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

我的研究是在理论物理学的启发下的高维几何学。自爱因斯坦1915年的工作以来，物理学家一直在寻找一种理论，在数学上将引力与量子力学统一起来。一个有希望的候选者是M理论，它使用以特殊方式弯曲的7维形状来描述宇宙。这种形状被称为G2流形。为了使物理理论与现实相一致，我们需要这些G2流形具有某些锥状点。这些被称为G2 conifolds。虽然我们知道成千上万的光滑G2流形（没有锥状点）的例子，但仍然没有证据证明真正的G2锥真的存在。从我和洛泰的物理论证和严格的数学工作来看，它们肯定会大量存在。长期目标是了解G2流形的属性和结构，就像我们了解卡拉比-丘流形一样，卡拉比-丘流形是具有类似属性的6维形状，这些属性已经得到了更好的理解。这两个物体都是物理学中大统一理论的候选者。在数学上，G2流形是非常有趣的，因为尽管它们与卡-丘流形有许多共同的性质，但由于技术原因，G2流形不能使用与卡-丘流形相同的工具来研究，即经典代数几何。这是因为，G2流形不是像卡-丘流形那样由复数局部建模，而是由一个只能存在于7维中的特殊数字系统局部建模。由于经典工具不可用，我们必须使用分析方法来研究G2流形，例如非线性偏微分方程。正是由于这个原因，G2流形和G2锥的数学分析在技术上是如此困难。我的研究目标之一是构建G2 conifolds的第一个例子，为它们的存在提供严格的证明。这是一个需要解决的非常重要的问题，因为它将为M理论作为我们物理宇宙模型的可行性提供数学证明。我建议使用的方法是最近发表的一种方法的推广，这种方法是我和乔伊斯构造光滑的G2流形的方法，它涉及到将一个特殊的空间族粘在形状上，这些空间族是爱因斯坦相对论方程的解。我研究的另一个目标是理解所有可能的G2流形的集合（模空间），它本身就是一个高维形状。研究光滑变形的G2流形可以发展锥状点的方式涉及到考虑到达边界的模空间上的曲线。我计划通过分析模空间本身的曲率来研究这个问题。建立这个曲率的上界给出了关于锥状点形成的定量信息，并对相关的物理学施加了限制。

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.