Conference on Low-Dimensional Topology

低维拓扑会议

• 批准号: 0450806
• 负责人: Vyacheslav Krushkal
• 金额: $ 1.4万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-11-15 至 2005-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0450806&HistoricalAwards=false
• 关键词: Conference; Low; Dimensional; Topology

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.

本次会议将集中在低维拓扑结构的主要方法之间的连接。伟大的成功来自经典的拓扑3-流形程序（表面，叶理和层压），从里奇流，弗洛尔同调，并从代数不变量在纽结理论和拓扑场论。在4维空间中，唐纳森、塞伯格-威滕和奥斯瓦斯-萨博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是不相连的，与显着的例外几何化的3流形，很少出现一般的结构特征。这与高维拓扑形成对比，在高维拓扑中，1950 - 1980年间的巨大进步揭示了由同伦理论、丛和稳定代数（K和L理论）主导的深刻且相对统一的结构。我们希望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。里奇流能耦合到四维流形上的2-形式或其他规范论对象吗？具有特殊动力学的非奇异流是否提供了Oszvath-Szabo“细胞”Floer同源性的“奇异同源性”模拟？与叶理和叠层相关的特殊度量对里奇流有影响吗？Taubes之间的关系塞伯格-威滕和格罗莫夫不变量的辛流形提出了希望，几何代表塞伯格-威滕可以找到更普遍的。到目前为止，这一点并不成功，但也许是时候重新考虑这个问题了。我们期待着来自不同领域的专家和年轻研究人员之间的交流，并带来新的观点。拓扑学旨在了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。世纪早期的工作集中在第2和第3维度上，期望更高的维度会越来越复杂，甚至可能无法理解。奇怪的是，事实并非如此：4以上的维度实际上更容易，并且在1950年至1980年之间开发了一个深入的系统理论。在过去的30年里，焦点又回到了第三和第四维度。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，仍然缺乏类似于在更高维度中实现的统一观点。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。