Conference on Low-Dimensional Topology

本次会议将重点讨论低维拓扑主要研究方法之间的联系。经典的拓扑3流形规划（曲面、叶状和层状）、里奇流、花同调以及结论和拓扑场论中的代数不变量都取得了巨大的成功。在维数4中，Donaldson、Seiberg-Witten和Oszvath-Szabo理论揭示了光滑流形的结构。然而，这些不同的线程仍然在很大程度上没有连接，并且，除了3流形的几何化之外，很少出现一般的结构特征。这与高维拓扑学相反，高维拓扑学在1950-1980年间的巨大进步揭示了由同伦理论、束和稳定代数（K和L理论）主导的深度和相对均匀的结构。我们希望在一系列专家和新研究人员之间的热烈讨论将有助于揭示低维的深层结构。里奇流可以耦合到一个2形式或其他计量理论对象上的4流形？具有特殊动力学的非奇异流是否提供了Oszvath-Szabo“细胞”花同源的“奇异同源”类比？与叶状和层压相关的特殊度量是否对里奇流有影响？Taubes关于辛流形的Seiberg-Witten和Gromov不变量之间的关系，为更普遍地找到Seiberg-Witten的几何表示带来了希望。到目前为止，这还没有成功，但也许是时候重新审视这个问题了。我们期待着广泛的专家和具有新鲜观点的年轻研究人员之间的互动。拓扑学旨在理解局部看起来像普通欧几里得空间但其整体形状可能相当复杂的物体的结构。20世纪早期的研究主要集中在二维和三维，人们期望更高的维度会变得越来越复杂，甚至可能超出人们的理解范围。奇怪的是，事实并非如此：4以上的维度实际上更容易，一个深入的系统理论在1950年至1980年左右发展起来。在过去的30年里，焦点又回到了第3和第4维度。这是数学中最活跃的领域之一，与几何、分析、代数和物理有着深刻的关系。然而，一个类似于在更高维度中所实现的统一视角仍然缺乏。会议的目的是促进研究该主题不同方面的研究人员之间的互动，希望统一的观点将开始出现。

This conference will focus on connections between main approaches to low dimensional topology. Great successes have come from the classical topological 3-manifold program (surfaces, foliations and laminations), from the Ricci flow, Floer homology, and from algebraic invariants in knot theory and topological field theory. In dimension 4 the Donaldson, Seiberg-Witten, and Oszvath-Szabo theories have revealed structure of smooth manifolds. However these different threads remain largely unconnected and, with the notable exception of geometrization of 3-manifolds, few general structural features have emerged. This is in contrast with high dimensional topology where great advances between 1950-1980 revealed deep and relatively uniform structure dominated by homotopy theory, bundles, and stable algebra (K and L theory). We hope that a spirited discussion among a range of experts and new researchers will help reveal the deep structure of low dimensions. Can the Ricci flow be coupled to a 2-form or other gauge-theoretic object on a 4-manifold? Do nonsingular flows with special dynamics provide a ``singular homology'' analog of the Oszvath-Szabo ``cellular'' Floer homology? Do special metrics associated to foliations and laminations have implications for the Ricci flow? Taubes' relation between Seiberg-Witten and Gromov invariants of symplectic manifolds raised hopes that geometric representatives for Seiberg-Witten could be found more generally. So far this has been unsuccessful but perhaps it is time to revisit the question. We look forward to interactions among a wide range of experts and young researchers with fresh viewpoints.Topology aims to understand the structure of objects that locally look like the ordinary Euclidean space but whose global shape may be rather complicated. Work in the early 20th century focused on dimensions 2 and 3, with the expectation that higher dimensions would be increasingly complicated and possibly beyond comprehension. Strangely this was not the case: dimensions above 4 are actually easier and a deep systematic theory was developed between 1950 and about 1980. For the last 30 years the focus has returned to dimensions 3 and 4. This has been one of the most active areas of mathematics, with deep relations to geometry, analysis, algebra and physics. However a unifying perspective analogous to the one achieved in higher dimensions is still lacking. The aim of the conference is to promote interactions among researchers working on different aspects of the subject, in hopes that unifying perspectives will begin to emerge.