Geometric analysis of special structures in high dimensions inspired from physics; including singularities, torsion, and geometric evolution

受物理学启发的高维特殊结构的几何分析；

• 批准号: RGPIN-2019-03933
• 负责人: Karigiannis, Spiro
• 金额: $ 1.53万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=678168
• 关键词: Geometric; analysis; special; structures; dimensions

My research is in higher dimensional geometry inspired by theoretical physics. Since Einstein's work in 1915, physicists have been searching for a theory that mathematically unifies gravity with quantum mechanics. A promising candidate is M-theory, that describes the universe using a 7-dimensional shape that is curved in a special way. Such shapes are called G2 manifolds. For the physical theory to be consistent with reality we need these G2 manifolds to have certain cone-like points. These are called G2 conifolds. Although we know thousands of examples of smooth G2 manifolds (without cone-like points), there is still no proof that proper G2 conifolds really exist. They definitely are expected to exist in abundance, both from physical arguments and rigorous mathematical work of myself and Lotay.******The long-term goal is to understand the properties and structure of G2 manifolds as well as we understand Calabi-Yau manifolds, which are 6-dimensional shapes with similar properties that are much better understood. Both objects are candidates for grand unified theories in physics. Mathematically, G2 manifolds are very interesting because although they share many common properties with Calabi-Yau manifolds, for technical reasons G2 manifolds cannot be studied using the same tools that have been successful for Calabi-Yau manifolds, namely classical algebraic geometry. This is because, rather than being locally modelled by the complex numbers as are the Calabi-Yau manifolds, the G2 manifolds are instead locally modelled by an exceptional number system that can exist only in 7 dimensions. Because classical tools are not available, we must instead study G2 manifolds using methods of analysis, such as nonlinear partial differential equations. It is precisely for this reason that the mathematical analysis of G2 manifolds and G2 conifolds is so technically difficult.******One objective of my research is to construct the first ever examples of G2 conifolds, providing rigorous proof of their existence. This is a very important problem to solve, as it would give mathematical justification for the feasibility of M-theory as a model of our physical universe. The method I propose to use is a generalization of a recently published method of constructing smooth G2 manifolds of myself and Joyce, which involves glueing onto the shape a particular family of spaces that are solutions to Einstein's equations of relativity.******Another objective of my research is to understand the set of all possible G2 manifolds (the moduli space), which is itself a shape of high dimension. Studying the ways a smoothly deforming G2 manifold can develop cone-like points involves considering curves on the moduli space that reach the boundary. I plan to investigate this question by analyzing the curvature of the moduli space itself. Establishing upper bounds on this curvature gives quantitative information about the formation of cone-like points and imposes restrictions on the associated physics.********

我的研究处于受理论物理学启发的较高维几何形状。自从爱因斯坦（Einstein）于1915年的工作以来，物理学家一直在寻找一种用量子力学统一重力的理论。一个有前途的候选人是M理论，它使用以特殊方式弯曲的7维形状来描述宇宙。这样的形状称为G2歧管。为了使物理理论与现实一致，我们需要这些G2流形才能具有某些类似锥体的点。这些称为G2 Conifolds。尽管我们知道成千上万的平滑G2歧管示例（没有锥形的点），但仍然没有证据表明正确的G2 Conifold确实存在。从物理论证和我自己和lotay的严格数学工作中，它们肯定会存在丰富的存在。这两个物体都是物理学统一理论的候选人。从数学上讲，G2歧管非常有趣，因为尽管它们与Calabi-yau歧管具有许多共同特性，但由于技术原因G2歧管无法使用与Calabi-yau歧管成功的工具一起研究G2歧管，即经典代数几何学。这是因为，G2歧管不是由局部建模的，而是由仅在7个维度中存在的特殊数字系统进行局部建模。由于不可用的经典工具，因此我们必须使用分析方法（例如非线性偏微分方程）来研究G2歧管。正是由于这个原因，对G2歧管和G2 Conifolds的数学分析在技术上是如此困难。这是一个非常重要的问题，因为它将为M-Bealing作为我们物理宇宙的模型的可行性提供数学上的理由。 The method I propose to use is a generalization of a recently published method of constructing smooth G2 manifolds of myself and Joyce, which involves glueing onto the shape a particular family of spaces that are solutions to Einstein's equations of relativity.******Another objective of my research is to understand the set of all possible G2 manifolds (the moduli space), which is itself a shape of high dimension.研究平稳变形的G2歧管可以形成锥形点的方式涉及考虑到达边界的模量空间上的曲线。我计划通过分析模量空间本身的曲率来调查这个问题。在此曲率上建立上限提供了有关形成锥形点的定量信息，并对相关物理施加限制。**********