Arrangementer - Side 6

This will be a colloqium-style talk, with pictures, about the classifying spaces and automorphism groups of manifolds, and the relation to surgery theory and algebraic K-theory.   

På vegne av NFR-prosjektet Topologi i Norge arrangeres det et topologi-møte ved Universitetet i Oslo torsdag 3. desember og fredag 4. desember 2015.  Ytterligere opplysninger kommer på denne siden ettersom de blir tilgjengelige.

In this talk I will present the Real algebraic K-theory construction of Hesselholt and Madsen, and discuss some on-going joint work with Ib Madsen. Real algebraic K-theory is a functor that to a ring A with anti-involution associates a genuine C_2-equivariant spectrum KR(A). Here C_2 denotes the cyclic group of order two. The underlying spectrum of KR(A) has the homotopy type of K(A), the usual K-theory space of A in the sense of Quillen, and the C_2-fixed point spectrum is weakly equivalent to the Hermitian K-theory of A. I will talk about generalizations of known theorems for algebraic K-theory to KR, including delooping results, "fundamental" theorems and group completion.

Modular forms are certain complex-analytic functions on the upper-half plane. They can also be interpreted as giving linear-algebraic invariants of elliptic curves, perhaps equipped with some extra structure, and in this way they reveal their algebraic-geometric nature. One of the most fundamental modular forms is the Dedekind eta function. However, it seems that only recently has it been pinned down precisely what extra structure on an elliptic curve is needed to define eta. Namely, Deligne was able to express this extra structure in terms of the 2- and 3-power torsion of the elliptic curve. Deligne's proof, apparently, is computational. In this talk I'll describe a conjectural reinterpretation of Deligne's result, together with some supporting results and a hint at a possible conceptual proof. The reinterpretation is homotopy theoretic, the key being to think of an elliptic curve as giving a class in framed cobordism. This directly connects the number "24" which often appears in the study of eta to the 3rd stable stem in topology. 

I will discuss joint work in progress with David Gepner, computing the ring of endomorphisms of the equivariant motivic sphere spectrum, for a finite group. The result is a combination of the endomorphism ring of the equivariant topological sphere spectrum (which equals the Burnside ring by a result of Segal) and that of the motivic sphere spectrum (which equals the Grothendieck-Witt ring of quadratic forms by a result of Morel). This computation is a corollary of a tom Dieck style splitting for certain equivariant motivic homotopy groups.   

This is a work we had done jointly with Garkusha (after Voevodsky) arXiv:1409.4372. Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives is introduced and studied. To any smooth algebraic variety X in Sm/k, the framed motive M_fr(X) is associated in that category . Also, for any smooth scheme X in Sm/k an explicit quasi-fibrant motivic replacement of its suspension P1-spectrum is given. Moreover, it is shown that the bispectrum (M_fr(X),M_fr(X)(1),M_fr(X)(2), ... ), each term of which is a twisted framed motive of X, has motivic homotopy type of the suspension bispectrum of X. We also construct a compactly generated triangulated category of framed bispectra SH_fr(k) and show that it reconstructs the Morel-Voevodsky category SH(k). As a topological application, it is proved that the framed motive M_fr(pt)(pt) of the point pt = Speck evaluated at pt is a quasi-fibrant model of the classical sphere spectrum whenever the base field k is algebraically closed of characteristic zero.   

The goal of this talk is to present some recent computations of the Picard groups of several spectra of topological modular forms. The first part of the talk will introduce the toolbox, which consists of descent theory and a technical lemma allowing us to compare stable and unstable information in spectral sequences. This is joint work with Akhil Mathew.   

Jeg vil definere Singerkonstruksjonen R_+(M) og gjennomføre Adams-Gunawardena-Millers bevis av Lins teorem.  

Jeg vil snakke om de endelige underalgebraene A(n) i Steenrodalgebraen, analysere A(n)-modulstrukturen til den kontinuerlige kohomologien til Tatekonstruksjonen, og skissere Lin-Davis-Mahowald-Adams' bevis av Lins teorem. 

Jeg avslutter reduksjonen av Segalformodningen til et algebraisk spørsmål om en Ext-ekvivalens, og vil se i mer detalj på hvordan Pontryagin--Thom-konstruksjonen brukes til å bevise Wirthmüller- og Adams transfer-ekvivalensene i stabil ekvivariant homotopiteori. 

Jeg vil vise hvordan Segalformodningen kan omformuleres, ved hjelp av Tatekonstruksjonen, norm-restriksjonssekvensene, Warwick dualitet og Adams' transferekvivalens, til en form som lettere lar seg bevise v.h.a. den algebraiske Singerkonstruksjonen. Dette er det andre i en serie foredrag som sikter mot å gi et bevis for en versjon av Segalformodningen i motivisk homotopiteori. 

Motivert av Atiyah og Segals kompletteringsteorem for ekvivariant topologisk K-teori formulerte Graeme Segal en tilsvarende formodning om ekvivariant stabil kohomotopi. Jeg vil minne om hva teoremet og formodningen sier, og vise hvordan Segalformodningen kan omformuleres, ved hjelp av norm-restriksjonssekvensene, til en form som lettere lar seg bevise v.h.a. den algebraiske Singerkonstruksjonen. Tanken er at dette skal være det første av en serie foredrag som sikter mot å gi et bevis for en versjon av Segalformodningen i motivisk homotopiteori. Dette var temaet for Thomas Gregersens PhD-avhandling fra 2012.

If B is a sub-Hopf algebra of the mod 2 Steenrod algebra, the category of B-modules has subcategories of modules local or colocal with respect to certain Margolis homologies, and corresponding localization and colocalization functors. The Picard groups of these subcategories are sufficient to detect the Picard group of the whole category and contain modules of geometric interest. General results obtained along the way allow us to begin to attack the analogous questions for E(2) and A(2)-modules. Applications include better descriptions of polynomial algebras as modules over the Steenrod algebra, and of the values of certain generalized cohomology theories on the classifying spaces of elementary abelian groups.   

Et Eilenberg-Mac Lane rom K(G,n) er bestemt av å ha homotopigruppe G konsentrert i dimensjon n (og 0 ellers), og representerer n-te kohomologi med koeffisienter i G opp til homotopi. Barkonstruksjonen på K(G,n) gir et nytt rom BK(G,n)=K(G,n+1) som selv er en topologisk gruppe. Diagonalen G--->GxG og multiplikasjonen GxG--->G induserer en multiplikasjon på mod p homologigrupper, og gjør denne til en koalgebraisk ring (eller Hopf algebra). Jeg vil konstruere en spektralsekvens av Hopf algebraer (bar spektralsekvensen) som beregner homologi av K(G,n+1) gitt homologi av K(G,n) (for homologiteorier med en Künneth-formel), og vil så beregne mod p-homologi av K(Z,3)=BK(Z,2) som Hopf algebra.   

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.   

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.