Time and place: May 10, 2017 2:15 PM - 4:00 PM, B 738

Topological cyclic homology is a variant of negative cyclic homology which was introduced by Bökstedt, Hsiang and Madsen. They invented topological cyclic homology to study algebraic K-theory but in recent years it has become more and more important as an invariant in its own right. We present a new formula for topological cyclic homology and give an entirely model independent construction. If time permits we explain consequences and further directions.

Time and place: May 3, 2017 2:15 PM - 4:00 PM, B 738

Joint work with Bjørn I. Dundas. We prove that algebraic K-theory, topological Hochschild homology and topological cyclic homology satisfy cubical and cosimplicial descent at connective structured ring spectra along 1-connected maps of such ring spectra.

Time and place: Apr. 26, 2017 2:15 PM - 4:00 PM, B 738

 In this talk all spaces and spectra will be localised at 2. Many E-infinity ring spectra turn out to be `finitely generated' in the sense that there is finite CW spectrum and a map from the free E-infinity ring spectrum generated by it inducing an epimorphism in mod 2 homology. This turns out to be an interesting condition and I will discuss some examples such as HZ, kO, kU, tmf and tmf_1(3). One long term goal of this work is to produce `ultra-generalised Brown-Gitler spectra' and I will discuss this idea if there is time.

Time and place: Apr. 3, 2017 2:15 PM - 4:00 PM, B 637

Given a knot K in the 3-sphere, we use Heegaard Floer correction terms to give lower bounds on the first Betti number of (orientable and non-orientable) surfaces in the 4-ball with boundary K. An amusing feature of the non-orientable bound is its superadditivity with respect to connected sums. This is joint work with Marco Marengon. If time permits, I will discuss relations with deformations of singularities of curves (joint work with József Bodnár and Daniele Celoria). 

Time and place: Mar. 15, 2017 2:15 PM - 4:00 PM, B 738

In the 80's Bökstedt introduced THH(A), the Topological Hochschild homology of a ring A, and a trace map from algebraic K-theory of A to THH(A). This trace map, along with the circle action on THH, have since been used extensively to make calculations of algebraic K-theory. When the ring A has an anti-involution Hesselholt and Madsen have promoted the spectrum K(A) to a genuine Z/2-spectrum whose fixed points is the K-theory of Hermitian forms over A. They also introduced Real topological Hochschild homology THR(A), which is a genuine equivariant refinement of THH, and Dotto constructed an equivariant refinement of Bökstedt's trace map. I will report on recent joint work with Dotto, Patchkoria and Reeh on models for the spectrum THR(A) and calculations of its RO(Z/2)-graded homotopy groups. 

Time and place: Feb. 22, 2017 2:15 PM - 4:00 PM, B 738

The classical s-cobordism theorem classifies completely h-cobordisms from a fixed manifold, but it does not tell us much about the relationship between the two ends. In the talk I will present some old and new results about this. I will also discuss how this relates to a seemingly different problem: what can we say abobut two compact manifolds M and N if we know that MxR and NxR are diffeomorphic? This is joint work with Slawomir Kwasik, Tulane, and Jean-Claude Hausmann, Geneva.

Time and place: Feb. 22, 2017 12:15 PM - 2:00 PM, B 637

I will survey the connection between the space H(M) of h-cobordisms on a given manifold M, several categories of spaces containing M, Waldhausens algebraic K-theory A(M), and the algebraic K-theory of the suspension ring spectrum S[?M] of the loop space of M. The results extend the h-cobordism theorem of Smale and the s-cobordism theorem of Barden, Mazur and Stallings to a parametrized h-cobordism theorem, valid in a stable range established by Igusa, first discussed by Hatcher and finally proved and published by Waldhausen, Jahren and myself.

Time and place: Feb. 1, 2017 2:15 PM - 4:00 PM, B 738

The Barratt nerve BSd X of the Kan subdivision Sd X of a simplicial set X \in sSet is a triangulation. The Barratt nerve is defined as taking the poset of non-degenerate simplices, thinking of it as a small category and then finally taking the nerve.Waldhausen, Jahren and Rognes (Piecewise linear manifolds and categories of simple maps) named this construction 'the improvement functor' because of the homotopical properties and because its target is non-singular simplicial sets. A simplicial set is said to be 'non-singular' if its non-degenerate simplices are embedded. There is a least drastic way of making a simplicial set non-singular called 'desingularization', which is a functor D:sSet -> nsSet that is left adjoint to the inclusion. The functor DSd^2 is the left Quillen functor of a Quillen equivalence where the model structure on sSet is the standard one where the weak equivalences are those that induce weak homotopy equivalences and the fibrations are the Kan fibrations. I will talk about the main steps of the proof that the natural map DSd X -> BX is an isomorphism for regular X. This implies that DSd^2 is a triangulation and that the improvement functor is less ad hoc than it may seem. Furthermore, I will explain how the result provides evidence that any cofibrant non-singular simplicial set is the nerve of some poset.   

Time and place: Jan. 10, 2017 10:15 AM - 12:00 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 AM - 1:00 PM, B 738

A continuation of part I.

Time and place: Dec. 6, 2016 10:15 AM - 12:00 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 AM - 12:00 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 AM - 12:00 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 AM - 12:00 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 AM - 12:00 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 PM - 4:00 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 PM - 3:15 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 AM - 12:00 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 AM - 12:00 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 PM - 4:00 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 PM - 4:00 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 AM - 12:00 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 AM - 12:00 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 PM - 3:15 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 PM - 3:15 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.