren fortsetter seminaret om 4-mangfoldigheter i uke 8

(ikke seminar i uke 7).

Bjørn Jahren fortsetter seminaret om 4-mangfoldigheter.

Dette semesteret arrangerer vi et seminar om 4-mangfoldigheter, et felt som burde interessere både topologer og algebraiskgeometere. Dessuten, siden vi nå er i samme avdeling, bør vi jo ha noe felles aktivitet. Et overordnet mål er å studere komplekse flater opp til diffeomorfi. Det konkrete programmet vil bli til underveis, men emner vi regner med å komme innom inkluderer - Topologisk klassifikasjon av enkeltsammenhengende, kompakte   4-mangfoldigheter (Freedman) - Donaldsons teorem om snittformen til differensiable 4-mangfoldigheter - Enriques-Kodaira klassifikasjon av komplekse flater - Seiberg-Witten teori.

Vi starter opp mandag 28. januar 1415-1600 i B70 med at Bjørn snakker om snittformen og den topologiske teorien. Bjørn Jahren og Geir Ellingsrud

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. 

Abstract: Klassifikasjon av mangfoldigheter leder ofte til spørsmål om h-kobordente mangfoldigheter er homeomorfe eller ikke. Jeg skal presentere noen nye resultater av Slawomir Kwasik og meg selv om dette viktige problemet. 

Jeg vil snakke om en flettet monoidal diagram kategori av B-rom, Quillen ekvivalent til simplisielle mengder, der kommutative monoider svarer til doble løkkerom i simplisielle mengder. For enhver liten flettet monoidal kategori har jeg konstruert en kommutativ B-roms monoide slik at homotopigrensen av den er svakt ekvivalent til nerven av den flettede monoidale kategorien.Dette gir en modell for alle doble løkkerom siden Fiedorowicz, Stelzer og Vogt viser at alle doble løkkerom er svakt ekvivalente til en nerven av en flettet monoidal kategori i en artikkel fra september 2011. 

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.