EPSRC Reference: 
EP/N022513/1 
Title: 
The Combinatorics of Mirror Symmetry 
Principal Investigator: 
Kasprzyk, Dr AM 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
University of Nottingham 
Scheme: 
EPSRC Fellowship 
Starts: 
01 June 2016 
Ends: 
31 May 2021 
Value (£): 
550,902

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Logic & Combinatorics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Geometry is essentially the study of shapes in space. By allowing certain transformations of the space we can concentrate on different features of interest. For example, in topology we allow very flexible transformations: space is treated like a rubber sheet that can be stretched and deformed freely. When a topologist says that a doughnut is equivalent to a coffee cup they are remarking on what's important to them: to a topologist the number of holes in an object can never be changed, but beyond that the transformations are so general that very little else is preserved. The rules of transformation can also be extremely strict: in Euclidean geometry we can only perform rigid transformations such as rotation. Here it makes no sense to equate a doughnut with a coffee cup, and even two coffee cups will often be considered to have different shapes. Different rules emphasise different properties.
Algebraic geometry occupies the middle ground between the most rigid of geometries (Euclidean geometry) and the most flexible (the rubber sheets of topology). Here the focus is on special spaces called manifolds. We can draw parallels with a landscape: a manifold is similar to a smooth desert, with gently undulating hills and valleys; it is the opposite of, say, a chasm where the topography undergoes sudden and unexpected changes. Manifolds come in three types, distinguished by their curvature. Think of a sphere: if you draw a triangle on the surface of a sphere and add up the angles you'll get more than 180 degrees. This is an example of positive curvature. Now think of a plane: here we know that the angles of a triangle sum to exactly 180 degrees. A plane is an example of zero curvature. For an example of negative curvature think of a saddle, curving upwards at two ends and downwards in the middle. If you draw a triangle on a saddle and sum the angles it will come to less than 180 degrees. Manifolds with positive curvature are called Fano; those with zero curvature are called CalabiYau; and those with negative curvature are called general type.
Manifolds play an important role in theoretical physics. Classically a particle moving between two points A and B will follow a straight line. In string theory the particle is replaced with a string moving through a manifold. The analogue of a line joining points A and B becomes a twodimensional surface swept out by the string as it moves through space. Unlike in classical physics where there is a unique line connecting A and B, in string theory these strings can trace out many different surfaces, and the key to understanding the manifold the string lives in is to count the number of possible surfaces. This counting is done by the mathematics of GromovWitten theory.
Mirror Symmetry predicts a remarkable phenomenon: when the manifold is Fano (and so has positive curvature) the numbers given by GromovWitten theory  that is, the counts of the number of different paths a string can trace as it moves through space  can be reproduced by seemingly unrelated mathematical objects called Laurent polynomials. By understanding how to interpret the mathematics of Laurent polynomials in terms of geometry we will learn to see the structure of Fano manifolds in new ways. In turn this will reveal previously unexplored commonalities between different areas of mathematics. The main aim of this proposal is to use Laurent polynomials in order to describe all possible Fano manifolds, in a similar way to how topologists describe all possible shapes in terms of the number of holes.
There are also strong hints that Mirror Symmetry describes the geometry of a broader class of spaces: the terminal Fano varieties. Although not manifolds  these spaces have singularities, which you can imagine as sharppointed hills space  they are "close" to being manifolds. This proposal will investigate this connection, and use Laurent polynomials to classify the terminal Fano varieties in three dimensions.

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