Research in Geometry and Topology

大部分的建议是致力于大规模几何的外部自同构群的自由群的有限秩和相关的外层空间。在过去的15年中开发的映射类组理论作为一个指导，这种方法的大部分已经在过去的2-3年中实施，但困难的问题仍然存在，主要是由于没有“足够”的预测类似于地下预测。 理解这一点可以导致自由群的自同构群的拟等距刚性。该建议的结论与几个较小的部分涉及具体问题的第二个同源的Torelli群，同源增长的伪Anosov同胚在有限的覆盖，和几何的二次形式，因为看到从圆满的收缩。主题的建议是研究的拓扑和几何性质的几个对象，自然出现在数学。例如，考虑具有固定圈数的所有可能的有限图的集合。这个集合形成了一个空间，称为外层空间。它有一种自然的几何形状，其内在特征是它不对称（与城市中的单行道不同，从A到B可能比从B到A更快）。然而，这种几何形状也显示出双曲线特征。在双曲几何中，如果考虑连接每对三个点A、B、C的最短路径，则总是存在“拥塞点”T，所有这三条路径都接近T。建议的一部分是进一步探讨这一现象。

Most of the proposal is dedicated to the large scale geometry of the outer automorphism group of a free group of finite rank and of the associated Outer space. Mapping class group theory developed in the last 15 years serves as a guide and much of this approach has already been implemented in the last 2-3 years, but the hard questions remain, mostly due to the fact that there aren't "enough" projections analogous to subsurface projections. Understanding this could lead to quasi-isometric rigidity of automorphism groups of free groups. The proposal concludes with several smaller sections involving specific questions about the second homology of Torelli groups, homology growth of pseudo-Anosov homeomorphisms in finite covers, and the geometry of quadratic forms as seen from the well rounded retract.The subject of the proposal is the study of topological and geometric properties of several objects that naturally appear in mathematics. For example, consider the collection of all possible finite graphs with a fixed number of cycles. This collection forms a space, called Outer space. There is a natural geometry associated with it, whose intrinsic feature is that it is not symmetric (not unlike in a city with one-way streets, where it might be quicker to get from A to B than from B to A). Nevertheless, this geometry displays hyperbolic features as well. In hyperbolic geometry, if one considers shortest paths joining each pair of three points A,B,C there is always a "point of congestion" T with all three of these paths coming close to T. A part of the proposal is to explore this phenomenon further.