Geometric Topology in Three and Four Dimensions; August 2009, Davis, CA

三维和四维几何拓扑；

• 批准号: 0905638
• 负责人: Maggy Tomova
• 金额: $ 2.5万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-05-01 至 2010-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905638&HistoricalAwards=false
• 关键词: Geometric; Topology; Three; Four; Dimensions

In 3-manifold topology several important conjectures were resolved in the last two years. Hass--Thompson--Thurston and independently Bachman provided examples of a manifold with two distinct Heegaard splitting whose lowest genus common stabilization had genus equal to the sum of the two genera, resolving a long standing question. Another striking development is the proof by Qui and Scharlemann of the Gordon conjecture showing that the sum of two Heegaard splittings is stabilized if and only if one of the original Heegaard splittings was stabilized. Kroneheimer and Mrowka used the connection between contact structures on a 3-manifold and the induced symplectic structure on certain related 4-manifolds to prove Property $P$. Yi Ni used the connection between Heegaard Floer homology and sutured manifolds in his proof of the fibered knot conjecture. All of these results are very new and the techniques used are likely to open the doors to many other long standing problems in low dimensional topology. The goal of this conference is to disseminate these ideas, and in particular to introduce junior topologists to these new developments.Low dimensional topology studies the structure of 3 and 4 dimensional manifolds. These are objects that locally look like 3 and 4 dimensional balls. Many physical objects of interest to other sciences can be studied via techniques stemming from low dimensional topology. Examples of such objects are DNA molecules, proteins, and even our universe. Partly because of its wide applications low dimensional topology has attracted the attention of many mathematicians and new discoveries are being made at breathtaking pace. The goal of the proposed conference is to bring together leading researchers in low-dimensional topology and allow for an exchange of information and ideas.

在3-流形拓扑几个重要的代数在过去两年中得到解决。哈斯-汤普森-瑟斯顿和独立巴赫曼提供的例子，一个多方面的两个不同的Heegaard分裂其最低属共同稳定了属等于总和的两个属，解决了一个长期存在的问题。另一个引人注目的发展是证明奎和沙勒曼的戈登猜想表明，总和两个Heegaard分裂是稳定的，当且仅当一个原来的Heegaard分裂是稳定的。Kroneheimer和Mrowka利用3-流形上的接触结构与某些相关4-流形上的诱导辛结构之间的联系来证明性质$P$。易尼使用之间的联系Heegaard弗洛尔同调和缝合流形在他的证明纤维结猜想。所有这些结果都是非常新的，所使用的技术可能会打开大门，许多其他长期存在的问题，在低维拓扑结构。本次会议的目的是传播这些想法，特别是介绍初级拓扑学家这些新的发展。低维拓扑学研究的结构3和4维流形。这些物体局部看起来像三维和四维球。许多其他科学感兴趣的物理对象可以通过源于低维拓扑的技术来研究。 这样的物体的例子是DNA分子，蛋白质，甚至我们的宇宙。部分是因为它的广泛应用，低维拓扑吸引了许多数学家的注意，新的发现正在以惊人的速度进行。拟议会议的目标是汇集低维拓扑学的主要研究人员，并允许交换信息和想法。