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EPSRC Reference: EP/D007518/1
Title: Spaces of stability conditions in algebraic geometry
Principal Investigator: Bridgeland, Professor T
Other Investigators:
Researcher Co-Investigators:
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Department: Pure Mathematics
Organisation: University of Sheffield
Scheme: Standard Research (Pre-FEC)
Starts: 01 November 2005 Ends: 28 February 2009 Value (£): 134,098
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Physics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
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Summary on Grant Application Form
Algebraic geometry is the study of solutions of polynomial equations. Thus for example the equation x^2 + y^2 = 1 describes a circle in the plane, the equation xy = 1 describes a hyperbola, and so on. If we take lots of equations in lots of variables we can obtain very complicated shapes in this way. These shapes are called varieties. In general they can be of any dimension, not one-dimensional like the two examples given above. It is hard to imagine what a four-dimensional shape might look like, but we can always use abstract algebra to help us when our intuition fails. In fact algebraic geometry is a huge subject, with a long history, and many interesting varieties have been described and studied. Along the way, lots of general results have been proved which help us to understand varieties in general.Recently physicists have got interested in algebraic geometry. That's because they have a new theory of the world called string theory. One consequence of this theory is that the world really has 10 dimensions (or 11, depending on who you ask and when). The reason we don't see these extra dimensions is that they're curled up really small. So when we move around we can only move in three planes (left/right, up/down, backwards/forwards). We also usually count time as a fourth dimension. The other dimensions predicted by string theorists are invisible to us. But nonetheless the shape in which the extra dimensions are curled up is very important as it leads to the fundamental forces such as gravity and electromagnetism. So string theorists are very interested in higher-dimensional shapes, and varieties are a good way of defining these.The research project we're proposing aims to try to understand a piece of algebraic geometry which came directly out of string thoery. To every variety we can associate another curved shape called the space of stability conditions. These spaces are important in string theory, but we don't really understand them very well mathematically yet. So the best thing to do is just to sit down and compute them in some examples, so we can get a feel for what's going on. Hopefully that might give us some ideas about what these spaces look like in general.
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Organisation Website: http://www.shef.ac.uk