Seminars - Page 2

In this second talk, I will define Chekanov's version of Legendrian contact homology (LCH) for Legendrian knots in R3. I will begin with an example, showing that LCH is more sensitive than the classical invariants. This will use a linearized version of the homology. In the second part of the talk I will focus on the proof that the differential indeed squares to zero, and also say something about invariance under Legendrian Reidemeister moves. This is intended to be a smooth introduction to the next talk, where we will consider Legendrian contact homology defined for Legendrians in arbitrary 1-jet spaces. This case is more delicate, and we have to understand the concept of Gromov compactness for pseudo-holomorphic curves to prove that we get a differential graded algebra associated to each Legendrian, whose homology will give a Legendrian invariant.

Let G be a finite (abstract) group and let k be a field of characteristic zero. We prove that for a non-singular projective G-variety X over k, and a non-singular G-invariant subvariety Y of dimension >= 3, which is a scheme-theoretic complete intersection in X, the pullback map PicG(X) -> PicG(Y) is an isomorphism. This is an equivariant analog of the Grothendieck-Lefschetz theorem for Picard groups.   

A Cartan-Eilenberg system is an algebraic structure introduced as a model of the diagram obtained by taking the homology of all subquotients in a filtered chain complex. There are two exact couples and a single spectral sequence associated with such a system, and one may thus apply Boardman's theory of convergence to either exact couple. After reviewing parts of this theory, I will clarify the convergence situation in a Cartan-Eilenberg system and in particular present new work on a simpler interpretation of Boardman's whole plane obstruction group.   

I will give a series of talks about Legendrian contact homology, an invariant of Legendrian submanifolds in 1-jet spaces, defined by a count of pseudo-holomorphic curves. In this first lecture I will give a brief and gentle introduction to symplectic and contact geometry, with focus on Lagrangian and Legendrian submanifolds. No previous knowledge about the subject is needed, except for elementary knowledge about differentiable manifolds.   

I will discuss the differential structure in the mod 2 Adams spectral sequence for tmf, leading to its E_\infty-term.  These calculations were known to Hopkins-Mahowald; in their current guise they are part of joint work with Bruner.

I will report on work in progress on calculations of the motivic homotopy groups of MGL (the algebraic cobordism spectrum) over number fields. It is known that pi_{2n,n}(MGL) is the Lazard ring, and pi_{-n,-n}(MGL) is Milnor K-theory of the base field. We will calculate all of pi_{*,*}(MGL) with the slice spectral sequence (motivic Atiyah-Hirzebruch spectral sequence) over a number field. I will give a brief review of the the tools and sketch the main parts of the calculation: The input from motivic cohomology, the use of C_2-equivariant Betti realization and comparison with Hill-Hopkins-Ravenel to determine the differentials, and settle most of the hidden extensions. 

I will discuss the algebra structure of the E_2-term of the mod 2 Adams spectral sequence for tmf, given by the cohomology Ext_{A(2)}(F_2, F_2) of A(2).  We (Bruner & Rognes) use Groebner bases to verify the presentation given by Iwai and Shimada, with 13 generators and 54 relations. Thereafter I will discuss the relationship between differentials and Steenrod operations in the Adams spectral sequence for E_\infty ring spectra.  

I will discuss machine computations in a finite range, using Bruner's ext-program, of Ext over A, the mod 2 Steenrod algebra, and over A(2), the subalgebra of A generated by Sq^{^1}, Sq^{^2} and Sq^{^4}. These are the E_2-terms of the mod 2 Adams spectral sequences for S and tmf, respectively.

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.

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.

 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.

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). 

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. 

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.

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.

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.   

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. 

A continuation of part I.

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.  

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.  

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.  

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)   

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.  

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.