| EPSRC Reference: |
EP/L018772/1 |
| Title: |
"Algebraic Structures in Local Mirror Symmetry" |
| Principal Investigator: |
Pomerleano, Dr D |
| Other Investigators: |
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| Department: |
Mathematics |
| Organisation: |
Imperial College London |
| Scheme: |
EPSRC Fellowship |
| Starts: |
01 October 2014 |
Ends: |
30 September 2017 |
Value (£): |
236,950
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| EPSRC Research Topic Classifications: |
| Algebra & Geometry |
Mathematical Physics |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
My proposed research is interdisciplinary between two area of pure mathematics known as geometry and algebra and is inspired by mathematical physics. My research actually concerns two types of geometry, which are on the surface, quite different. Symplectic geometry is the study of Hamiltonian systems. Algebraic geometry is the study of polynomial equations. Mirror symmetry is a duality between symplectic geometry and algebraic geometry, which was discovered by string theorists. This duality, while very natural from a string theory point of view, is a rich mathematical puzzle--- on the face of it, there is simply no mathematical explanation for this duality.
Nevertheless, mathematicians have made some progress in understanding the physicists' predictions. Classical mirror symmetry is principally concerned with "compact manifolds," small geometries which are meant to represent hidden microscopic dimensions in our physical universe. A tremendous mathematical insight of the last few years is that in order to understand mirror symmetry for compact manifolds, it is essential to understand it for "non-compact" or unbounded geometries as well. In physics, this is called "local mirror symmetry." Moreover, these open manifolds have rich geometry and are interesting in their own right.
My proposed research aims to use local mirror symmetry as a tool for understanding algebraic and symplectic geometry. The work is divided into three questions, which roughly speaking aim to:
1) Develop algebro-geometric models for symplectic invariants of local spaces.
2) Find hidden algebraic equivalences in the symplectic geometry of a wide class of non-compact spaces
known as "conic bundles."
3) Develop a new field of algebraic singularity theory for "hybrid models", which is inspired by "Gromov-Witten theory" in symplectic geometry.
Our work will also have consequences for the mathematical understanding of mirror symmetry. For example, in answering the third question, subtle questions concerning the relationship between mirror symmetry for open manifolds and mirror symmetry for closed manifolds must be answered.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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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 |
| Summary |
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| Date Materialised |
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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.imperial.ac.uk |