The Homological Minimal Model Programme (Extension)

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A surprisingly large proportion of the natural world, with its incredibly rich structure, systems and complex lifeforms, can be understood using scientific principles that are either underpinned, controlled, or can be approximated by, mathematical objects called polynomials. These fundamental objects are built on solid, long-standing mathematical foundations, and have the advantage of being able to describe relationships with both precision and with grace. Their deceptively simple form, however, often masks a deep and complicated underlying geometry, which in turn often exhibits very counter-intuitive behaviour.This proposal lies within the framework of polynomials, and their attached geometric varieties. It seeks to answer a series of open questions in birational surgeries, their classification, and enumerative questions by using newly constructed noncommutative invariants, and using the additional structure that these encode to distinguish geometric objects to a much finer degree. The first part of this proposal seeks classification of geometric structures via noncommutative techniques, and in process proposes an ADE classification of certain Jacobi algebras. This, and previous work, strongly suggests results in other areas, and the second part of the project involves these, from generating sets of the pure braid groups, to new combinatorial tilings of the plane which conjecturally control many structures in both algebra and geometry, with strong links to Coxeter groups. The third part unifies and generalises into a wider framework, which involves understanding enumerative and structural questions for both singular flops, and for flips.