| EPSRC Reference: |
EP/N029828/1 |
| Title: |
Universal Moduli of Bundles on Curves |
| Principal Investigator: |
Martens, Dr J |
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| Department: |
Sch of Mathematics |
| Organisation: |
University of Edinburgh |
| Scheme: |
First Grant - Revised 2009 |
| Starts: |
01 September 2016 |
Ends: |
31 August 2018 |
Value (£): |
99,485
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
Over the last few decades the concept of a moduli space has become central in modern geometry. Roughly speaking, a moduli space is a geometric space that parameterizes instances of various kinds of objects in geometry, algebra, and physics. Somewhat surprisingly, the geometry of these moduli spaces themselves can be very rich, and encode the answers to all kinds of questions one can ask about the original objects.
Perhaps one of the most venerable moduli spaces, but still a central topic of study in modern mathematics, is Riemann's moduli space that parameterizes Riemann surfaces, also known as smooth complex algebraic curves. Riemann surfaces are two-dimensional surfaces that have a notion of angle, but not of length. Mathematician's caught the first glimpses of this moduli space in the 19th century, but it took almost a hundred years to place its construction on sound footing. Typical for such an important object is that it can be understood from a number of different viewpoints: in algebraic geometry, topology or differential geometry.
An unfortunate aspect of the naïve approach to the moduli space is that it yields a space that is not compact, which means that one can "run out of it". This defect was resolved however in a very beautiful way by Deligne and Mumford in the late 1960s: one can compactify (or complete) this space by generalizing the kind of objects a little bit, in particular by allowing the Riemann surfaces to develop certain mild singularities. Not only does this overcome the non-compactness, but the resulting larger moduli space is extremely well-behaved in all sorts of ways.
Another classical moduli space is the Jacobian variety of a fixed Riemann surface - which in modern parlance parameterizes line bundles on that Riemann surface. A special kind of abelian variety, it is the home of the Theta-functions, central objects in number theory and string theory alike. A natural generalization of the Jacobian variety seeks to classify non-abelian analogues of line bundles, a question that is strongly motivated by gauge theories such as the Standard Model in theoretical particle physics.
This proposal centers around the following basic question: what happens to these moduli spaces of non-abelian bundles when one allows the Riemann surface to degenerate, as in the Deligne-Mumford compactification? This is a very natural thing to ask, which has a surprising number of links with representation theory, quantum mechanics and conformal field theory. To be more precise, one would like to construct a modular compactification of the "universal" moduli space of bundles on smooth curves, where both are allowed to move simultaneously, that behaves in a similarly nice way as the Deligne-Mumford compactification.
We are proposing a new line of attack on this problem, based on recent developments initiated by the PI and others in the last few years, and aim to investigate its consequences and links with other disciplines.
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Potential use in non-academic contexts |
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.ed.ac.uk |