Moduli spaces from a topological point of view

A mason begins a new project by first taking stock of the materials available - bricks and stones come in various shapes and sizes. He or she then must recall the rules governing how the bricks may be assembled and stacked. There is a general principal here: creating a complicated composite object like a house or a tower from elementary pieces like bricks requires that one first answer two questions.1. What are the elementary building blocks available? 2. How are these allowed to be assembled?Although mathematicians work with purely intellectual and abstract objects, often they, like masons and engineers, must build complicated objects from simpler ones. In mathematics, the answers to both of the above questions are often encapsulated in the form of a intricate mathematical object known as a 'moduli space'. It is a geometric object that we can visualise, and its geometric and topological properties -- the curvature and size and shape -- encode a wealth of important information.For example, a mathematical building block could be the solution set to an equation, such as the parabola y = x^2. A moduli space is the object which tells us how things like the solution set of an equation can be extruded and bent and wrapped around to make more complicated objects; it can tell us about the shapes of all possible parabolas at once. Moduli spaces are the universal blueprints describing simultaneously all possible composites which can be assembled from the basic building blocks.Two of the most fundamental types of moduli spaces in mathematics are the `moduli spaces of algebraic curves', and the `moduli spaces of abelian varieties'. These sets of blueprints are created by the intricate machinery of algebraic geometry and they are used in many branches of mathematics, as well as in theoretical physics. The problem is that we cannot fully read them yet. My research program brings the powerful tools of topology to bear on these moduli spaces from algebraic geometry. The tools of one field illuminate the creations of another, and a better understanding of the structure of these moduli spaces could lead to results in many mathematical fields, such as number theory or even theoretical physics. It is an example of the interconnectedness of the mathematical universe. The novelty and advantage of using topological tools here is that topology is designed to organise and filter information; it ignores the the local structure and sees only the underlying global skeletal structure. Focusing attention on only the global structure allows a flexibility of models, and this flexibility can reveal patterns and properties that were previously invisible. This is what led to the proof of the Mumford conjecture by Madsen and Weiss, which described much of the topological structure of the moduli spaces of algebraic curves. I will apply these and other techniques to the moduli spaces of abelian varieties and related spaces.

Research in pure mathematics has had enormous cultural and economic impact in the UK through contributions to the sciences, engineering, finance, and medicine. There are two identifiable potential direct research beneficiaries outside pure mathematics, although both are highly speculative at present. In addition, there are several educational and outreach activities in which I will engage during the fellowship. * Phylogeneticists Phylogenetic statistics has emerging academic, environmental and commercial importance. Certain mathematical objects dealt with in my proposal appear to describe the geometry of the space of phylogenetic trees and thereby affect the efficiency and effectiveness of statistical algorithms. If the possibility emerges that my results can be directly relevant in phylogenetics then I will seek out connections with appropriate academic statisticians, and through them I will make further connections outside of academia where applicable. * DNA and protein structural biochemists Naturally occurring knotting of protein and DNA strands can determine aspects of their biological behaviour within cells. Thus understanding 3-dimensional topology and knot theory can have important applications in biology. Part of my proposal aims to study the space of 3-dimensional handlebodies , which is an important ingredient in knot theory. Results here could potentially contribute to DNA and protein structural biochemistry, and perhaps even the development of new medical techniques. * Impact of the PhD student My fellowship proposal includes a PhD studentship. The impact of the student will be towards maintenance, continuity and renewal of the British mathematical tradition, the topology community in particular, and the British higher education system as a whole. If the student leaves academia then the transferrable skills acquired during the PhD course will be a valuable contribution to the British workforce. People with PhDs in mathematics are highly employable in many areas such as education, computing, finance, consultancy and operations research. * MathCymru Lectures During the fellowship I will continue my involvement with the MathCymru programme (funded by the Welsh Assembly and TechniQuest). This programme sends practising mathematicians in to primary and secondary schools in Wales to give lectures meant to inspire students in their study of maths and other technological subjects. The impacts of this programme include increased university attendance, greater interest in STEM subjects, and benefits to underprivileged Welsh communities. The reduced teaching load afforded me by the fellowship will allow me to maintain a high level of commitment in working with MathCymru. * Resource website for postgraduates I am the current maintainer of the website, `The Moduli of Curves Resource Page' (http://www.aimath.org/WWN/modspacecurves/ ), published as part of the proceedings of an American Institute of Mathematics workshop held in 2004. This website is a brief mathematical encyclopaedia for the study of moduli spaces of curves and related topics. The site is currently used in training postgraduate mathematics students working in this area. During my fellowship tenure I plan to undertake an extensive revision and expansion of this resource. The impact of this activity will be in aiding postgraduate mathematical education in the UK and internationally. * Undergraduate mathematical modelling contest During the period of the fellowship I will continue to organise and coach teams of undergraduates for competition in the COMAP Mathematical Contest in Modeling - a large international mathematical modelling contest conducted through the internet. The skills developed in this activity include technical writing, team-work, presentation, and a range of other important and highly transferrable skills. University graduates with these skills are valuable members of today's workforce.