Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaf 37). Also available on the Internet.

Hurwitz numbers are a weighted count of degree d ramified covers of curves with specified ramification profiles at marked points on the codomain curve. Isomorphism classes of these covers can be included as a dense open set in a moduli space, called a Hurwitz space. The Hurwitz space has a forgetful morphism to the moduli space of marked, stable curves, and this morphism encodes the Hurwitz numbers. Mikhalkin has constructed a moduli space of tropical marked, stable curves, and this space is a tropical variety. In this paper, I construct a tropical analogue of the Hurwitz space in the sense that it is a connected, polyhedral complex with a morphism to the tropical moduli space of curves such that the degree of the morphism encodes the Hurwitz numbers. / text

We construct classical tilting objects in derived categories of equivariant sheaves on quasi-projective varieties, which give equivalences with derived categories of modules over algebras. Our applications include a conceptual explanation of the importance of the McKay quiver associated to a representation of a finite group G and the development of a McKay correspondence for the cotangent bundle of the projective line. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-09-04 14:42:25.099

In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.

Inaug.-Diss.--Rheinische Friedrich-Wilhelms-Universität, 1992. / Includes bibliographical references (p. 91-92).

We construct a semi-orthogonal decomposition for the equivariant derived category of a hypersurface associated to the sum of two potentials. More specifically, if $f,g$ are two homogeneous poynomials of degree $d$ defining smooth Calabi-Yau or general type hypersurfaces in $\mathbb{P}^n$, we construct a semi-orthogonal decomposition of $D[V(f\oplus g)/\mu_d]$. Moreover, every component of the semi-orthogonal decomposition is explicitly given by Fourier-Mukai functors.

This work is devoted to investigating the behaviour of invariant algebraic curves for the two dimensional Lotka-Volterra systems and examining almost a geometrical approach for finding invariant algebraic surfaces in three dimensional Lotka-Volterra systems. We consider the twenty three cases of invariant algebraic curves found in Ollagnier (2001) of the two dimensional Lotka-Volterra system in the complex plane and then we explain the geometric nature of each curve, especially at the critical points of the mentioned system. We also investigate the local integrability of two dimensional Lotka-Volterra systems at its critical points using the monodromy method which we extend to use the behaviour of some of the invariant algebraic curves mentioned above. Finally, we investigate invariant algebraic surfaces in three dimensional Lotka- Volterra systems by a geometrical method related with the intersection multiplicity of algebraic surfaces with the axes including the lines at infinity. We will classify both linear and quadratic invariant algebraic surfaces under some assumptions and commence a study of the cubic surfaces.

This thesis provides an introduction to the theory of tropical mutation schemes, and computes explicit examples. Tropical mutation schemes generalize toric geometry. The study of toric varieties is a popular area of algebraic geometry, due to toric varieties' strong combinatorial interpretations. In particular, the characters and one-parameter subgroups of the rank $r$ algebraic torus form a pair of dual lattices of rank $r$, isomorphic to $\mathbb{Z}^r$. We can then construct toric varieties from fans in these lattices, and compactifications of the algebraic torus are parametrized by full dimensional convex polytopes. A tropical mutation scheme is a finite collections of lattices, equipped with bijective piecewise-linear functions between each pair of lattices, where these functions satisfy certain compatibility conditions. They generalize lattices in the sense that a lattice can be viewed as the trivial tropical mutation scheme. We also introduce the space of points of a tropical mutation scheme, which is the set of functions from a tropical mutation scheme to $\mathbb{Z}$ which satisfy a minimum condition. A priori, the structure of the space of points of a tropical mutation scheme is unknown, but in certain cases can be identified by the elements of another tropical mutation scheme, inducing a dual pairing between the two tropical mutation schemes. When we have a strict dual pairing of tropical mutation schemes, we can sometimes construct an algebra to be a detropicalization of the pairing. In the trivial case, the coordinate ring of the algebraic torus is a detropicalization of a single lattice and its dual. Thus, when we can construct a detropicalization for a non-trivial strict dual pairing, we recover much of the useful combinatorics from the toric case. This thesis shows that all rank 2 tropical mutation schemes on two lattice charts are autodual, meaning there is a dual pairing between the tropical mutation scheme and its own space of points. Furthermore, we construct a detropicalization for these tropical mutation schemes. We end the thesis by reviewing open questions and future directions for the theory of tropical mutation schemes. / Thesis / Master of Science (MSc) / Informally, algebraic geometry is the study of solution sets to systems of polynomial equations, called algebraic varieties. Such systems are ubiquitous across the sciences, being found as biological models, optimization problems, revenue models, and much more. However, it is a difficult problem in general to ascertain salient properties of the solutions to these systems. One type of algebraic variety which is easier to work with is a toric variety. These varieties can be associated to simpler mathematical objects such as lattices, polytopes and fans, and important geometric properties of the variety can then be obtained via analyzing properties of these simpler objects. This thesis introduces the notion of a tropical mutation scheme, which is a generalization of a lattice. A broader class of algebraic varieties can be associated with tropical mutation schemes in a similar manner to how toric varieties are associated with lattices. We then compute this association explicitly in the case of the simplest non-trivial examples of a tropical mutation scheme, rank 2 tropical mutation schemes with 2 charts.

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