In 1991, Davis and Januszkiewicz opened up the study of toric varities to algebraic topologists and homotopy theorists by proposing the related notion of TORIC MANIFOLDS. These admit particularly well-structured actions by high dimensional tori, and investigation of their homological properties creates a beautiful interplay between geometry, combinatorics, and commutative ring theory. The original definition led implicitly to the existence of stable complex structures on toric manifolds, and therefore to an anologue of Hirzebruch's famous question about algebraic varieties; is it possible to represent every complex cobordism class by a (necessarily connected) toric manifold? In work with Victor Buchstaber during 1997 we began the study of this problem, and we have recently obtained a fully affirmative answer. I propose to offer an audience-friendly outline of the constructions, concentrating on geometric and combinatorial issues. In particular, I would like to emphasise some endangered aspects of stable complex structure theory, and to mention an interesting notion of connected sum for appropriate simple polytopes.

Broué's Abelian Defect Group Conjecture predicts that in many situations of interest in modular representation theory, the module categories of related blocks (i.e., direct factors of the group algebras) A and B of group algebras should have equivalent derived categories. Except in very special cases, the origin of this equivalence is still mysterious, and so it is of interest to try to verify the conjecture in examples.

The method of doing this that has been used most often is to construct a "tilting complex": this is the object in the derived category of B that should correspond to the free module for A. The definition of a tilting complex is a just a translation of some categorical properties of the free module, and if there is such a complex, it is known that there is an equivalence of derived categories.

In all but the simplest examples, this is a difficult procedure to carry out, because the required tilting complex will have a very complicated structure.

In this talk I will describe a new method of constructing equivalences. This involves finding the objects of the derived category of B that correspond to the simple A-modules. These objects are typically much less complicated than the tilting complex, and the tilting complex can be constructed from these objects by a procedure somewhat reminiscent of Bousfield localization.

The chromatic splitting conjecture describes L_{n-1} L_{K (n)} X (which we abbreviate by writing R (X)) for X finite, in terms of localizations of lower periodicity. We consider X of different chromatic `type;' (recall X of type n means K(n)_*(X) is not 0 but K(n-1)_*(X) is 0). For X of type n or higher, R (X) is contractible, so the simplest interesting case is when X is of type n-1.

Following a suggestion of Mike Hopkins, we approach this conjecture by considering a spectral sequence derived from taking a BP-Adams resolution of X and then smashing with an inverse system that produces L_{K (n)}X. This is similar to (but simpler than) techniques used by Goerss in his paper on the homology of inverse limits. The E_2 term of the spectral sequence involves derived functors of inverse limits in the category of comodules over BP_* BP, and the spectral sequence converges to BP_* L_{K (n)}X. If X is of type n-1, inverting v_{n-1} computes BP_* R (X). In generic situations (p large with respect to n) we can then deduce that L_{n-1}X splits off of R (X).

We can draw some conclusions about homotopy groups when X is not of type n-1, and we can also use these techniques to gain information about the Tate homology of L_{K (n)}X.

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