New Geometric Structures from String Theory

我们拟议的研究旨在发现支撑物质基本性质的数学结构、宇宙的起源和时空的量子结构，其中每一个都将是科学的一场革命，其可能的后果与物质被证明是由原子组成时一样深远。正如微分几何对爱因斯坦的引力理论至关重要一样，我们正在研究的数学可能是弦理论公式革命的重要组成部分，可能产生深远的影响。长期以来，理论物理一直是激发几何学重要新方向的核心。弦理论的最新研究指出，存在着丰富的一类引人注目的新几何结构。这项研究的目的是发展这些结构的基本统一数学，为使用弦理论和规范理论的工具和思想进行几何研究的新领域奠定基础。很难找到一种既美观又易于处理的新型几何。希钦提出的广义几何就是这样一个新的例子，和其他例子一样，它与物理学的思想有着密切的联系。我们的建议的首要重点将是更大的一套想法，这是其中的一部分，包括扩展几何，加倍几何，通量几何，非几何空间和全息结构。我们预计广泛的应用在数学科学从几何和拓扑代数和数论和数学物理。

Our proposed research is aimed towards discovering the mathematical structures underpinning the fundamental nature of matter, the origins of the universe and the quantum structure of spacetime each of which would be a revolution in science, with possible consequences as far-reaching as they were when matter was shown to be composed of atoms. Just as differential geometry was vital for Einstein's theory of gravity, the mathematics we are investigating could be an essential part of a revolution the formulation of string theory, with possible far-reaching consequences. Theoretical physics has long been central in motivating vital new directions in geometry. Recent research in string theory points to the existence of a rich class of remarkable new geometrical structures. The aim of the research is to develop the underlying unifying mathematics of these structures, laying the foundations for new areas of geometric study using tools and ideas from string theory and gauge theory. It is rare to find a new type of geometry that is both beautiful and tractable. Generalized geometry, introduced by Hitchin, is one such new example, and like others is strongly linked with ideas from physics. The overarching focus of our proposal will be the much larger set of ideas of which this is part, including extended geometries, doubled geometries, flux geometries, non-geometric spaces and holographic structures. We anticipate wide ranging applications across mathematical sciences from geometry and topology to algebra and number theory and mathematical physics.