Title:  Spin Cobordism and Wedge Quasitoric Manifolds

Abstract:  Quasitoric manifolds are smooth 2n-manifolds admitting a "nice" action of the compact n-torus so that the quotient of this action yields a (combinatorially) simple polytope.  They are a generalization of smooth projective toric variaties and much is known about these manifolds in terms of complex cobordism theory.  In fact they were used by Buchstaber and Panov to show that every cobordism complex class contains a (connected) quasitoric manifold.

Far less is known about spin quasitoric manifolds and spin cobordism which requires the calculation of KO-characteristic classes.  We consider a procedure developed to investigate topological data for spin quasitoric manfolds which utilizes a wedge polytope operation on the quotient polytope.  We'll discuss a list of results concerning these "wedge" quasitoric manifolds, including such topics as Bott manifolds, the connected sum, the Todd genus and lastly, specific criteria in terms of combinatorial data allowing for the calculation of KO-characteristic classes of spin quasitoric manifolds.

Title:  Subfunctors of Extension Functors

Abstract:  In this talk we examine subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enevloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalo and Enochs and has proven to be beneficial for module theory as well as for representation theory. First we will focus on subfunctors of Ext and their properties. We show how the class of precoverings give us subfunctors of Ext. Later, we investigate the sunfunctor of Hom called ideals. The definition of cover and envelope carry over to the ideals naturally. Classical conditions for existence theorems for covers led to similar approaches in the ideal case. Even though some theorems such as Salce's Lemma were proven to extend  to ideals, most of the theorems do not directly apply to the new case. We show how Eklof-Trlifaj's result can partially be extended to the ideals generated by a set. Moreover by relating the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in A_2 we obtain a sufficient condition for the existence of covering ideals in a more general setting and finish with applying this result to the class of phantom morphisms.

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.