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.
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 Lazard ring MU*(-) ⊗L ℤ. This is joint work with Amit Hogadi.
Plan: Introduction to the talks. The motivic Steenrod algebra and its dual, Hopf algebroids, Ext, the canonical Adams resolution and identification of the E2-page of the motivic Adams spectral sequence. If there is time, a change of rings theorem. Most of the material is taken from:
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.
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.
Using the machinery from the two previous lectures we apply the motivic Adams spectral sequence to some spectra (Hℤ, KGL, KQ) over certain base fields (ℂ, ℝ, 𝔽q, ℚp). This will illustrate some common techniques for calculations with the motivic Adams spectral sequence. We will discuss the cobar complex, the ρ-Bockstein spectral sequence and base change.
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.
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 η-local sphere over the complex numbers and discuss how these calculations may work over other base fields.
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.
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.
The Bass-Quillen conjecture states that every vector bundle over 𝔸nR 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 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 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 𝔸1-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 𝔸1-contractible (Joint work with Jean Fasel, Université Grenoble-Alpes).
Calculating the residues for rational integrals in complex variables is a classical problem in mathematics. It is directly related to questions on algebraic cycles, their cohomology classes, and the Abel-Jacobi map. I will discuss the analogs of these questions for algebraic cobordism cycles. In particular, I will present new versions of an Abel-Jacobi map, a regulator map and Arakelov arithmetic cobordism. This in part joint work with Michael Hopkins.
The advances on the Milnor- and Bloch-Kato conjectures have led to a good understanding of motivic cohomology and algebraic K-theory with finite coefficients. However, important questions remain about rational motivic cohomology and algebraic K-theory, including the Beilinson-Soulé vanishing conjecture. We discuss how the speaker's "connectivity conjecture" for the stable rank filtration of algebraic K-theory leads to the construction of chain complexes whose cohomology groups may compute rational motivic cohomology, and simultaneously satisfy the vanishing conjecture. These "rank complexes" serve a similar purpose as Goncharov's candidates for motivic complexes, but have the advantage that they have a precise relation to rational algebraic K-theory. (Here are lecture notes from an earlier talk on the same subject.)
Given a Nisnevich sheaf (on smooth schemes of finite type) of spectra, there exists a universal process of making it 𝔸1-invariant, called 𝔸1-localization. Unfortunately, this is not a stalkwise process and the property of being stalkwise a connective spectrum may be destroyed. However, the 𝔸1-connectivity theorem of Morel shows that this is not the case when working over a field. We report on joint work with Johannes Schmidt and sketch our approach towards the following theorem: Over a Dedekind scheme with infinite residue fields, 𝔸1-localization decreases the stalkwise connectivity by at most one. As in Morel’s case, we use a strong geometric input which is a Nisnevich-local version of Gabber’s geometric presentation result over a henselian discrete valuation ring with infinite residue field.
A continuation of part I. Here are my lecture notes.
In this talk I will explain how the use of functors defined on the category I of finite sets and injections makes it possible to replace E-infinity objects by strictly commutative ones. For example, an E-infinity space can be replaced by a strictly commutative monoid in I-diagrams of spaces. The quasi-categorical version of this result is one building block for an interesting rigidification result about multiplicative homotopy theories: we show that every presentably symmetric monoidal infinity-category is represented by a symmetric monoidal model category. (This is based on joint work with C. Schlichtkrull, with D. Kodjabachev, and with T. Nikolaus)
We compute the generalized slices (as defined by Spitzweck-Østvær) of the motivic spectrum KQ in terms of motivic cohomology and generalized motivic cohomology, obtaining good agreement with the situation in classical topology and the results predicted by Markett-Schlichting.
Hopkins, Kuhn, and Ravenel proved that, up to torsion, the Borel-equivariant cohomology of a G-space with coefficients in a height n-Morava E-theory is determined by its values on those abelian subgroups of G which are generated by n or fewer elements. When n=1, this is closely related to Artin's induction theorem for complex group representations. I will explain how to generalize the HKR result in two directions. First, we will establish the existence of a spectral sequence calculating the integral Borel-equivariant cohomology whose convergence properties imply the HKR theorem. Second, we will replace Morava E-theory with any L_n-local spectrum. Moreover, we can show, in some sense, a partial converse to this result: if an HKR style theorem holds for an E_\infty ring spectrum E, then K(n+j)_* E=0 for all j\geq 1. This partial converse has applications to the algebraic K-theory of structured ring spectra.
Framed correspondences were invented and studied by Voevodsky in the early 2000-s, aiming at the construction of a new model for motivic stable homotopy theory. Joint with Ivan Panin we introduce and study framed motives of algebraic varieties basing on Voevodsky's framed correspondences. Framed motives allow to construct an explicit model for the suspension P1-spectrum of an algebraic variety. Framed correspondences also give a kind of motivic infinite loop space machine. They also lead to several important explicit computations such as rational motivic homotopy theory or recovering the celebrated Morel theorem that computes certain motivic homotopy groups of the motivic sphere spectrum in terms of Milnor-Witt K-theory. In these lectures we shall discuss basic facts on framed correspondences and related constructions.
A continuation of part I.
We lift the classical theorem of Arnol'd on homological stability for configurations spaces of the plane to the motivic world. More precisely, we prove that the schemes of unordered configurations of points in the affine line satisfy stability with respect to the motivic t-structure on mixed Tate motives. See https://arxiv.org/abs/1511.09031.