An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. We are interested in the topology of the complement of an arrangement. If the arrangement is unimodular, we provide a combinatorial presentation for a differential graded algebra (DGA) that is a model for the complement, in the sense of rational homotopy theory. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers. If the arrangement is chordal, then this model is a Koszul algebra. In this case, studying its quadratic dual gives a combinatorial description of the Q-nilpotent completion of the fundamental group and the minimal model of the complement of the arrangement. This dissertation includes previously unpublished co-authored material.

This dissertation uses methods from homological algebra and computational commutative algebra to study four problems. We use Hilbert function computations and classical homology theory and combinatorics to answer questions with a more applied mathematics content: splines approximation, hyperplane arrangements, configuration spaces and coding theory. In Chapter II we study a problem in approximation theory. Alfeld and Schumaker give a formula for the dimension of the space of piecewise polynomial functions (splines) of degree d and smoothness r. Schenck and Stiller conjectured that this formula holds for all d 2r + 1. In this chapter we show that there exists a simplicial complex such that for any r, the dimension of the spline space in degree d = 2r is not given by this formula. Chapter III is dedicated to formal hyperplane arrangements. This notion was introduced by Falk and Randell and generalized to formality by Brandt and Terao. In this chapter we prove a criteria for formal arrangements, using a complex constructed from vector spaces introduced by Brandt and Terao. As an application, we give a simple description of formality of graphic arrangements in terms of the homology of the flag complex of the graph. Chapter IV approaches the problem of studying configuration of smooth rational curves in P2. Since an irreducible conic in P2 is a P1 (so a line) it is natural to ask if classical results about line arrangements in P2, such as addition-deletion type theorem, Yoshinaga criterion or Terao's conjecture verify for such configurations. In this chapter we answer these questions. The addition-deletion theorem that we find takes in consideration the fine local geometry of singularities. The results of this chapter are joint work with H. Schenck. In Chapter V we study a problem in algebraic coding theory. Gold, Little and Schenck find a lower bound for the minimal distance of a complete intersection evaluation codes. Since complete intersections are Gorenstein, we show a similar bound for the minimal distance depending on the socle degree of the reduced zero-dimensional Gorenstein scheme. The results of this chapter are a work in progress.

We are studying the Tutte Polynomial of hyperplane arrangements. We discuss some previous work done to compute these polynomials. Then we explain our method to calculate the Tutte Polynomial of some arrangements more efficiently. We next discuss the details of the program used to do the calculation. We use this program and present the actual Tutte Polynomials calculated for the arrangements E6, E7, and E8. / text

A recent result of Salvetti and Settepanella gives, for a complexified real arrangement, an explicit description of a minimal CW decomposition as well as an explicit algebraic complex which computes local system homology. We apply their techniques to discriminantal arrangements in two dimensional complex space and calculate the boundary maps which will give local system homology groups given any choice of local system. This calculation generalizes several known results; examples are given related to Milnor fibrations, solutions of KZ equations, and the LKB representation of the braid group. Another algebraic object associated to a hyperplane arrangement is the module of derivations. We analyze the behavior of the derivation module for an affine arrangement over an infinite field and relate its derivation module to that of its cone. In the case of an arrangement fibration, we analyze the relationship between the derivation module of the total space arrangement and those of the base and fiber arrangements. In particular, subject to certain restrictions, we establish freeness of the total space arrangement given freeness of the base and fiber arrangements.

<p> This thesis deals with combinatorial properties of hyperplane arrangements. In particular, we address a conjecture of Deza, Terlaky and Zinchenko stating that the largest possible average diameter of a bounded cell of a simple hyperplane arrangement is not greater than the dimension. We prove that this conjecture is asymptotically tight in fixed dimension by constructing a family of hyperplane arrangements containing mostly cubical cells. The relationship with a result of Dedieu, Malajovich and Shub, the conjecture of Hirsch, and a result of Haimovich are presented.</p> <p> We give the exact value of the largest possible average diameter for all simple arrangements in dimension two, for arrangements having at most the dimension plus two hyperplanes, and for arrangements having six hyperplanes in dimension three. In dimension three, we strengthen the lower and upper bounds for the largest possible average diameter of a bounded cell of a simple hyperplane arrangements.</p> <p> Namely, let ΔA(n, d) denote the largest possible average diameter of a bounded cell of a simple arrangement defined by n hyperplanes in dimension d. We show that • ΔA(n, 2) = 2[n/2] / (n-1)(n-2) for n ≥ 3, • ΔA(d + 2, d) = 2d/d+1, • ΔA(6, 3) = 2, • 3 - 6/n-1 + 6([n/2]-2) / (n-1)(n-2)(n-3) ≤ ΔA(n, 3) ≤ 3 + 4(2n^2-16n+21) / 3(n-1)(n-2)(n-3) • ΔA (n, d) ≥ 1 + (d-1)(n-d d)+(n-d)(n-d-1) for n ≥ 2d. We also address another conjecture of Deza, Terlaky and Zinchenko stating that the minimum number Φ0A~(n, d) of facets belonging to exactly one bounded cell of a simple arrangement defined by n hyperplanes in dimension d is at least d (n-2 d-1). We show that • Φ0A(n, 2) = 2(n - 1) for n ≥ 4, • Φ0A~(n, 3) ≥ n(n-2)/3 +2 for n ≥ 5. We present theoretical frameworks, including oriented matroids, and computational tools to check by complete enumeration the open conjectures for small instances. Preliminary computational results are given.</p> / Thesis / Master of Science (MSc)

In this dissertation, a method for producing multinets from a net in P^3 is presented. Multinets play an important role in the study of resonance varieties of the complement of a complex hyperplane arrangement and very few examples are known. Implementing this method, numerous new and interesting examples of multinets are identified. These examples provide additional evidence supporting the conjecture of Pereira and Yuzvinsky that all multinets are degenerations of nets. Also, a complete description is given of proper weak multinets, a generalization of multinets.

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev completion of its fundamental group G and the cohomology groups of X with coefficients in rank one local systems. In this thesis, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. The relative completion S_p is a prosolvable group that generalizes the classical Malcev completion; when p is the trivial representation, S_p is the Malcev completion of G. The group S_p is tightly controlled by the cohomology groups H^1(X,L_{p^k}) with coefficients in the irreducible local systems L_{p^k} associated to the representation p.The pronilpotent Lie algebra u_p of the prounipotent radical U_p of S_p has been described by Hain. If p is the trivial representation, then u_p is the holonomy Lie algebra, which is well-known to be quadratically presented. In contrast, we show that when X is the complement of the braid arrangement in complex two-space, there are infinitely many representations p from G into (C^*)^2 for which u_p is not quadratically presented.We show that if Y is a subtorus of the character torus T containing the trivial character, then S_p is combinatorially determined for general p in Y. We do not know whether S_p is always combinatorially determined. If S_p is combinatorially determined for all characters p of G, then the characteristic varieties of the arrangement X are combinatorially determined.When Y is an irreducible subvariety of T^N, we examine the behavior of S_p as p varies in Y. We define an affine group scheme S_Y over Y such that if Y = {p}, then S_Y is the relative Malcev completion S_p. For each p in Y, there is a canonical homomorphism of affine group schemes from S_p into the affine group scheme which is the restriction of S_Y to p. This is often an isomorphism. For example, if there exists p in Y whose image is Zariski dense in G_m^N, then this homomorphism is an isomorphism for general p in Y. / Dissertation

Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e. the connected components of the maximal induced subgraphs of G on which an eigenvector ψ does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of ψ in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures.

No description available.

We study a multivariate version of the Bernstein–Sato polynomial, the so-called Bernstein–Sato ideal, associated to an arbitrary factorization of an analytic germ <i>f - f</i>₁···<i>f</i>_r. We identify a large class of geometrically characterized germs so that the <i>D</i>_X,x[<i>s</i>₁,...,<i>s</i>_r]-annihilator of <i>f</i>^s₁¹···<i>f</i>^s_r^r admits the simplest possible description and, more-over, has a particularly nice associated graded object. As a consequence we are able to verify Budur’s Topological Multivariable Strong Monodromy Conjecture for arbitrary factorizations of tame hyperplane arrangements by showing the zero locus of the associated Bernstein–Sato ideal contains a special hyperplane. By developing ideas of Maisonobe and Narvaez-Macarro, we are able to find many more hyperplanes contained in the zero locus of this Bernstein–Sato ideal. As an example, for reduced, tame hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial contained in [−1,0) are combinatorially determined; for reduced, free hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial are all combinatorially determined. Finally, outside the hyperplane arrangement setting, we prove many results about a certain <i>D</i>_X,x-map ∇_<i>A</i> that is expected to characterize the roots of the Bernstein–Sato ideal.