Abstract:
We design a probabilistic algorithm for computing endomorphism rings of ordinary elliptic curves defined over finite fields that we prove has a subexponential runtime in the size of the base field, assuming solely the generalized Riemann hypothesis. Additionally, we improve the asymptotic complexity of previously known, heuristic, subexponential methods by describing a faster isogeny-computing routine.

Abstract:
Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity analysis rest on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases.

Abstract:
In this paper, we investigate the Quillen model structure defined by Bisson and Tsemo in the category of directed graphs Gph. In particular, we give a precise description of the homotopy category of graphs associated to this model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category Ho(Gph). This enables us to show that Ho(Gph) is equivalent to the category cZSet of periodic Z-sets, and to show that two finite directed graphs are almost-isospectral if and only if they are homotopy-equivalent in our sense.

Abstract:
The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two Freyd-Kelly factorization systems based on Folding, Injecting, and Covering graph morphisms.

Abstract:
We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing endomorphism rings of abelian surfaces over finite fields, and we use our results to extend a completeness result of Murabayashi and Umegaki to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.

Abstract:
The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number.

Abstract:
Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, $C^*$-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which the weak equivalences are those graph morphisms which induce bijections on the set of walks. We determine the resulting homotopy category. We also introduce a "finite-level" homotopy category which respects the natural topology on the set of walks. To each graph we associate a basal graph, well defined up to isomorphism. We show that the basal graph is a homotopy invariant for our model structure, and that it is a finer invariant than the zeta series of a finite graph. We also show that, for finite walkable graphs, if $B$ is basal and separated then the walk spaces for $X$ and $B$ are topologically conjugate if and only if $X$ and $B$ are homotopically equivalent for our model structure.

Abstract:
The jarosite family of minerals contain antiferromagnetically coupled Fe3+ ions that make up the kagome network. This geometric arrangement of the Fe3+ ions causes magnetic frustration that results in exotic electronic ground states, e.g. spin glasses and spin liquids. Synthesic research into jarosites has focused on producing near perfect stoichiometry to eliminate possible magnetic disorder. An new oxidative synthesis method has been devel-oped for the potassium, sodium, rubidium and ammonium jarosites that leads to high Fe coverage. We show through the identification of a meta-stable intermediate, using powder X-ray diffraction, how near perfect Fe coverage arises using this method. Understanding this new mechanism for jarosite formation suggests that is it possible to synthesis hydronium jarosite, an unconventional spin glass, with a very high Fe coverage.

Abstract:
Highly frustrated systems have macroscopically degenerate ground states that lead to novel properties. In magnetism its consequences underpin exotic and technologically important effects, such as, high temperature superconductivity, colossal magnetoresistence, and the anomalous Hall effect. One of the enduring mysteries of frustrated magnetism is why certain experimental systems have a spin glass transition and its exact nature, given that it is not determined by the strength of the dominant magnetic interactions. There have been some suggestions that real systems possess disorder of the magnetic sites or bonds that are responsible. We show that the spin glass transition in the model kagome antiferromagnet hydronium jarosite arises from a spin anisotropy. This weaker energy scale is much smaller than that of the magnetic exchange, yet it is responsible for the energy barriers that are necessary to stabilise a glassy magnetic phase at finite temperature. The resultant glassy phase is quite unlike those found in conventional disordered spin glasses as it is based on complex collective rearrangements of spins called "spin folds". This simplifies hugely theoretical treatment of both the complex dynamics characteristic of a spin glass and the microscopic nature of the spin glass transition itself.