Geometry and Topology in Samos

这是一项为2012年6月11日至16日在希腊萨莫斯举行的“萨莫斯的几何和拓扑学”会议提供资金的提案。这次会议的焦点是流形的分类，包括拓扑和几何方面。中心刚性猜想是Farrell-Jones猜想，它推广了Novikov和Borel刚性猜想。这次会议将把使用几何方法研究拓扑问题的研究人员和使用拓扑方法研究几何问题的研究人员聚集在一起。前者的例子是高维拓扑学家，后者的例子是几何群论家和微分几何。拓扑学通常被描述为“橡胶几何”：如果一个空间可以通过拉伸和压缩(但不撕裂或穿孔)变形成另一个空间，则认为空间是相同的。一个基本的问题是决定两个空间是否可以相互变形。实现这一点的主要方法是开发为空间分配可计算不变量的方法：如果两个空间可以相互变形，那么相关的不变量必须重合。反之，如果两个空间具有不同的不变量，则它们不能相互变形。这种不变量的一个例子是空间的同伦群。然后，人们可以问同伦群是否足够好的不变量来确定空间。换句话说，如果我们有两个同伦群相同的空间，它们能变形成彼此吗？拓扑学中的核心猜想之一是Borel猜想，它预言对于某一类空间，前一个问题的答案是“是的”。这个问题一直是高维拓扑学的中心问题，在这个问题上的工作涉及到从代数、几何和动力学等各个领域的复杂技术。会议计划召集国际专家，他们的工作涉及博雷尔猜想(和相关问题)，以期规划未来对这些主题的研究方向。

This is a proposal to provide funding for the conference "Geometry and Topology in Samos" to be held in Samos, Greece during June 11-16, 2012. The focus of this conference is the classification of manifolds, including both topological and geometric aspects. The central rigidity conjecture is the Farrell-Jones Conjecture, which generalizes both the Novikov and Borel rigidity conjectures. The conference will bring together researchers who use geometric methods to study topological problems with researchers who use topological methods to study geometric problems. Examples of the former are high-dimensional topologists and examples of the latter are geometric group theorists and differential geometers.Topology is often described as "rubber geometry": spaces are considered the same if one can be deformed into another via stretching and compressing (but without tearing or puncturing). A basic problem is to decide whether two spaces can be deformed into each other. The primary method for doing this is to develop ways to assign computable invariants to the space: if two spaces can be deformed into each other, then the associated invariants have to coincide. And conversely, if two spaces have different invariants, then they cannot be deformed into each other. An example of such invariants are the homotopy groups of a space. One can then ask whether the homotopy groups are good enough invariants to determine spaces. In other words, if if we have two spaces whose homotopy groups are the same, can they be deformed into each other? One of the central conjectures in topology is the Borel Conjecture, which predicts that for a certain class of spaces, the answer to the previous question is "yes". This problem has been central to high-dimensional topology, and work on it has involved sophisticated techniques from areas as diverse as algebra, geometry, and dynamics. The conference plans on bringing together international experts whose work touches upon the Borel conjecture (and related questions), with a view towards charting the course of future research on these topics.