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，并引发了一些最困难的科学问题，例如天体力学中的多体问题。这个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)在时间上移动，而不是整个系统的全局运动。这似乎是自相矛盾的，事实上，人们预计它主要发生在远离应用的情况下。然而，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大它的极限是有意义的。任何经典机械系统中额外复杂性的证据，即使它只直接影响少数情况，最终也会改变我们对这类系统总体复杂性的看法。这个项目的第二部分涉及数学中的一种现象，这种现象源于它目前与弦理论的密切思想交流：即出现复杂的显式函数(通常是一个变量的函数)。从更定性的拓扑学角度来看，人们希望将需要在此类函数中编码的信息量降至最低。例如，如果函数本身解决了一个微分方程，则可以从有限数量的信息中恢复它们。弦理论在提供这样的描述方面非常有效，但这个项目的目标是(在特殊情况下)进行更直接和更简单的描述。它应该被视为“非对易几何”的练习，这是数学家们为超越传统空间概念(这是当代数学和物理学中的一大挑战)而进行思考的一种方式。辛流形具有丰富的内部结构。这可以从变分的观点(容量，霍费尔范数，谱不变量)，或从弦理论和镜像对称性来处理。然而，已知的基本不变量是数或同调类的集合(Gromov-Witten不变量)。除此之外还有其他信息(拉格朗日子流形、Fukaya范畴)，但它不能直接用作分类工具。该项目打算通过观察作用于Fukaya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，例如通量，然后将它们导出到其他情况。该方法被设计成应用于与映射环面有关的一类特定辛流形。另一个主要话题是使用Lefschetz铅笔计算Fukaya范畴的方法。虽然在这个方向上有许多以前的工作，但它仅限于确切的(或单调的)情况，并没有解决理解Calabi-Yau案件中出现的无穷级数的挑战。PI的目的是以一种比镜像对称的标准框架(Gauss-Manin连接、镜像映射)提供的更直接的方式来描述这些系列；这种描述也将更加一般，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.