RUI: Flows of G2-Structures on Manifolds

最经久不衰的科学问题之一是理解宇宙的属性和支配它的规律。自从爱因斯坦的广义相对论以来，人们就知道物理学和宇宙几何学之间存在着非常密切的联系。然而，最新的物理理论，如超弦理论和M理论，表明四维时空的物理性质可以用隐藏的六维或七维空间的几何来描述。在这个项目中，PI将研究这些理论中出现的一种特殊的七维空间的性质，即具有G2结构的流形。该项目的主要目标是进一步发展G2结构的变形或“流动”理论，这将影响不同类别的G2结构之间的平稳过渡。PI预计这将为无扭转G2结构的存在提供充分条件，而G2结构是微分几何中最重要的开放问题之一。(G2结构在流形的每个点上定义了向量的‘点积’--两个向量相乘以获得标量的一种方式--以及允许一个向量乘以两个向量以产生第三个向量的‘叉积’。无扭转的G2-结构是这些点和叉积最大程度兼容的结构。)作为该项目的一部分，PI将培训本科生研究助理，并让他们参与实验代数与几何实验室的活动，PI是该实验室的联合主任。该实验室的活动将结合培训、指导、研究和推广，在不同的层面上促进数学--从K-12，更广泛的社区，一直到研究生水平。在他之前的工作中，PI引入了修改的共闭G2结构的拉普拉斯同流-纠正了标准拉普拉斯同流的非抛物性。PI还引入了八元数丛对G2结构的描述，将G2结构的扭转解释为八离子连接，并将同一度量类别中G2结构的选择解释为规范的选择。然后，具有无散度扭转的G2结构被解释为能量泛函的临界点，并类似于库仑规范。这个项目建立在这个先前的工作基础上。主要目的是证明标准拉普拉斯同流解的存在性。这将通过使用调和映射技术首先证明在固定度量类内具有无散度挠率的G2-结构的存在性来完成。这将被用作拉普拉斯同流的规范固定条件，这将使它与已知存在的修正的同流有关。这个项目的第二个目标是研究和构造G2-结构的其他重要流，目的是在适当的条件下证明这些流的存在性和稳定性，并分析这些流在齐次流形上的行为。这个项目的第三个目标是研究八元数束值微分形式的性质，并获得Dolbeault上同调的八元数值类似物。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the most enduring scientific problems is to understand the properties of the universe and the laws that govern it. Ever since Einstein's theory of general relativity, it was understood that there is a very close link between the physics and the geometry of the universe. However, most recent physical theories, such as superstring theory and M-theory, show that the physical properties of 4-dimensional spacetime may be described in terms of the geometry of hidden six- or seven-dimensional spaces. In this project, the PI will study the properties of a particular kind of seven-dimensional spaces which appear in these theories, known as manifolds with a G2-structure. The main goal of this project is to further develop the theory of deformations or `flows' of G2-structures which will effect smooth transitions between different classes of G2-structures. The PI anticipates that this will pave the way towards giving sufficient conditions for existence of torsion-free G2-structures, which is one of the most significant open problems in differential geometry. (A G2-structure defines, at each point of the manifold, a `dot product' of vectors -- a way of multiplying two vectors to get a scalar -- as well as a `cross product' that allows one to multiply two vectors to produce a third vector. Torsion-free G2-structures are those for which these dot and cross products are maximally compatible.) As part of the project, the PI will train undergraduate research assistants and will involve them in the activities of the Experimental Algebra & Geometry Lab, of which the PI is a co-director. The lab's activities will combine training, mentoring, research, and outreach to promote mathematics at various levels -- starting at K-12, the broader community, and going all the way to graduate level.In his prior work, the PI introduced the modified Laplacian coflow of co-closed G2-structures that rectified the non-parabolicity of the standard Laplacian coflow. The PI also introduced a description of G2-structures in terms of octonion bundles that interpreted the torsion of a G2-structure as an octonionic connection and the choice of a G2-structure within the same metric class as a choice of gauge. G2-structures with divergence-free torsion were then interpreted as critical points of an energy functional and as an analogue of the Coulomb gauge. This project builds upon this prior work. The main goal is to prove existence properties for the standard Laplacian coflow. This will be accomplished by first proving the existence of G2-structures with divergence-free torsion within a fixed metric class using harmonic map techniques. This will then be used as a gauge-fixing condition for the Laplacian coflow, which will relate it to the modified coflow for which existence is known. The second objective of this project is to study and construct other significant flows of G2-structures, with the aim of proving existence and stability under appropriate conditions and analyzing the behavior of these flows on homogeneous manifolds. The third objective of this project is to study the properties of octonion-bundle-valued differential forms and to obtain an octonion-valued analogue of Dolbeault cohomology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.