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

数学中的许多伟大成就都来自于看似不相交的研究领域之间的联系，允许在一个领域发展的技术和见解为另一个领域的问题带来新的启发。这方面的一个例子是使用非交换代数来研究几何。结合代数和几何的见解往往允许结果被扩展到更自然的概括水平，打破了几何设置所施加的限制，并从几何中产生有趣的代数结构。这种方法在研究奇点的分解方面特别成功，一个例子是有理曲面奇点的最小分解，它具有作为重构代数的非交换解释。这些有理曲面奇点的最小解析度的另一个特点是它们有一个特别迷人和美丽的几何变形理论，但目前还没有从非对易的观点来理解。重建代数的变形理论预计将内在地联系到几何的情况下，所以应该反映其有趣的功能，同时提供新的见解，从非交换的观点。本研究旨在了解这样的例子，通过建立一个桥梁之间的几何和非交换变形理论。这将涉及开发技术来构建非交换代数的变形，并产生从非交换代数恢复几何变形作为模空间的方法。它还将涵盖一般情况，例如移动到光滑品种的设置之外，这将在最小模型程序中的3折叠构建等领域产生广泛的新应用。

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.