Seminarer - Side 2
The so-called Koras-Russell threefolds are a family of topologically
contractible rational smooth complex affine threefolds which played an
important role in the linearization problem for multiplicative group
actions on the affine 3-space. They are known to be all diffeomorphic to
the 6-dimensional Euclidean space, but it was shown by Makar-Limanov in
the nineties that none of them are algebraically isomorphic to the affine
3-space. It is however not known whether they are stably isomorphic or not
to an affine space. Recently, Hoyois, Krishna and Østvær proved that many
of these varieties become contractible in the unstable A^1-homotopy
category of Morel and Voevodsky after some finite suspension with the
pointed projective line. In this talk, I will explain how additional
geometric properties related to additive group actions on such varieties
allow to conclude that a large class of them are actually A^1-contractible
(Joint work with Jean Fasel, Université Grenoble-Alpes).
In this talk, we will present some applications of the "transfer" to
algebraic K-theory, inspired by the work of Thomason. Let A --> B be a
G-Galois extension of rings, or more generally of E-infinity ring spectra
in the sense of Rognes. A basic question in algebraic K-theory asks how
close the map K(A) --> K(B)^hG is to being an equivalence, i.e., how close
K is to satisfying Galois descent. Motivated by the classical descent
theorem of Thomason, one also expects such a result after "periodic"
localization. We formulate and prove a general lemma that enables one to
translate rational descent statements as above into descent statements
after telescopic localization. As a result, we prove various descent
results in the telescopically localized K-theory, TC, etc. of ring
spectra, and verify several cases of a conjecture of Ausoni-Rognes. This
is joint work with Dustin Clausen, Niko Naumann, and Justin Noel.
The Bass-Quillen conjecture states that every vector bundle over A^n_R is
extended from Spec(R) for a regular noetherian ring R. In 1981, Lindel
proved that this conjecture has an affirmative solution when R is
essentially of finite type over a field. We will discuss an equivariant
version of this conjecture for the action of a reductive group. When R =
C, this is called the equivariant Serre problem and has been studied by
authors like Knop, Kraft-Schwarz, Masuda-Moser-Jauslin-Petrie. In this
talk, we will be interested in the case when R is a more general regular
ring. This is based on joint work with Amalendu Krishna
In Part 2 we will delve into the worlds of derived and spectral algebraic
geometry. After reviewing some basic notions we will explain how motivic
homotopy theory can be extended to these settings. As far as time permits
we will then discuss applications to virtual fundamental classes, as well
as a new cohomology theory for commutative ring spectra, a brave new
analogue of Weibel's KH
In Part 2 we will delve into the worlds of derived and spectral algebraic
geometry. After reviewing some basic notions we will explain how motivic
homotopy theory can be extended to these settings. As far as time permits
we will then discuss applications to virtual fundamental classes, as well
as a new cohomology theory for commutative ring spectra, a brave new
analogue of Weibel's KH
We consider extensions of Morel-Voevodsky's motivic homotopy theory to the
settings of derived and spectral algebraic geometry. Part I will be a
review of the language of infinity-categories and the setup of
Morel-Voevodsky homotopy theory in this language. As an example we will
sketch an infinity-categorical proof of the representability of Weibel's
homotopy invariant K-theory in the motivic homotopy category.
We will discuss the motivic May spectral sequence and demonstrate how to use it to identify Massey products in the motivic Adams spectral sequence. We will then investigate what is known about the motivic homotopy groups of the eta-local sphere over the complex numbers and discuss how these calculations may work over other base fields.
Certain 3-dimensional lens spaces are known to smoothly bound 4-manifolds with the rational homology of a ball. These can sometimes be useful in cut-and-paste constructions of interesting (exotic) smooth 4-manifolds. To this end it is interesting to identify 4-manifolds which contain these rational balls. Khodorovskiy used Kirby calculus to exhibit embeddings of rational balls in certain linear plumbed 4-manifolds, and recently Park-Park-Shin used methods from the minimal model program in 3-dimensional complex algebraic geometry to generalise Khodorovskiy's result. The goal of this talk is to give an accessible introduction to the objects mentioned above and also to describe a much easier topological proof of Park-Park-Shin's theorem.
In the nineties, Deninger gave a detailed description of a conjectural cohomological interpretation of the (completed) Hasse-Weil zeta function of a regular scheme proper over the ring of rational integers. He envisioned the cohomology theory to take values in countably infinite dimensional complex vector spaces and the zeta function to emerge as the regularized determinant of the infinitesimal generator of a Frobenius flow. In this talk, I will explain that for a scheme smooth and proper over a finite field, the desired cohomology theory naturally appears from the Tate cohomology of the action by the circle group on the topological Hochschild homology of the scheme in question.
The motivic Adams spectral sequence is a general tool for calculating homotopy groups of a motivic spectrum X. We will investigate the construction of the motivic Adams spectral sequence, determine the second page of the spectral sequence, and identify what it converges to in good cases. If time permits, we will show how to use the motivic Adams spectral sequence to obtain explicit calculations of the motivic homotopy groups of spheres and other spectra.
Bloch constructed higher cycle class maps from higher Chow groups to Deligne cohomology and étale cohomology. I will define a map from the motivic Eilenberg-Mac Lane spectrum to the spectrum representing Deligne cohomology in the motivic stable homotopy category over C such that it gives Bloch's higher cycle class map on cohomology. The map is induced by the map from Voevodsky's algebraic cobordism spectrum MGL to the Hodge-filtered complex cobordism spectrum defined by Hopkins-Quick. This extends a result of Totaro showing that the usual cycle class map to singular cohomology factors through complex cobordism modulo the coefficients of the Lazard ring MU^{2*} tensor_L Z. This is joint work with Amit Hogadi.
Abstract:
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.
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.