Cones and positivity in algebraic geometry

This project will address some basic issues in algebraic geometry, a central field of research in pure mathematics.

The objects we study in algebraic geometry are so-called algebraic varieties, meaning solution sets of systems of polynomial equations. Algebraic varieties are of great interest in many parts of mathematics, including number theory and topology, but also in a range of applications: mathematical physics, where they provide mathematical models for physical objects; control theory and motion planning, where they represent the possible states of a system subject to algebraic constraints; and many others. Getting a good high-level picture of the structure of algebraic varieties is therefore of immense theoretical and practical interest.

This project consists of two interrelated approaches to the problem of understanding the structure of algebraic varieties.

The first approach is to investigate an important hypothesis called the "Morrison--Kawamata cone conjecture". If true, this conjecture would provide very precise information about the way in which certain classes of algebraic varieties can be related to each by algebraic mappings. Our research will develop an inductive method, allowing to use information about smaller varieties to deduce information about many larger varieties. This will greatly increase the scope of what is known about the conjecture: to date it has only been proved for very special types of varieties, but our inductive method will yield results in much greater generality.

The second approach will focus on the question: given an algebraic variety, how can we describe the collection of all smaller algebraic varieties contained inside it? In the past thirty years, immense progress has been made on the problem of understanding the curves (that is, one-dimensional varieties) contained inside a given variety, and this has greatly improved our understanding of the general picture of all algebraic varieties. In this component of our project, we will tackle the problem of understanding all "larger" subvarieties in a given variety --- for example, the surfaces inside a variety of dimension four. Progress on this question will yield a new set of tools for understanding, distinguishing, and classifying algebraic varieties.

The proposed research promises to have significant impacts in a range of areas.

As explained in the "Academic Beneficiaries" document, this research will be of great relevance to research groups in algebraic geometry, especially birational geometry. In particular, the research will be of value to researchers at a number of UK institutions: it will inform and complement work on a number of related problems, and will serve as a guide for future research in the area. Taken together, this body of work will contribute substantially towards preserving the UK's standing as a world leader in the field of algebraic geometry. A tradition of excellence in algebraic geometry, including Cayley, Hodge, Atiyah, Reid and others, is a cornerstone of the UK's profile as a world leader in the mathematical sciences; preserving this tradition, and so continuing to attract the best students and researchers to UK institutions, is therefore of great strategic and economic importance. In addition, students trained in this environment of excellence will develop valuable portfolios of skills in critical thinking and quantitative understanding; in many cases these skills will be applied productively in key sectors such as finance and computing, benefitting the national economy as a whole.

As well as other researchers and students in algebraic geometry, this project will provide benefits to a range of "end-users" of algebraic geometry. These includes researchers in other academic disciplines such as optimisation, statistics, biology and control theory, together with industrial users of algebraic geometry in fields such as cryptography, computer graphics and modelling. The importance and range of applications of this kind continues to grow rapidly, with researchers such as Sturmfels (University of California, Berkeley), Sommese (University of Notre Dame) and Sottile (Texas A&M University) among the leaders in this development. The proposed research is closely connected to issues of central importance in these areas --- for example, the problem of "patch tuning" in algebraic modelling requires a clear understanding birational self-maps of projective spaces, a problem which falls within the scope of the cone conjecture. Our results will help to build up the theoretical framework that is necessary to guide the further developments of these applications.