Using the character theory of Hopkins-Kuhn-Ravenel and the total power operation in complex cobordism of tom Dieck, we develop a theory of power operations in Landweber-exact cohomology theories. We give a description of the total power operation in terms of the theory of subgoups of formal group laws developed by Lubin. We apply this machinery in two cases. For the cohomology theory Eh, we obt...

The notion of local cohomology was prominent in the summer school; in this volume it puts in explicit appearances in the papers of Greenlees and Huneke/Taylor, and figures at least peripherally in material related to the paper of Adem. The aim of this short account is to introduce local cohomology and, in a very sketchy way, tie together some of its various manifestations. There are some real d...

We describe the homotopy type of the analytic Milnor fiber in terms of the special fiber of a strictly semi-stable reduction, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. In the appendix, we consider the de Rham cohomology of the analytic Milnor fiber of an isolated singularity, and we re-interpret some class...

Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschet...

For a local non-Archimedean field K we construct GLd+1(K)equivariant coherent sheaves VOK on the formal OK -scheme X underlying the symmetric space X over K of dimension d. These VOK are OK -lattices in (the sheaf version of) the holomorphic discrete series representations (in Kvector spaces) of GLd+1(K) as defined by P. Schneider. We prove that the cohomology H(X,VOK ) vanishes for t > 0, for ...

We study the cohomology of the group consisting of all C∞diffeomorphisms of the line, which are C1-flat to the identity at the origin. We construct non-trivial two second real cohomology classes and uncountably many second integral homology classes of this group. 1. Notations and main results We denote by a1 the Lie algebra of all formal vector fields on R with the Krull topology. For k ≥ 0, we...

For a multiplicative cohomology theory E, complex orientations are in bijective correspondence with multiplicative natural transformations to E from complex bordism cohomology MU . If E is represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientation MU → E to a map respecting this extra structure, based on work of Arone–Lesh. The spa...

String theorists (notably Witten [W]) recently introduced the notion of a “quantum” deformation of the cohomology ring of a smooth projective variety X. This quantum deformation, or quantum cohomology ring, as it is often called, is an algebra over a formal-power-series ring which specializes to the ordinary cohomology ring, and which is defined in terms of intersection data (the Gromov-Witten ...

In this article, we introduce equivariant formal deformation theory of Lie triple systems. We an cohomology systems and using study the