This thesis is concerned with the study of the geometry and derived categories associated to the moduli problems of local systems and Higgs bundles in positive characteristic. As a cornerstone of our investigation, we establish a local system analogue of the BNR correspondence for Higgs bundles. This result (Proposition 4.3.1) relates flat connections to certain modules of an Azumaya algebra on the family of spectral curves. We prove properness over the semistable locus of the Hitchin map for local systems introduced by Laszlo–Pauly (Theorem 4.4.1). Moreover, we show that with respect to this Hitchin map, the moduli stack of local systems is étale locally equivalent to the moduli stack of Higgs bundles (Theorem 4.6.3) (with or without stability conditions). Subsequently, we study two-dimensional examples of moduli spaces of parabolic Higgs bundles and local systems (Theorem 5.2.1), given by equivariant Hilbert schemes of cotangent bundles of elliptic curves. Furthermore, the Hilbert schemes of points of these surfaces are equivalent to moduli spaces of parabolic Higgs bundles, respectively local systems (Theorem 5.3.1). The proof for local systems in positive characteristic relies on the properness results for the Hitchin fibration established earlier. The Autoduality Conjecture of Donagi–Pantev follows from Bridgeland–King–Reid’s McKay equivalence in these examples. The last chapter of this thesis is concerned with the con- struction of derived equivalences, resembling a Geometric Langlands Correspondence in positive characteristic, generalizing work of Bezrukavnikov–Braverman. Away from finitely many primes, we show that over the locus of integral spectral curves, the derived category of coherent sheaves on the stack of local systems is equivalent to a derived category of coherent D-modules on the stack of vector bundles. We conclude by establishing the Hecke eigenproperty of Arinkin’s autoduality and thereby of the Geometric Langlands equivalence in positive characteristic.

This thesis is concerned with Calabi-Yau threefolds and vector bundles upon them, which are the basic mathematical objects at the centre of smooth supersymmetric compactifications of heterotic string theory. We begin by explaining how these objects arise in physics, and give a brief review of the techniques of algebraic geometry which are used to construct and study them. We then turn to studying multiply-connected Calabi-Yau threefolds, which are of particular importance for realistic string compactifications. We construct a large number of new examples via free group actions on complete intersection Calabi-Yau manifolds (CICY's). For special values of the parameters, these group actions develop fixed points, and we show that, on the quotient spaces, this leads to a particular class of singularities, which are quotients of the conifold. We demonstrate that, in many cases at least, such a singularity can be resolved to yield another smooth Calabi-Yau threefold, with different Hodge numbers and fundamental group. This is a new example of the interconnectedness of the moduli spaces of distinct Calabi-Yau threefolds. In the second part of the thesis we turn to a study of two new `three-generation' manifolds, constructed as quotients of a particular CICY, which can also be represented as a hypersurface in dP6 x dP6, where dP6 is the del Pezzo surface of degree six. After describing the geometry of this manifold, and especially its non-Abelian quotient, in detail, we show how to construct on the quotient manifolds vector bundles which lead to four-dimensional heterotic models with the standard model gauge group and three generations of particles. The example described in detail has the spectrum of the minimal supersymmetric standard model plus a single vector-like pair of colour triplets.

A suitable subcategory of affine Azumaya algebras is defined and a functor from this category to the category of Zariski structures is constructed. The rudiments of a theory of presheaves of topological structures is developed and applied to construct examples of structures at a generic parameter. The category of equivariant algebras is defined and a first-order theory is associated to each object. For those theories satisfying a certain technical condition, uncountable categoricity and quantifier elimination results are established. Models are shown to be Zariski structures and a functor from the category of equivariant algebras to Zariski structures is constructed. The two functors obtained in the thesis are shown to agree on a nontrivial class of algebras.

In the setting of tropical mathematics, geometric objects are rich with inherent combinatorial structure. For example, each polynomial $p(x,y)$ in the tropical setting corresponds to a tropical curve; these tropical curves correspond to unbounded graphs embedded in $\R^2$. Each of these graphs is dual to a particular subdivision of its Newton polytope; we classify tropical curves by combinatorial type based on these corresponding subdivisions. In this thesis, we aim to gain an understanding of the likeliness of the combinatorial type of a randomly chosen tropical curve by using methods from polytope geometry. We focus on tropical curves corresponding to quadratics, but we hope to expand our exploration to higher degree polynomials.

Wang Yi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Some Preparations --- p.10 / Chapter 3 --- Proof of Theorem 1.1 --- p.24 / Chapter 4 --- Location of Blow-up Points (for n=2) --- p.26 / Chapter 5 --- Location of Blow-up Points (for General n) --- p.35 / Chapter 6 --- Asymptotic behavior of solutions near blow-up point --- p.46 / Chapter 7 --- Appendix --- p.57 / Bibliography --- p.61

Since their conception by Collins in 1975, Cylindrical Algebraic Decompositions (CADs) have been used to analyse the real algebraic geometry of systems of polynomials. Applications for CAD technology range from quantifier elimination to robot motion planning. Although of great use in practice, the CAD algorithm was shown to have doubly exponential complexity with respect to the number of variables for the problem, which limits its use for large examples. Due to the high complexity of CAD, much work has been done to improve its performance. In this thesis new advances will be discussed that improve the practical efficiency of CAD for a variety of problems, with a new complexity result for one set of algorithms. A new invariance condition, truth table invariance (TTICAD), and two algorithms to construct TTICADs are given and shown to be highly efficient. The idea of restricting the output of CADs, allowing for greater efficiency, is formalised as sub-decompositions and two particular ideas are investigated in depth. Efficient selection of various formulation choices for a CAD problem are discussed, with a collection of heuristics investigated and machine learning applied to assist in choosing an optimal heuristic. The mathematical expression of a problem is shown to be of great importance, with preconditioning and reformulation investigated. Finally, these advances are collected together in a general framework for applying CAD in an efficient manner to a given problem. It is shown that their combination is not cumulative and care must be taken. To this end, a prototype software CADassistant is described to help users take advantage of the advances without knowledge of the underlying theory. The effects of the various advances are demonstrated through a guiding example originally considered by Solotareff, which describes the approximation of a cubic polynomial by a linear function. Naïvely applying CAD to the problem takes 916.1 seconds of construction (from which a solution can easily be derived), which is reduced to 20.1 seconds by combining various advances from this thesis.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Includes bibliographical references. / Ph.D.

A polyhedron P is a subset of a rational vector space V bounded by hyperplanes. If we fix a lattice in V , then we may consider the exponential integral and sum, two meromorphic functions on the dual vector space which serve to generalize the notion of volume of and number of lattice points contained in P, respectively. In 2007, Berline and Vergne constructed an Euler-Maclaurin formula that relates the exponential sum of a given polyhedron to the exponential integral of each face. This formula was "local", meaning that the coefficients in this formula had certain properties independent of the given polyhedron. In this dissertation, the author finds a new construction for this formula which is very different from that of Berline and Vergne. We may 'perturb' any polyhedron by tranlsating its bounding hyperplanes. The author defines a ring of differential operators R(P) on the exponential volume of the perturbed polyhedron. This definition is inspired by methods in the theory of toric varieties, although no knowledge of toric varieties is necessary to understand the construction or the resulting Euler-Maclaurin formula. Each polyhedron corresponds to a toric variety, and there is a dictionary between combinatorial properties of the polyhedron and algebro-geometric properties of this variety. In particular, the equivariant cohomology ring and the group of equivariant algebraic cycles on the corresponding toric variety are equal to a quotient ring and subgroup of R(P), respectively. Given an inner product (or, more generally, a complement map) on V , there is a canonical section of the equivariant cohomology ring into the group of algebraic cycles. One can use the image under this section of a particular differential operator called the Todd class to define the Euler-Maclaurin formula. The author shows that this formula satisfies the same properties which characterize the Berline-Vergne formula.

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to the receptive fields of place cells. Lastly we describe several purely geometric results related to neural codes.

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be linear fractional transformations when in $U^2$ and $B^2$. We will then study a bit of topology, specifically the topological groups relevant to the group of isometries of hyperbolic $n$-space, $I(H^n)$. Finally we will combine what we know about hyperbolic $n$-space and topological groups in order to study fundamental regions, fundamental domains, Dirichlet domains, and quotient spaces. Using examples in $U^2$, we will then illustrate how useful fundamental domains are when it comes to visualizing the geometry of quotient spaces.