Events - Page 3

Time and place: Jan. 10, 2017 10:15 AM12:00 PM, B 738 NHA

Triangulated categories of motives over schemes are sort of the "universal derived categories" among various derived categories obtained by various cohomology theories like l-adic cohomology. Ayoub constructed them using the A1-homotopy equivalences and étale topology. I will introduce the construction of triangulated categories of motives over fs log schemes. Fs log schemes are kinds of "schemes with toroidal boundary," and A1-homotopy equivalences and étale topology are not enough to obtain all homotopy equivalences between fs log schemes. I will explain what extra homotopy equivalences and topologies are neeeded. 

Time and place: Dec. 8, 2016 11:15 AM1:00 PM, B 738

A continuation of part I.

Time and place: Dec. 6, 2016 10:15 AM12:00 PM, B 738

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.  

Time and place: Nov. 29, 2016 10:15 AM12:00 PM, B 738

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.  

Time and place: Nov. 22, 2016 10:15 AM12:00 PM, B 738

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.  

Time and place: Nov. 1, 2016 10:15 AM12:00 PM, B 638

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)   

Time and place: Oct. 25, 2016 10:15 AM12:00 PM, B 638

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.  

Time and place: Oct. 21, 2016 2:15 PM4:00 PM, B 637 NHA

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.

Time and place: Oct. 14, 2016 2:15 PM3:15 PM, B 738

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).

Time and place: Oct. 11, 2016 10:15 AM12:00 PM, B 738

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.

Time and place: Oct. 4, 2016 10:15 AM12:00 PM, B 738

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

Time and place: Sep. 29, 2016 2:15 PM4:00 PM, B 735

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

Time and place: Sep. 29, 2016 2:15 PM4:00 PM, B 735

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

Time and place: Sep. 27, 2016 10:15 AM12:00 PM, B 738

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.

Time and place: Sep. 20, 2016 10:15 AM12:00 PM, B 738

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.  

Time and place: Sep. 14, 2016 2:15 PM3:15 PM, B 637

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.  

Time and place: Sep. 8, 2016 2:15 PM3:15 PM, B637 NHA

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.  

Time and place: Sep. 6, 2016 10:15 AM12:00 PM, B 738

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. 

Time and place: Aug. 16, 2016 10:15 AM, B 738 NHA

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.

Time and place: Apr. 29, 2016 12:15 PM2:00 PM, B 637 (B 63)
Time and place: Feb. 5, 2016 12:15 PM2:00 PM, B 637 NHA

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.   

Time and place: Dec. 1, 2014 10:15 AM12:00 PM, B 62 NHA

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. 

Time and place: Nov. 20, 2014 2:15 PM4:00 PM, B1036 NHA

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.   

Time and place: Nov. 18, 2014 10:15 AM12:00 PM, B935 NHA

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.   

Time and place: Nov. 17, 2014 10:15 AM12:00 PM, B62 NHA

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.