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

模空间、非交换代数和变形理论之间的相互作用。

基本信息

批准号：
EP/M017516/1
负责人：
Joseph Karmazyn
金额：
$ 28.35万
依托单位：
University of Bath
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FM017516%2F1
关键词：
Interactions between Moduli Spaces Non

项目摘要

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.

数学领域的许多巨大成功都源于看似互不相交的研究领域之间的联系，使一个领域开发的技术和见解能够为另一个领域的问题提供新的视角。其中一个例子是使用非交换代数来研究几何。将代数和几何洞察力结合起来通常可以将结果扩展到更自然的概括水平，突破几何设置所施加的限制，并从几何中产生有趣的代数结构。这种方法在奇点分辨率的研究中特别成功。有理表面奇点的最小分辨率提供了一个例子，它具有作为重建代数的非交换解释。这些有理表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迷人和美丽的几何变形理论，然而目前这还不能从非交换的角度来理解。重构代数的变形理论预计与几何情况有内在联系，因此应该反映其有趣的特征，同时从非交换的角度提供新的见解。本研究旨在通过在几何案例之间建立一座桥梁来理解此类示例。几何和非交换变形理论。这将涉及开发构造非交换代数变形的技术，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，例如移出平滑品种的设置，这将在最小模型程序中构建 3 倍等领域产生广泛的新应用。