9625654 Gompf This project deals with the interplay between symplectic and contact geometry and low-dimensional topology. A main focus of the investigation is on the topology of Stein surfaces, or complex affine varieties that are also smooth 4-manifolds. These are automatically Kaehler, hence symplectic, and when their topology is finite, they have natural boundaries that are contact 3-manifolds. The principal investigator is studying Stein surfaces by means of their handle decompositions, addressing questions such as the following: Which 3-manifolds arise as boundaries of Stein surfaces, i.e., which 3-manifolds admit holomorphically fillable contact structures? Which homotopy classes of 2-plane fields on a given 3-manifold can be realized in this manner? How many Stein surfaces can have a given 3-manifold as boundary? Through the known connections between symplectic and 4-manifold topology (such as Seiberg-Witten theory), this should relate to the classification problem for smooth 4-manifolds as well as to other problems, such as the slice genus of classical knots. One of the cornerstones of modern mathematics and physics is the notion of a "manifold," that is, a space that looks locally like ordinary Euclidean space. Curves are 1-dimensional manifolds, and surfaces are 2-dimensional. The space in which we live is 3-dimensional, and the universe, space-time, is an example of a 4-manifold. A basic question of topology is what shapes such spaces can have. Ironically, the familiar dimensions 3 and 4 are by far the least understood -- even less understood than dimensions 5 and higher, where there is more "room" in which to work. Almost nothing was known about the shapes of 4-manifolds until 15 years ago, when the revolutionary breakthroughs of Freedman and Donaldson led to an explosive growth in knowledge and techniques. In the last few years, it has become apparent that topology in low dimensions (i.e., 3 and 4) is intimately linked to sympl ectic and contact geometry. These latter fields emerged from classsical physics (Hamiltonian mechanics), and in recent years they have become rapidly developing independent subjects with connections to various other areas of mathematics (for example, algebraic geometry). The principal investigator has been studying 4-manifolds since the initial breakthroughs in the field, and he is now examining the connections between low-dimensional topology and symplectic and contact geometry. Using topological techniques, he recently expanded the scope of symplectic geometry by showing that symplectic structures are much more common than originally believed. He is currently producing new examples of contact structures, and new tools for analyzing them. He anticipates that continued investigation of the interplay between these three areas should lead to further insight into all three. ***
