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

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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.