Lefschetz Fibrations, Mapping Tori, and Dynamics on Moduli Spaces of Objects

动力系统（随时间变化的系统的数学模型）描述了许多影响我们的过程，并引起了科学中一些最难的问题，例如天体力学中的多体问题。该项目的第一部分旨在探索一种全新的动力学，在这种动力学中，系统的状态（如粒子的位置）随时间移动，而不是整个系统的全局运动。这似乎是矛盾的，事实上，人们期望它主要发生在远离应用程序的情况下。尽管如此，已经证明这种现象在数学上是可能的，而且因为动力系统思维提供了如此强大的直觉，所以想要尽可能地扩展其极限是有道理的。任何在经典机械系统中存在额外复杂性的证据，即使它只直接影响到少数情况，最终也会改变我们对这类系统复杂性的看法。这个项目的第二部分涉及数学中的一种现象，这种现象源于目前与弦理论的密切思想交流：即复杂显式函数（通常是一个变量）的出现。从更定性的拓扑角度来看，人们希望最小化需要在这些函数中编码的信息量。例如，如果函数本身解微分方程，则可以从有限的信息中恢复它们。弦理论已经非常有效地提供了这样的特征，但这个项目的目的是（在一个特殊的情况下）更直接和更简单的描述。它应该被视为“非交换几何”的练习，这是数学家们为超越传统的空间概念（这是当代数学和物理学的一大挑战）做准备的方式。辛流形具有丰富的内部结构。这可以从变分的观点（容量、霍弗范数、谱不变量）或弦理论和镜像对称来解决。然而，基本的已知不变量是一组数字，或同调类（Gromov-Witten不变量）。除此之外还有其他信息（拉格朗日子流形，深谷分类），但它不能直接用作分类工具。该项目打算通过观察作用于深谷范畴的动力系统来解决这种情况。这个想法是从几何考虑开始的，比如通量，并将它们导出到其他情况。该方法被设计用于与映射环面相关的辛流形的特定类。另一个主要主题是使用Lefschetz铅笔计算Fukaya分类的方法。虽然之前在这个方向上有大量的工作，但它仅限于精确（或单调）的情况，并且没有解决理解Calabi-Yau案例中出现的无穷级数的挑战。PI的目的是用一种比镜像对称的标准框架（高斯-马宁连接，镜像映射）更直接的方式来描述这些序列；这种描述也将更加普遍，因为它最终独立于镜像对称的考虑。

Dynamical systems (mathematical models of systems changing in time) describe many processes that affect us, and have given rise to some of the hardest questions in science, such as the multi-body problem in celestial mechanics. The first part of this project aims to explore an entirely new kind of dynamics, in which the states of the system (such as positions of particles) move around in time, without a global motion of the entire system. This seems paradoxical, and indeed one expects it to happen mostly in situations that are far from applications. Nevertheless, it has been shown that the phenomenon is mathematically possible, and because dynamical systems thinking provides such a powerful intuition, it makes sense to want to stretch its limits as far as possible. Any evidence of additional complexity in classical mechanical systems, even if it directly affects only a few cases, ultimately changes how we think of the complexity of such systems in general. The second part of this project deals with a phenomenon in mathematics which arises from its current close exchange of ideas with string theory: namely, the appearance of complicated explicit functions (typically, of one variable). From the viewpoint of topology, which is more qualitative, one hopes to minimize the amount of information that needs to be encoded inside such functions. For instance, if the functions themselves solve a differential equation, they can be recovered from a finite amount of information. String theory has been very effective in providing such a characterization, but this project aims (in a special case) for a more direct and simpler description. It should be viewed as an exercise in "noncommutative geometry", which is the mathematicians' way to prepare ourselves for thinking beyond conventional notions of space (which is one of the big challenges in contemporary mathematics and physics).Symplectic manifolds have a rich internal structure. This can be approached from a variational viewpoint (capacities, Hofer norms, spectral invariants), or from string theory and mirror symmetry. Nevertheless, the basic known invariants are a collection of numbers, or homology classes (Gromov-Witten invariants). There is information beyond that (Lagrangian submanifolds, Fukaya categories), but it is not directly amenable to being used as a classification tool. The project intends to attack this situation by looking at dynamical systems acting on the Fukaya category. The idea is start with geometric considerations such as flux, and export them to other situations. The approach is designed to be applied to a specific class of symplectic manifolds, related to mapping tori. The other major topic is a way of computing Fukaya categories, using Lefschetz pencils. While there is a body of previous work in this direction, it is restricted to the exact (or monotone) situation, and does not address the challenge of understanding the infinite series that arise in the Calabi-Yau case. The PI's aim is to describe those series in a more direct way than is provided by the standard framework of mirror symmetry (Gauss-Manin connections, mirror maps); this description would then also be more general, since it is ultimately independent of mirror symmetry considerations.