Abstract:
The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces, notably Poincaré and Newton polynomials, and observe various salient features and geometrical patterns. 1. Introduction and Summary The subject of roots of monovariate polynomials is, without doubt, an antiquate one and has germinated an abundance of fruitful research over the ages. It is, therefore, perhaps surprising that any new statements could at all be made regarding such roots. The advent of computer algebra, chaotic phenomena, and random ensembles has, however, indeed shed new light upon so ancient a metier. Polynomials with constrained coefficients and form, though permitted to vary randomly, have constituted a vast field itself. As far back as 1782, Edward Waring, in relation to his famous problem on power summands, had shown that for cubic polynomials with random real coefficients, the ratio of the probability of finding nonreal zeros versus that of not finding non-real zeros is less than or equal to 2. Constraining the coefficients to be integers within a fixed range has, too, its own history. It was realised in [1] that a degree random polynomial with distributed evenly, the expected number of real roots is of order asymptotically in . This was furthered by [2] to be essentially independent of the statistics, in that has the same asymptotics (cf. also [3, 4]), as much for being evenly distributed real numbers, in , or as Gaussian distributed in . Continual development ensued (q.v. also [5]), notably by Littlewood [6], Erd？s and Turán [7], Hammersley [8], and Kac [9]. Indeed, a polynomial with coefficients only taking values as has come to be known as a Littlewood polynomial, and the Littlewood Problem asks for the the precise asymptotics, in the degree, of such polynomials taking values, with complex arguments, on the unit circle. The classic work of Montgomery [10] and Odlyzko [11], constituting one of the most famous computer experiments in mathematics (q. v. Section？？ 3.1 of [12] for some recent remarks on the distributions), empirically showed that the distribution of the (normalized) spacings between successive critical zeros of the Riemann zeta function is the same as that of a Gaussian unitary ensemble of random matrices, whereby infusing our subject with issues of uttermost importance. Subsequently, combining the investigation of zeros

Abstract:
Along the recently trodden path of studying certain number theoretic properties of gauge theories, especially supersymmetric theories whose vacuum manifolds are non-trivial, we investigate Ihara's Graph Zeta Function for large classes of quiver theories and periodic tilings by bi-partite graphs. In particular, we examine issues such as the spectra of the adjacency and whether the gauge theory satisfies the strong and weak versions of the graph theoretical analogue of the Riemann Hypothesis.

Abstract:
We present, in explicit matrix representation and a modernity befitting the community, the classification of the finite discrete subgroups of G_2 and compute the McKay quivers arising therefrom. Of physical interest are the classes of N=1 gauge theories descending from M-theory and of mathematical interest are possible steps toward a systematic study of crepant resolutions to smooth G_2 manifolds as well as generalised McKay Correspondences. This writing is a companion monograph to hep-th/9811183 and hep-th/9905212, wherein the analogues for Calabi-Yau three- and four-folds were considered.

Abstract:
We study closed string tachyon condensation on general non-supersymmetric orbifolds of C^2. Extending previous analyses on Abelian cases, we present the classification of quotients by discrete finite subgroups of GL(2; C) as well as the generalised Hirzebruch-Jung continued fractions associated with the resolution data. Furthermore, we discuss the intimate connexions with certain generalised versions of the McKay Correspondence.

Abstract:
In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student.

Abstract:
We investigate certain arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic. Parallel to the Plethystic Programme of counting the spectrum of operators from the syzygies of the complex geometry, we construct, based on the zeros of the vacuum moduli space over finite fields, the local and global Hasse-Weil zeta functions, as well as develop the associated Dirichlet expansions. We find curious dualities wherein the geometrical properties and asymptotic behaviour of one gauge theory is governed by the number theoretic nature of another.

Abstract:
With a bird's-eye view, we survey the landscape of Calabi-Yau threefolds, compact and non-compact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have been useful in the interaction between the physics and the mathematics, especially in string and gauge theories. A skein which runs through this review will be algorithmic and computational algebraic geometry and how, implementing its principles on powerful computers and experimenting with the vast mathematical data, new physics can be learnt. It is hoped that this inter-disciplinary glimpse will be of some use to the beginning student.

Abstract:
D-brane probes, Hanany-Witten setups and geometrical engineering stand as a trichotomy of the currently fashionable techniques of constructing gauge theories from string theory. Meanwhile, asymptotic freedom, finitude and IR freedom pose as a trichotomy of the beta-function behaviour in quantum field theories. Parallel thereto is a trichotomy in set theory of finite, tame and wild representation types. At the intersection of the above lies the theory of quivers. We briefly review some of the terminology standard to the physics and to the mathematics. Then we utilise certain results from graph theory and axiomatic representation theory of path algebras to address physical issues such as the implication of graph additivity to finiteness of gauge theories, the impossibility of constructing completely IR free string orbifold theories and the unclassifiability of N<2 Yang-Mills theories in four dimensions.

Abstract:
These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of D-brane gauge theories constructed from local Calabi-Yau spaces. Topics such as orbifolds, toric singularities, del Pezzo surfaces as well as chaotic duality will be covered.

Abstract:
We present the complete classification of smooth toric Fano threefolds, known to the algebraic geometry literature, and perform some preliminary analyses in the context of brane tilings and Chern-Simons theory on M2-branes probing Calabi-Yau fourfold singularities. We emphasise that these 18 spaces should be as intensely studied as their well-known counterparts: the del Pezzo surfaces. 1. Introduction A flurry of activity has, since the initial work of Bagger and Lambert [1–3] and Gustavsson [4], rather excited the community for the past two years upon the subject of supersymmetric Chern-Simons theories. It is by now widely believed that the world-volume theory of M2-branes on various backgrounds is given by a -dimensional quiver Chern-Simons (QCS) theory [5–26], most conveniently described by a brane tiling. Even though analogies with the case of D3-branes in Type IIB, whose world-volume theory is a -dimensional supersymmetric quiver gauge theory, are very reassuring, the story is much less understood for the M2 case. Much work has been devoted to the understanding of issues such as orbifolding, phases of duality, brane tilings, and dimer/crystal models and so forth. Nevertheless, the role played by the correspondence between the world-volume theory and the underlying Calabi-Yau geometry is of indubitable importance. Indeed, there is a bijection: the vacuum moduli space of the former is, tautologically, the latter, while the geometrical engineering on the latter gives, by construction, the former. This bijection, called, respectively, the “forward” and “inverse” algorithms [27, 28], persists in any dimension and can be succinctly summarised in Table 1. Table 1: Brane probes and associated world-volume physics in various backgrounds. A crucial feature for all the brane embeddings in Table 1 is that in the toric case they are all described by brane tilings. The first case, with CY2, is described by one-dimensional tilings, that is, brane intervals and thus brane constructions following the work in [29]. The second case is the well-established two-dimensional brane tilings which use dimer techniques to study supersymmetric gauge theories [30–32]. The third case is the newly proposed construction [13] of Chern-Simon theories. It is perhaps na？vely natural to propose three-dimensional tilings for the case of M2-branes probing CY4, but in fact, it turns out not to be as useful as it may initially seem. These three-dimensional tilings have been nicely advocated in the crystal model [33, 34]. The main issue perhaps is the current shortcoming of this model to