# EP/I003908/2 - Moduli spaces from a topological point of view

Research Perspectives grant details from EPSRC portfolio

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Research Perspectives grant details from EPSRC portfolio

Principal Investigator - College of Science, Swansea University

Start Date

10/2011

End Date

09/2015

Value

£362,707

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Summary and Description of the grant

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.

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Grant Event Details:

Name: Moduli spaces from a topological point of view - EP/I003908/2

Start Date: 2011-10-01T00:00:00+00:00

End Date: 2015-09-30T00:00:00+00:00

Organization: Swansea University

Description: 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 her ...