CAREER: Surface bundles and logic in geometric group theory

这个项目围绕着寻找离散群上的方程的解。这相当于理解一对离散组(表示方程的域和目标感兴趣的组)之间的同态集合。这个项目的前半部分是由S丛的理论所推动的，其中S是一个紧的可定向曲面。曲面丛在整个纯数学中产生。如果B是流形(或其他合理空间)，则B上的S丛的同构类(定向)集与从B的基本群到S的映射类群的同态集(共轭类)之间存在一一对应。PI的目的是当B的基本群是有限生成的(例如，当B是紧的)时，建立该集合的一般结构理论。在这个项目的后半部分(与H，Wilton联合工作)，我们研究到自由(更广泛地说是无挠双曲)群的同态。广义地讲，这是一类“负弯曲”的群体。这项研究的动机是这些群体的一阶逻辑。值得注意的是，负曲率几何对一阶逻辑有着深刻的影响。我们打算从算法的角度来研究这些一阶理论，并找到决定逻辑句子真假的一般决策过程。群体形成了研究数学对象对称性的自然语言。因此，它们出现在整个数学中，而且它们的研究受到数学的许多分支的影响。对称的基本性质很少，其中最重要的是它不会丢失关于空间的任何信息，所以它是可以撤销的。在这个项目中，我们将群视为内在感兴趣的对象，尽管我们提出的问题的动机来自拓扑学、几何学、逻辑学和计算机科学以及群论。广义地说，我们取群G上的一组方程，并试图理解这些方程的所有解的集合。在项目的前半部分，群G是曲面的映射类群，它捕捉了曲面(看起来像小集合中的平面的空间)的大部分对称性。通过研究曲面丛(任何点附近的空间分解成一对更小的集合，其中之一是曲面)，研究映射类群上的方程是几何学和拓扑学中的基本兴趣。通过对映射类群上方程的研究，给出了曲面丛参数化的一般理论。这个项目的后半部分从逻辑的角度研究群上的方程，并寻找判断逻辑语句是否为真或假的通用算法。该项目由拓扑学计划和基金会计划共同资助。

This project revolves around finding solutions to equations over discrete groups. This is equivalent to understanding sets of homomorphisms between a pair of discrete groups (the domain representing the equations and the target the group of interest). The first half of this project is motivated by the theory of S-bundles, where S is a compact orientable surface. Surface bundles arise throughout pure mathematics. If B is a manifold (or other reasonable space), then there is a one-to-one correspondence between the set of isomorphism classes (oriented) of S-bundles over B and the set of (conjugacy classes of) homomorphisms from the fundamental group of B to the mapping class group of S. The PI intends to develop a general structure theory for this set when the fundamental group of B is finitely generated (which will always happen when B is compact, for example). In the second half of this project (joint work with H, Wilton), we study homomorphisms to free (and more generally torsion-free hyperbolic) groups. Broadly speaking, this is the class of `negatively-curved' groups. This study is motivated by the first-order logic of these groups. It is quite remarkable that the geometry of negative curvature has profound implications for first-order logic. We intend to study these first-order theories from an algorithmic point of view, and find general decision processes which determine if a logical sentence is true or false. Groups form a natural language for studying symmetries of mathematical objects. As such, they arise throughout mathematics, and their study is informed by many branches of mathematics. There are very few basic properties of a `symmetry', the most important of which is that it doesn't lose any information about the space, so it can be undone. In this project, we treat the groups as objects of intrinsic interest, although the questions that we ask take their motivation from topology, geometry, logic and computer science as well as from within group theory. Broadly speaking, we take a collection of equations over a group G, and try to understand the set of all solutions to these equations. In the first half of the project, the group G is the mapping class group of a surface, which captures much of the symmetry of a surface (a space which looks like the plane in small sets). Studying equations over the mapping class group is of fundamental interest in geometry and topology, through the study of surface bundles (a space which nearby any point decomposes into a pair of smaller sets, one of which is a surface). The study of equations over the mapping class group gives a general theory parametrizing surface bundles. The second half of this project studies equations over groups from the point of view of logic, and looks for general algorithms which decide if logical sentences are true of false. This project is jointly funded by the Topology Program and the Foundations Program.