Fields Institute Program on Homotopy Theory and its Applications University of Western Ontario London, Ontario September, 2003

The conference is now over. This information on these pages will not be updated further.

Abstracts for all minicourses

Paul Balmer, Codescent theory

Gunnar Carlsson, Representation Theory and K-theory of fields

In this course we present a conjectural model for the homotopy type of the K-theory spectrum of a field. The model is constructed out of the K-theory spectrum of the representation theory of the absolute Galois group, using a homotopy invariant notion of completion. We will discuss possible lines of proof, as well as connections with other approaches, notably the motivic one.

Gunnar Carlsson, Computational Algebraic Topology: Point clouds, shape recognition, and algorithms

We will discuss several methods for applying algebraic topology to the problem of identifying features in high-dimensional data and for recognizing shapes in lower dimensions. Homology, applied to spaces constructed out of a space X, can recognize qualitative features in X such as edges, corners, cone points, etc, and can be used as a tool for recognizing the presence of such features which is insensitive to small deformations. It can also be used to recognize the presence of regions of higher curvature. We will be dealing with sets presented as "point cloud data", i.e. as large but finite sets sampled from the object. The methods will also lead to some interesting algorithmic questions for the computation of homology, to which we will give some answers.

Paul Goerss, Sheaves of modules and sheaves of spectra on the moduli stack of formal groups

The point of these two lectures is to make explicit the connection between the algebraic geometry of formal group laws and stable homotopy theory. The first lecture will develop the foundations, make explicit the connections between comodules and sheaves, and explore the chromatic decomposition of spectra from the point of view of the algebraic geometry. The second lecture will focus of the differential calculus of the moduli stack of formal groups, including its tangent space, the cotangent sheaf, the connections to the deformation theory of formal group laws, and to the problem of realizing algebra in homotopy theory.

Much of the material here is well-known, in some diffuse sense. For example, the first lecture here is not much more than an detailed explanation of the table in section 2 of the paper

Hopkins, M. J. and Gross, B. H., "The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory", Bull. Amer. Math. Soc. (N.S.), 30 (1994) No. 1, 76--86.

Many of these results have been proved by these and other methods by any number of people: Hopkins, Hovey, Morava, Ravenel, Strickland, and others I'm sure. The material on stacks in homotopy theory goes back to Strickland and Hopkins, and has been developed by Hollander, Jardine, Pribble, and others.

Ieke Moerdijk, Operads, Hopf algebras and homotopy theory

I will explain some basic constructions relating Hopf algebras and operads, and give three applications: In the first lecture, I show how the existence of a suitable Hopf algebra interval can be used to equip the category of operads with a a closed Quillen model structure; this applies to many contexts, e.g. topological operads and operads in dg vector spaces. In lecture 2, I will discuss the Boardman-Vogt resolution of an operad, originally designed for topological operads. This resolution applies to operads in any symmetric monoidal category possessing a suitable interval, and can be used to relate various existing resolutions. Finally, in the third lecture, I will discuss the Connes-Kreimer Hopf algebra and some of its variations from the point of view of operads.

Jack Morava, Recent developments in Galois theory

It is an old conjecture of Iwasawa that the Galois group Gal(Q¯/ Q_ab) of the algebraic closure Q¯ of the rationals over the cyclotomic field Q_ab (defined by adjoining all roots of unity to Q¯) is (pro)free. This conjecture has begun to emerge as a central issue in recent years, in algebraic geometry (via the theory of motives [3,7]), in quantum field theory (through Connes and Kreimer's Galois theory of renormalization of Feynman integrals [1,2] and Kontsevich's work on deformation quantization [5]), and in algebraic topology (via automorphisms of the little disk operad, and possibly in other ways [6]). This minicourse will attempt to scratch the surface of this vast body of work, starting from a basepoint in algebraic topology.

I.   Tannakian technology

Algebraic geometers have developed a sophisticated dictionary which translates between symmetric monoidal Abelian (or perhaps triangulated) categories, and representations of certain related groupschemes, or Hopf algebras. Being secretly frightened by torsion, they traditionally work in categories enriched over Q, but many of their central ideas have much broader applicability. This talk will be a sketchy introduction to various useful categories of motives and motivic groups; but as background I will try to clarify how the stable homotopy categories of algebraic topology fit into this picture, and how to think of automorphisms of cohomology theories (the Steenrod algebra, Adams operations, etc) as variations on the theme of motivic groups.

II.   The Grothendieck-Teichmüller group(s)

Late last century Drinfel'd [4], pursuing ideas of Grothendieck, constructed an embedding of the absolute Galois group Gal(Q¯/Q) into a group of automorphisms of the system of braid groups [3]. This subject has developed rapidly in two orthogonal directions, one profinite [7], the other pro-unipotent (ie, involving Lie algebras over Q) [3]. The latter theory is in some ways more radical, with connections to the study of the values of the Riemann zeta function at odd positive integers, and through that to algebraic K-theory of the integers (via the Borel regulators) and to the transcendetal integrals that appear in quantum knot invariants. I will try to use the (relatively geometrical) theory of mixed motives as a thread through this maze of ideas.

III.   Galois theory in physics

Connes and Kreimer have developed a beautiful approach to the classical theory of renormalization, starting with certain Hopf algebras defined combinatorially in terms of Feynman graphs and rooted trees, and ending with certain cosmic questions [1] about transcendental numbers. Somewhat similar ideas appear in Kontsevich's work on deformation quantization, with the group Gal(Q¯/Q) lurking in both cases in the background as unidicted coconspiritor. This talk will focus on Connes and Kreimer's construction of a representation of their renormalization Galois group, in the group of formal automorphisms of the line (which plays the role of motivic group for complex cobordism).

References

[1] P. Cartier, A mad day's work ... BAMS 38 (2001).

[2] A. Connes, Symetries Galoisiennes et renormalisation, available at math.QA/0211199.

[3] P. Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois groups over Q, MSRI Publ. 16 (1989).

[4] V. Drinfel'd, Quasitriangular quasiHopf algebras and a group closely related to Gal(Q¯/Q), Petersburg Math. Journal 2 (1991).

[5] M. Kontsevich, Operads and motives in deformation quantization, available at math.QA/9904055.

[6] J. Morava, The motivic Thom isomorphism, available at math.AT/0306151.

[7] L. Schneps, The Grothendieck-Teichmüller group: a survey, in The Grothendieck theory of dessins d'enfants, ed. L. Schneps, LMS notes 200 (1994).

Fabien Morel, Computation and conjectures on motivic stable homotopy groups of spheres

In this series of lectures, I'll mostly explain the main steps in computing the motivic pi₀ of the sphere spectrum as the Milnor-Witt K-theory of the base field. The first part will be introductory: we will recall the standard definitions of A¹-homotopy theory, and the basic examples. In the second part we will state and prove the stable A¹-connectivity theorem, and will explain some of its consequences, such as the existence of the homotopy t-structure for S¹-spectra. In the third part, we will introduce the Milnor-Witt K-theory of fields, and its unramified version, and prove the main theorem. In the last part, we will give some applications of the main theorem, will give some computations in rational stable homotopy theory and will also give some conjectures concerning higher stable homotopy groups of spheres.