Free Complexes over the Exterior Algebra with Small Homology

Let M be a graded module over a standard graded polynomial ring S. The Total Rank Conjecture by Avramov-Buchweitz predicts the total Betti number of M should be at least the total Betti number of the residue field. Walker proved this is indeed true in a large number of cases. One could then try to push this result further by generalizing this conjecture to finite free complexes which is known as the Generalized Total Rank Conjecture. However, Iyengar and Walker constructed examples to show this generalized conjecture is not always true. In this thesis, we investigate other counterexamples of the Generalized Total Rank Conjecture and some of their properties. Under the BGG correspondence, a finite free graded complex over the exterior algebra with small homology corresponds to a free complex over the polynomial ring with a small total Betti number. Therefore, we focus on examples of finite free complexes over the exterior algebra with small homology. The main examples we consider are Koszul complexes of quadrics, and we show the Koszul complex of one general quadric and the Koszul complex of two general quadrics have the smallest possible homology among complexes over the exterior algebra with the same graded Poincaré series. Finally while analyzing these Koszul complexes, we notice the dimension of their total homology has a nice asymptotic behavior and investigate under what conditions other complexes have this same asymptotic behavior. Adviser: Alexandra Seceleanu and Mark E. Walke

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page