The Dynamics of Space-Time

该提案有两个主要目标。第一个是洛伦兹动力学的系统发展。一个重要的里程碑将是对连通洛伦兹流形上允许非固有作用的连通李群的完整描述。在单李群的情况下，由于N. Kowalsky，有了明确的结果，但非半单李群的情况，目前还没有定论。该提案的第二个主要目标是将已知的刚性结果从几何扩展到具有切向几何和横向不变测度的叶状设置。许多数学家在这个方向上做了很多工作，但所用的程序通常是特别的，每个单独的定理都有不同的方法。这里的目标是找到一种方法，使这一过程系统化，并在某些情况下提供比以前可能的更强的叶状结果。该提案有两个主要目标。首先要理解混沌动力学在时空环境中发生的方式。广义相对论的一个原则是，我们的宇宙是一个时空流形，因此，了解哪些时空具有复杂（即“混沌”）的对称性应该是一些有趣的事情。第二个主要目标是使用一些复杂的动力学机制和一些复杂的几何机制的组合来理解称为“叶”的数学对象的集合。这些对象与动力物理有关，因为一组流动轨迹会产生叶理。然而，在我使用的化身中，叶状结构包含了大量的结构，使其在数学意义上丰富，但可能比大多数数学以外的科学家所研究的（目前！）要复杂得多。

The proposal has two main goals. The first is a systematic development of Lorentz dynamics. A major milestone would be a complete description of the connected Lie groups that admit a nonproper action on a connected Lorentz manifold. In the case of simple Lie groups, there are definitive results due to N. Kowalsky, but the non-semisimple case is, for now, wide open. The second main goal of the proposal is to extend known rigidity results from geometry to a foliated setting with tangential geometry and with transverse invariant measure. Much work in this direction has been done by many mathematicians, but the procedures used have often been ad hoc in nature, with a different method developed for each individual theorem. The goal here would be to find an approach that would systematize the process and that would give, in some cases, stronger foliated results than were previously possible. The proposal has two main goals. The first involves understanding the ways in which chaotic dynamics can happen in a space-time setting. It is a tenet of general relativity that our universe is a space-time manifold and it should therefore be of some interest to know which space-times have a complicated (that is, "chaotic") collection of symmetries. The second main goal is to use a combination of some sophisticated dynamical machinery and some sophisticated geometric machinery to understand a collection of mathematical objects called "foliations." These objects relate to dynamical physics, since the set of trajectories of a flow give rise to a foliation. However, in the incarnation that I am using, a foliation carries an enormous amount of structure, making it rich in a mathematical sense, but probably more complicated than what would be studied (currently!) by most scientists outside of mathematics.