Seminars - Page 3

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.   

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.   

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.   

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.