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

三流形拓扑在过去两年中解决了几个重要的猜想。Hass—Thompson—Thurston和独立的Bachman提供了具有两个不同Heegaard分裂的流形的例子，其最低属共同稳定的属等于两个属的和，解决了一个长期存在的问题。另一个引人注目的发展是Qui和Scharlemann对Gordon猜想的证明，证明两个Heegaard分裂的和是稳定的，当且仅当其中一个原始Heegaard分裂是稳定的。krooneheimer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证明了性质$P$。Yi Ni在证明纤维结猜想时，利用了Heegaard flower同调和缝合流形之间的联系。所有这些结果都是非常新的，所使用的技术很可能为低维拓扑中许多其他长期存在的问题打开大门。这次会议的目的是传播这些思想，特别是向初级拓扑学家介绍这些新的发展。低维拓扑研究三维和四维流形的结构。这些物体局部看起来像三维和四维的球。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。这类物体的例子包括DNA分子、蛋白质，甚至我们的宇宙。部分由于其广泛的应用，低维拓扑吸引了许多数学家的注意，新的发现正以惊人的速度出现。本次会议的目标是将低维拓扑领域的主要研究人员聚集在一起，并允许交换信息和思想。

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.