Talk titles and abstracts
Michael Andrews, Non-nilpotent self-maps in motivic and equivariant homotopy theory
Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self-maps on finite p-local spectra induce nonzero homomorphisms on BP-homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and although η:S1,1 → S0,0 induces zero on BP-homology, it is non-nilpotent. Work with Haynes Miller has led to a calculation of the homotopy of the η-local motivic sphere spectrum over C, proving a conjecture of Guillou and Isaksen. Continuing the story as in the classical case, we find that there is a non-nilpotent self-map on the cofiber of η. Motivically over R, this construction provides a Σ2-equivariant self-map which is the Adams' self-map on fixed points.
Aravind Asok, Algebraizing topological vector bundles
I will discuss the problem of when a topological vector bundle on a smooth complex affine variety admits an algebraic structure. A necessary condition that a topological vector bundle is algebraizable is that its integral Chern classes are algebraic, i.e., lie in the image of the cycle class map from Chow groups to integral cohomology. It is a folk conjecture (arguably one that can be attributed to P. Griffiths) that any topological vector bundle on a smooth complex affine variety with algebraic Chern classes is algebraizable. I will explain that this conjecture is true for smooth affine varieties of dimension ≤ 3, and show that it is false for smooth affine varieties of dimension ≥ 4. In particular, I will describe necessary and sufficient condition for algebraizability of rank 2 vector bundles on smooth complex affine 4-folds and an example that exhibits non-triviality of an obstruction beyond algebraicity of Chern classes. This is based on joint work with Jean Fasel and Mike Hopkins.
Bert Guillou, Eta and the structure of motivic Ext
Let A denote the mod 2 motivic Steenrod algebra over Spec(C). The motivic Adams spectral sequence has E2 term given by ExtA and converges to the motivic stable homotopy groups of spheres. The motivic Hopf map eta is not nilpotent, contrary to the classical case, and this is represented by an infinite h1-tower in ExtA, which computes the homotopy of the eta-local motivic sphere. We will also see that the h1-towers give the only classes appearing above the classical Adams vanishing line. The argument will lead to discussion of motivic type 2 complexes. This is joint work with Dan Isaksen.
Jeremiah Heller, Endomorphisms of the equivariant motivic sphere
For a finite group G, I will discuss a tom Dieck style splitting theorem for certain stable equivariant motivic homotopy groups. As a corollary, we obtain a computation of the group of endomorphisms of the equivariant motivic sphere spectrum. This is joint work with D. Gepner.
Mike Hill, Towards the equivariant Dyer-Lashof algebras
The Dyer-Lashof algebra describes the natural operations that show up in the homology of an infinite loop space. Equivariantly, extremely little is known about these. In this talk, I'll discuss joint work with Blumberg and with Lawson which provides a heuristic framework to understand the equivariant Dyer-Lashof algebras, and I will also demonstrate some basic stating computations.
Po Hu, Equivariant K-theory of compact Lie groups with involution
I will talk about joint work with I.Kriz and P.Somberg in which we computed the G ⋊ Z/2-equivariant K-theory of a compact Lie group G with involution. I will relate this to representation theory and, briefly, to symmetric spaces.
Igor Kriz, Calculations of "ordinary" RO(G)-graded cohomology over (Z/2)n
I will report on recent joint work with my student John Holler in which we calculated the RO(G)-graded coefficients of the Mackey cohomology with respect to the constant Z/2 coefficient system for G = (Z/2)n.
Mona Merling, Equivariant algebraic K-theory
A group action on the input ring or category induces an action on the algebraic K-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic K-theory is, for example, that the map of spectra with G-action induced by a G-map of G-rings is not equivariant. We will define a version of equivariant algebraic K-theory which encodes a group action on the input in a functorial way to produce a genuine algebraic K-theory G-spectrum, and we will discuss some properties of the resulting G-spectrum. For example, our construction recovers as particular cases equivariant topological real and complex K-theory, Atiyah's Real K-theory and statements previously formulated in terms of naive G-spectra for Galois extensions. We will also discuss possible approaches to equivariant A-theory.