Classification, Computation, and Construction: New Methods in Geometry

We are interested in geometrical shapes given as sets of solutions of many polynomial equations in many variables. More than the abstract study of shapes, it is writing down and solving explicit equations that lends our work its special flavour and power in applications. A first aim of our program is to vastly improve the understanding of the structures that control how we write these equations.

All our shapes can be classified into three different kinds: Fano (positively curved), Calabi-Yau (flat) and General Type (negatively curved). Fano and Calabi-Yau shapes play a special role in geometry as "atomic pieces" when breaking complex shapes down to simpler ones, and in physics as backgrounds for string theory. The second key aim of this research is to classify and write down the equations of these Fano and Calabi-Yau atomic pieces in 3, 4, and 5 dimensions, thereby producing a vast "periodic table" of all possible shapes.

String theory is a leading candidate for a "theory of everything". It postulates that the fundamental objects in physics are not point-like particles but strings. These strings move in a background that, in addition to space and time, has extra hidden dimensions curled up in (depending on the version of the theory) either a 3-dimensional Fano or Calabi-Yau shape (3 complex dimensions = 6 real dimensions). Thus, our classification of shapes is also an encyclopaedia of all the possible background geometries of string theory.

To tackle our classification, we will show how to associate a Fano shape to every reflexive polytope. A reflexive polytope is a geometric object closely related to the "Platonic solids" that we learn about in school. To give an idea of the size and complexity of the problem, the complete list of 4-dimensional reflexive polytopes is known and there are nearly half a billion of them. There is an operation on polytopes, called a mutation, and it is possible to mutate a polytope into another polytope. If two polytopes are related by mutation then they give rise to the same Fano shape, so we need to sort all polytopes up to mutation. In order to do this, we need to perform parallel computations on a massive scale distributed over a High Performance Computing cluster.

In string theory, it can happen that two mathematically very different background geometries produce the same physics. When this happens, the two geometries are said to be "mirror symmetric" to each other. Mirror symmetry is a most fascinating aspect of string theory, and our third key aim is to give the first truly transparent explanation of this phenomenon.

Our project requires us to integrate the scientific expertise of our three institutions, and enlist the specialist collaboration of many mathematicians nationally and internationally. In turn, the results and methods of our work will have significant applications in many areas of mathematics, science, and scientific computation.

In a very exciting spin-off project, we will study some avatars of Fano shapes in the world of congruences between coefficients of Fourier expansions of L-functions. This is a profound area of number theory, started by Ramanujan, that lies at the heart of Wiles' proof of Fermat's last theorem.

As scientists, we are inspired when we see the same structures arise independently and for separate reasons in different parts of mathematics and science - this reveals deep connections between seemingly unrelated scientific disciplines. An example was the discovery of group theory (the theory of symmetries) in mathematics and in quantum physics at the turn of the 20th century. There are several examples in our own work: Fano shapes correspond to lattice polytopes and to congruences of coefficients of L-functions; Calabi-Yau shapes arise independently in geometry and in string theory; cluster algebras are found at the same time in algebra and in the geometry of mirror symmetry.

ACADEMIC IMPACT BEYOND PURE MATHEMATICS

Our project will have impact in pure mathematics, string theory and scientific computation. This impact will happen through research results published in peer-reviewed Open Access journals; through workshops, conferences, and invited lecture series by leading international experts; and through new software and datasets that we will produce. The impact on string theory will mainly be through the datasets that we generate: see below. The impact in scientific computation will be through enhancements to existing Computational Algebra Systems (CAS), improving their ability to handle huge datasets ("Big Data") and distributed computation on HPC clusters.

SOFTWARE AND DATA

A significant portion of our research output will be in the form of publicly available databases storing new large classifications of geometric objects, and software to interface with these databases and to study the objects themselves. Both the software and the data will be unique resources in academia and science as tools to do research: to make experiments, guide the development of theory, and prove results. In particular, our databases of polytopes and Fano/LG pairs will be an important resource for physicists working in several flavours of string theory. Our datasets will be made available through the Graded Ring Database (http://www.grdb.co.uk); the data and software will be released under a public domain (CC0) license and made available through a public repository. To ensure greater impact, we will take advice from the string theory community, and from participants in our interface workshops, on the design of the databases and search interfaces.

THE PEOPLE PIPELINE

Our PDRAs will receive training that will give them the option of careers in academia, education, policy, and hi-tech industry. They will acquire uniquely broad research expertise combining different research methods, ideas and perspectives. Furthermore they will be trained in: supervision of undergraduate and postgraduate students, communication and presentation, research planning and management, computer algebra and software engineering, and outreach.

ECONOMY

Indirect economic impact via the People Pipeline and via contributions to Computational Algebra Systems. CASs have concentrated impact in knowledge-intensive, high-technology industrial research, a key wealth-creating sector of the UK economy. Our project partners at Singular have much experience applying computer algebra to problems from cryptography and industry, and this gives us a clear pathway for industrially-relevant research outputs from this program to reach end users outside academia.

CULTURAL AND SOCIETAL IMPACT

The Imperial team (AC, TC) are committed to a long-term effort to take their research to a broad audience in a direct way through collaborations with visual artists. In this context, we plan for two residencies by visual artists, who will produce art directly related to the program. This will lead to exhibitions in galleries and museums, public lectures, drawing workshops for the general public, and academic and art publications. In a separate art/science project, we will contract an artist to develop rules for foreshortening 4-dimensional objects, leading to the development of software for visualising and manipulating 4-dimensional polytopes on screen and with the Google Cardboard virtual reality kit. This will be a direct aid in our research and for mathematical dissemination, but it will also form the basis of artistic collaborations and public demonstrations.

We will make strong media outreach efforts, in particular via the Imperial College Press Office, timed around the artist exhibitions and major scientific milestones in the program. Our track record here shows that basic mathematical research of the kind that we are proposing has romantic appeal and attracts significant public interest.