Low-dimensional topology and geometry

At the core of low-dimensional topology has been the classification of knots and links in the 3-sphere and the classification of 3- and 4-dimensional manifolds (see Wikipedia for the definitions of basic topological terms). Beginning with the introduction of hyperbolic geometry into knots and 3-manifolds by W. Thurston in the late 1970s, geometric tools have become vital to the subject. Next came Freedman's (1) classification of simply connected topological 4-manifolds in 1981 followed by the gauge theory invariants of smooth 4-manifolds introduced by Donaldson (2) in 1982. The gauge theory invariants (2) were based on solutions to the Yang–Mills equations for connections on a complex 2-plane bundle over the 4-manifold X4. These results were striking, giving many smooth structures on many compact, closed, oriented 4-manifolds. Even more striking was the discovery of uncountably many exotic smooth structures on ordinary 4-space, R4. It is possible that all compact smooth 4-manifolds have many smooth structures and that all noncompact smooth 4-manifolds have uncountable smooth structures. In 1994, the Seiberg–Witten equations (3) were discovered, and they were a much simpler pair of equations to work with than the Yang–Mills equations. Within months, Taubes (4, 5) had shown that, in the case of a symplectic 4-manifold X4, the Seiberg–Witten invariants were equivalent to the Gromov–Witten invariants, which count the number of pseudoholomorphic curves in X4 that belong to certain 2-dimensional homology classes. The symplectic 4-manifold has a compatible, almost-complex structure [a lifting of the tangent bundle of X to a U(2) bundle]; the pseudoholomorphic curves are immersed real surfaces whose tangent planes are complex lines in the U(2) bundle, and the homology classes are chosen so that the compact moduli space of pseudoholomorphic curves is 0-dimensional and thus, finite. Counting pseudoholomorphic curves is a …