Let G be a simple graph on n vertices. LetH be either the complete graph Km or the complete bipartite graph Kr,s on a subset of the vertices in G. We show that G contains H as a subgraph if and only if βi,α(H) ≤ βi,α(G) for all i ≥ 0 and α ∈ Z. In fact, it suffices to consider only the first syzygy module. In particular, we prove that β1,α(H) ≤ β1,α(G) for all α ∈ Z if and only if G contains a ...

In this dissertation, we are concerned with decompositions of Betti diagrams over standard graded rings and the information about that ring and its modules that can be recovered from these decompositions. In Chapter 2, we study the structure of modules over short Gorenstein graded rings and determine a necessary condition for a matrix of nonnegative integers to be the Betti diagram of such a mo...

We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and Kodaira dimension at same time. Using give novel constructions of Calabi–Yau surfaces from complex general type completely resolve conjecture Stipsicz on existence exceptional sections in Lefschetz fibrations. Combining unchaining with others, all correspond to certain monodromy sub...

Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can be the kth Betti number bk(G) = rank Hk(G) providing that G has length ≤ N and bk(G) is finite? We prove that for every k ≥ 3 the maximum bk(N) of kth Betti numbers of all such groups is an extremely rapidly growing function of N . It grows faster that all f...

The conjectures of M. Boij and J. Söderberg [3], proven by D. Eisenbud and F.-O. Schreyer [8] (see also [7, 4]), link the extremal properties of invariants of graded free resolutions of finitely generated modules over the polynomial ring S = k[x1, . . . , xn] with the Herzog–Huneke–Srinivasan Multiplicity Conjectures. Here k is any field and S has the standard Z-grading. In the course of their ...

Let X be a finite connected CW -complex. Suppose that its fundamental group π is residually finite, i.e., there is a nested sequence . . . ⊂ Γm+1 ⊂ Γm ⊂ . . . ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that the p-th L2-Betti number of X is the limit of the sequence bp(Xm)/[π : Γm] where bp(Xm) is the (ordinary) p-th Betti number of the finite coveri...

Polygon spaces such as M = {(u1, . . . , un) ∈ S1 × . . . S1,∑ni=1 liui = 0}/SO(2), or the three-dimensional analogs N play an important rôle in geometry and topology, and are also of interest in robotics where the li model the lengths of robot arms. When n is large, one can assume that each li is a positive real valued random variable, leading to a randommanifold. The complexity of such manifo...

The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The h′vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the h′-vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In t...

This paper generalizes work of Lascoux and Jo zeeak-Pragacz-Weyman computing the characteristic zero Betti numbers in minimal free resolutions of ideals generated by 2 2 minors of generic matrices and generic symmetric matrices, respectively. In the case of 2 2 minors, the quotients of certain polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we co...

The k-parabolic subspace arrangement, introduced by Barcelo, Severs and White, is a generalization of the well known k-equal arrangements of type-A and type-B. In this paper we use the discrete Morse theory of Forman to study the homology of the complements of k-parabolic subspace arrangements. In doing so, we recover some known results of Björner et al. and provide a combinatorial interpretati...