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

数学中的许多伟大成就都源于看似互不相交的研究领域之间的联系，使一个领域中发展出来的技术和见解能够为另一个领域的问题带来新的曙光。其中一个例子是使用非对易代数来研究几何。将代数洞察力和几何洞察力结合起来，往往可以将结果扩展到更自然的泛化水平，打破几何设置的限制，从几何中产生有趣的代数结构。这种方法在奇点分解的研究中特别成功，例如具有非对易解释的有理曲面奇点的极小分解作为重构代数。这些有理曲面奇点的最小分辨率所具有的另一个特征是它们具有特别吸引人和美丽的几何形变理论，然而目前这一理论并不是从非对易的观点来理解的。重构代数的变形理论应该与几何情形有内在的联系，因此应该反映其有趣的特征，同时从非对易的角度提供新的见解。本研究试图通过在几何和非对易变形理论之间架起一座桥梁来理解这样的例子。这将涉及开发构造非交换代数的变形的技术，以及产生将非交换代数的几何变形恢复为模空间的方法。它还将涵盖一般情况，如移动到光滑品种的设置之外，这将在最小模型程序中构建3折叠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.