Interactions between Moduli Spaces, Non-Commutative Algebra, and Deformation Theory.

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Many great successes within mathematics arise from linking between seemingly disjoint fields of research, allowing techniques and insights developed in one area to shine a new light on problems in another. One example of this is the use of non-commutative algebra to study geometry. Combining both algebraic and geometric insight often allows results to be extended to more natural levels of generalisation, breaking out of restrictions imposed by geometric settings and producing interesting algebraic structures from the geometry. This approach has been particularly successful in the study of resolutions of singularities.An example is provided by minimal resolutions of rational surface singularities having a non-commutative interpretation as reconstruction algebras. Another feature that these minimal resolutions of rational surface singularities possess is that they have a particularly fascinating and beautiful geometric deformation theory, however currently this is not understood from a non-commutative viewpoint. The deformation theory of the reconstruction algebras is expected to be intrinsically linked to the geometric case and so should mirror its interesting features while offering new insights from a non-commutative viewpoint.This research seeks to understand examples such as this by building a bridge between the geometric and non-commutative deformation theory. This will involve developing techniques to construct deformations of non-commutative algebras and producing methods of recovering geometric deformations from non-commutative ones as moduli spaces. It will also encompass general situations, such as moving outside the setting of smooth varieties, which will generate a wide range of new applications in areas such as the construction of 3-folds in the minimal model program.