See also the Brave New Rings seminar page.
(this is based on joint work with Gilmour, and Reinhard)
André-Quillen homology (for simplicial rings) and the analogue for commutative S-algebras known as topological André-Quillen homology can be calculated for cellular S-algebras in terms of the cell structures.
I will explain how this works and then mention applications to the study of minimal atomic p-local commutative S-algebras and simplicial commutative algebras over a Noetherian local ring, generalising analogous results of Baker, May, Pereira for p-local spaces and spectra and Alshumrani for chain complexes over a Noetherian local ring.
A logarithmic structure on a connective commutative S-algebra A is a map alpha : M --> Omega^infty A from a pointed E_infty monoid to the multiplicative E_infty monoid on A, that maps the preimage of GL_1(A) by an equivalence to GL_1(A). It specifies extra geometric data on Spec(A), somewhat like a divisor. Following Kato, Barwick and others, I wish to explore topological examples, like A = ku, with a view to comparing the algebraic K-theory of ku with a logarithmic structure generated by the Bott element, with the algebraic K-theory of KU.
We will discuss Andre-Quillen homology in the setting of algebras over simplicial algebras and use it to define a cellular structure in this category.
There is a fruitful analogy between the homological studies of complements of plane arrangements and spaces of knots (whose 0-dimensional cohomology classes are exactly the knot invariants).
We shall describe in parallel a geometric realization (based on work of G. Ziegler and R. Zhivaljevich) of the Goresky-MacPherson formula for cohomology of complements of arrangements, on one hand, and an effective algorithm for constructing combinatorial formulas for knot invariants, on the other.
Is it possible to determine the Quillen equivalence type of a stable model category just from the triangulated structure of its homotopy category? We will apply this question to the K-local stable homotopy category. Localised at 2, the K-local stable homotopy category has a unique model indeed. However, for odd primes this does not hold, and we will describe an exotic model for this case.
I give a simple model for the bicategory of 2-groups and lax functors between them and explain why it is useful in the study of (2-)group actions on stacks.
I will talk about joint work with Andrew Stacey. This describes the structure of the unstable cohomology operations for a suitable cohomology theory in terms of "plethories" (also known as "Tall-Wraith monoids"). It can be viewed as a monoidal reinterpretation of a description due to Boardman, Johnson and Wilson. One advantage of this approach is that composition of operations is encoded in a natural way.
Let S be a surface of genus g with r boundary components. We study the mapping class group MC(S), i.e. the connected components of the group of orientation preserving diffeomorphisms of S fixing the boundary poinwise. It is a classical result by Ivanov that the homology of MC(S) in degree n with integral coefficients does not depend on r or g for g>2n. This is proved by comparing S with the surface obtained from S by gluing on a "pair of pants"; an operation that changes g and/or r by one. Later, Harer proposed an improved bound, roughly 2g>3n, but with rational coefficients. In this talk I will explain how the proof of such a stability result goes, and indicate what kind of modifications is needed to improve the original stability result.
We outline a calculation, for regular primes p, of the mod p cohomology module over the Steenrod algebra, and some of the p-primary homotopy groups, of the spectrum of the title. The applications are to the homotopy type of the space of concordances of highly-connected manifolds. We also discuss the linearization map A(*) --> K(Z) in degree 5.
The problem of determining when a smooth compact manifold admits a positive-scalar-curvature (psc) Riemannian metric is comparatively well understood. However, even for the n-sphere, surprisingly little work has been done to date concerning the topological stucture of the moduli space of all psc metrics modulo diffeomorphisms. In this talk, I will present some new results concerning the rational homotopy groups of this space for the n-sphere with n>4. My approach uses results on higher analytical/topological torsion due to Hatcher, Igusa, and Goette.
We define and discuss étale cobordism, a quite new cohomology theory for schemes which is related to algebraic cobordism, in analogy to algebraic and étale K-theory. As for K-theory, étale and algebraic cobordism with finite coefficients should agree after inverting a Bott element. We discuss this conjecture and a proof for smooth schemes over algebraically closed fields. The key ingredients for the constructions are a stable étale realization functor for motivic spectra and the stable homotopy theory of simplicial profinite sets.
The history of topological methods in algebraic geometry started almost 60 years ago when André Weil formulated his famous conjectures. A lot of scheme invariants of cohomological nature were introduced since that time and many classical problems were solved in that way. However, until mid-90's, it was unclear whether cohomology theories on schemes can be represented by objects of some category playing the same role as the category of spectra does in topology. The situation changed revolutionarily after works of Voevodsky and collaborators. Their approach gave us a lot of new cohomology theories on schemes and also raised many interesting questions.
We are going to look at one old result from algebraic K-theory (Suslin's rigidity theorem) from this new point of view. We'll consider it as the particular case of a more general statement holding for a wide class of cohomology theories. In order to describe this class, we introduce a notion of an orientable theory in the algebraic context. Orientable theories inherit some nice properties from topology. For example, such a classical topological result as Poincaré duality also holds for orientable theories on algebraic varieties. Besides algebraic K-theory, important examples of orientable theories include motivic cohomology and algebraic cobordism.
Department page Old pageJohn Rognes / December 12th 2007