The Combinatorics of Mirror Symmetry

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 Calabi-Yau; 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 two-dimensional 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 Gromov-Witten theory.

Mirror Symmetry predicts a remarkable phenomenon: when the manifold is Fano (and so has positive curvature) the numbers given by Gromov-Witten 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 sharp-pointed 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.

A next-generation computational algebra system:

The significance of computer algebra to applications of major international importance cannot be overstated. The prototype Computational Algebra System (CAS) emerging from this project has the potential for huge benefits to the UK academic and industrial sectors. CASs are essential and widely-used tools throughout science and engineering; applications include problems in data analysis, operational research, network analysis, robotics, coding theory, and cryptography; users range from Microsoft Research and Google Labs to government agencies such as GCHQ.

There are two urgent and unavoidable challenges faced by users of CASs in both science and industry: a massive expansion in problem sizes, and the growing need to manipulate huge datasets. No existing CAS will scale to the next generation of problem sizes, and no existing system can cope with the demands of Big Data. Tomorrow's CASs must have parallel processing and database fluency built in from the outset if they are to succeed in overcoming these challenges. A key outcome of this proposal will be a next-generation CAS with these core principles at the heart of its design.

The focus of this system is on computational algebra and discrete mathematics (where, despite a clear need, there are no appropriately-capable products on the market) rather than on numerical methods or finite element models (where existing products, having embraced parallelism, do a much better job). The CAS will have applications across science and engineering, to Big Data problems and network analysis, and to coding theory, cryptography and national security.

To ensure that the new system directly targets the requirements of industrial end-users, time has been allocated in the final year to obtain and tackle real-world "challenge problems". These challenge problems, coming from potential clients and at the limit of what is possible with current systems, will illustrate the broad applicability of the CAS. They will include problems related to cryptography and cryptanalysis, bioinformatics, and optimisation. At the conclusion of the project we will be in an excellent position to secure near-market commercialisation funding, for example via EPSRCs Pathways to Impact scheme. There is also strong potential to attract research and development funding from global businesses, including Google, whose new language 'Go' forms the heart of the CAS.

A mature CAS will take at least a decade to produce, but the first step will be realised by this project. It will position the UK at the forefront of an emerging technology, and development over the coming years will result in a generation of highly skilled students and researchers with expertise essential to tomorrow's information economy.

Public engagement:

This proposal includes a workshop on visualising higher-dimensional polytopes, to be led by artist Gemma Anderson. We will develop online interactive art to illustrate the themes of this research.

Eight local A-Level students will participate in a two-week program of research investigating the deformation theory of surfaces through combinatorics. As part of this they will use the CAS Sage - and hence the programming language Python, a key employment skill - to explore cutting-edge mathematical questions. During a second project, six A-Level students will engage in a two-week workshop to develop programming skills and experience working with and visualising Big Data. This will provide the students with highly sought-after experience.

A community of experts:

An important part of this proposal is a Go users' workshop for Nottingham, Derby, and Loughborough, helping to establish a community within which new programmers and experts can exchange ideas and experience. The visibility this provides may help attract new investors and start-up companies to the region.