Gromov-Witten Theory

首席研究员将研究Gromov-Witten理论中的一些主题。这些主题由一个共同的主题联系在一起：由Givental介绍的Gromov- Witten理论的新视角，它揭示了与辛线性代数，几何量化和环群理论的惊人联系。这项研究的目的之一是了解这种观点所揭示的结构的几何起源；第二个也是主要的目标是应用这个视角提供的新的计算工具：辛商Gromov—Witten不变量的计算2. 高格Gromov—Witten不变量的计算3. 非半简单Frobenius流形的结构理论和4。量子k理论和量子超上同调中的各种结构和计算问题。在过去的十年里，为了理解弦理论的一些数学结果，人们对Gromov- Witten不变量进行了深入研究。当弦在空间中运动时，它会掠过一个表面：空间的Gromov- Witten不变量描述了空间中满足某些条件的表面的数量——例如，人们可以坚持认为这些表面经过许多不同的点，或者它们与特定的曲线相交。Gromov-Witten不变量给出了关于空间形状的重要信息。它们在镜像对称中起着至关重要的作用，镜像对称是一个用数学形式表达两种不同弦理论等价的概念圈。镜像对称表明，我们可以通过研究空间X'上的微分方程（称为X的镜像）来计算空间X中的曲面（即计算X的Gromov—Witten不变量）。这从数学和物理的角度来看都很有趣：一个多世纪以来，数学家们一直在研究表面计数和求解微分方程，却没有意识到这两门学科之间有任何联系。迄今为止，弦理论的一些最具说服力的证据并非来自实验，而是来自对其数学预测的逐步验证。

DMS-03401275Tom CoatesThe principal investigator will study a number of topics in Gromov-Witten theory. These topics are linked by a common theme: a new perspective on Gromov--Witten theory, introduced by Givental, which reveals surprising connections with symplectic linear algebra, geometric quantization and the theory of loop groups. One objective of this research is to understand the geometric origin of the structures revealed by this point of view; the second and main objective is to apply the new computational tools that this perspective provides: 1. To the calculation of Gromov--Witten invariants of symplectic quotients; 2. To the computation of higher-genus Gromov--Witten invariants; 3. To the structure theory of non-semisimple Frobenius manifolds; and 4. To various structural and computational questions in quantum K-theory and quantum extraordinary cohomology.Gromov--Witten invariants have been studied intensively for the past decade in an effort to understand some of the mathematical consequences ofstring theory. As a string moves through space, it sweeps out a surface: the Gromov--Witten invariants of a space describe the number of these surfaces in the space which satisfy certain conditions --- for example,one could insist that the surfaces pass through a number of different points, or that they intersect with a particular curve. Gromov-Witten invariants give important information about the shape of a space. They play a vital role in Mirror Symmetry, a circle of ideas which expresses in mathematical form the equivalence of two different string theories.Mirror symmetry suggests that we can count surfaces in a space X (i.e. compute Gromov--Witten invariants of X) by studying differential equations on a space X' called the mirror of X. This is interesting from the point of view of both mathematics and physics: mathematicians had studied surface-counting and solving differential equations for more than a century without realising that the subjects had anything to do with one other, and some of the most convincing evidence for string theory to date comes not from experiment but from the gradual verification of its mathematical predictions.