Low-dimensional topology and the complex of curves

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Topology in dimension three is the most accessible field ofmathematics; this is because three-dimensional space is the realm ofeveryday experience. Established by Poincare in a series of inspiringarticles in the late 1800's much of the work in low-dimensionaltopology was combinatorial in nature. In 1979 Thurston revolutionizedthe field by revealing deep connections to many other, more geometric,areas of mathematics.The geometric theme in low-dimensional topology has expanded to include coarse geometry. One striking success of this theme was the work of Masur and Minsky [1999, 2000]; they introduced the idea of using coarse geometry to understand the complex of curves. The complex of curves, defined by Harvey, is a combinatorial analogue of Teichmuller space.The novelty proposed here is to use the methods of Masur and Minsky tostudy and solve combinatorial problems, some first encountered byPoincare, in low-dimensional topology. The themes running through theproposal include handlebodies and Heegaard diagrams, the recognitionproblem for the three-sphere, the structure of the mapping classgroup, and the cobordism group of automorphisms of surfaces.