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.
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.
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[ΩM] 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 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 about 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.
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.
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 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.
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.
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.
An introductory lecture.
Guillou and Isaksen, with input from Andrews and Miller, have calculated the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map eta over the fields R and C. We will review these known results and show how to calculate the motivic stable homotopy groups of the two-complete eta-inverted sphere spectrum over fields of cohomological dimension at most two with characteristic different from 2 and the field of rational numbers.
I will give a series of lectures on topological modular forms. In the first lecture I will review (the moduli stack of) formal groups, complex bordism, (the moduli stack of) elliptic curves, elliptic cohomology, and the Goerss-Hopkins-Miller theorem leading to the construction of the topological modular forms spectra TMF, Tmf and tmf.
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.
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 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 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 differential structure in the Adams spectral sequence, leading to its E_\infty-term, and (in a later lecture) the extension problems leading to the ring \pi_*(tmf) of topological modular forms. These calculations were known to Hopkins-Mahowald; in their current guise they are part of joint work with Bruner.