The Homological Minimal Model Program

One of the more startling discoveries of twentieth century mathematics was the realization that, in contrast to the theory of surfaces, and in contrast to many results and the prevailing wisdom, in dimension three geometric minimal models may be singular. Thus the `best' answers, indeed the spaces we are aiming for, may themselves have singularities.

This project aims to understand these structures from a homological viewpoint, and will give new information even in the well-studied cases. Current derived category techniques focus on situations where the minimal models are smooth, whereas this project will push both our theoretical understanding and our computational ability deep into the singular setting.

The first part of this project develops the homological algebra surrounding reconstruction algebras and maximal modification algebras, strengthening them and extending them in many new directions. These new techniques will then be applied to algebraic geometry, and will recover the existence of geometric minimal models in dimension three as a special case. The techniques, however, will give much more, and in the main body of this work we use this extra data to obtain information regarding contraction of curves and flops; this then allows us to run aspects of the minimal model program in an algorithmic manner.

The primary impact of this project will be in developing and maintaining the UK knowledge base in pure mathematics. It will build on the existing strength of UK mathematics, in areas such as geometry and noncommutative algebra, whilst at the same time it will develop other areas (for example commutative algebra) where, compared to our international competitors, the UK is currently under-represented. Funding for this proposal will also help train one post-doctoral research assistant, whose resulting expertise will further add to the UK knowledge pool.

As an intradisciplinary proposal in mathematics, ultimately the main impact of this proposal will be felt through the resulting interdisciplinary transfer of information and skills. The proposal centres around problems in algebraic geometry, but the methods proposed arise from, and will have impact in, noncommutative structures, homological algebra, commutative algebra, representation theory, and many other areas.

As my most recent work (outlined in the Case for Support) demonstrates, this interdisciplinary transfer is already beginning to emerge. This fellowship would provide the ideal framework in which I can maximise this impact, to the benefit of both UK mathematics and mathematics world-wide.