Title:  Homological Algebra with Filtered Module

Abstract:  Classical homological algebra begins with the study of projective and injective modules.  In this talk I will discuss analogous notions of projectivity and injectivity in a category of filtered modules.  In particular, projective and injective objects with respect to the restricted class of strict morphisms are defined and characterized.  Additionally, an analogue to the injective envelope is discussed with examples showing how this differs from the usual notion of an injective envelope.

A long-standing problem in cobordism theory has been to find convenient manifolds to represent cobordism classes.  For example, Hirzebruch asked which complex cobordism classes can be represented by smooth connected algebraic varieties in the late 1950’s.  In this talk, I will describe a toric version of this question.  After a brief introduction to toric varieties, I will discuss certain combinatorial obstructions to a complex cobordism class containing a smooth projective toric variety.  Up to dimension six, I will completely describe the cobordism classes containing such varieties.  In addition, the role of toric varieties in the polynomial ring structure of complex cobordism will be examined.  More specifically, I will construct smooth projective toric varieties as polynomial ring generators in most dimensions.  I will also present overwhelming evidence suggesting that a smooth projective toric variety generator exists in every dimension.