Mathematical Sciences: Algebraic Geometry and Seiberg-Witten Invariants

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Friedman Friedman studies the new 4-manifold invariants introduced by Seiberg and Witten and their connections with algebraic geometry in the case of algebraic surfaces. For many interesting surfaces, there is little contact between the algebraic geometry of the surface and Seiberg-Witten theory. However, for ruled surfaces, there is a close connection between Seiberg-Witten theory and certain questions in classical algebraic geometry. If the ruled surface is a product, then the description of the Seiberg-Witten moduli spaces is related to Brill-Noether theory for curves, in both its deformation-theoretic and enumerative guises. If the ruled surface is more general, the Seiberg-Witten moduli spaces are connected to questions concerning stable rank two vector bundles over algebraic curves. One problem is whether the Hilbert scheme of a general ruled surface is smooth. This problem is related to a famous example of Mumford, which says that every irreducible curve on a general ruled surface has positive self-intersection. A second question in classical algebraic geometry is to study various enumerative loci in the moduli space of stable bundles where the Hilbert scheme is not smooth or does not have the expected dimension. Another set of questions involves studying the classification of symplectic 4-manifolds, which in many ways should be similar to the classification of algebraic surfaces. Algebraic geometry is an old branch of mathematics which studies the solutions to polynomial equations in the plane or in space. It is important to study the solutions in complex numbers to such equations, in order to see more of the geometry of the solutions. Over the last fifty years, new methods in algebra, analysis, and topology have led to deep results in this theory. Since an algebraic surface depends on two complex numbers and hence four real numbers, it is a four-dimensional object and thus is important physically, since the universe of space-time has four dimensions. Recent ideas from physics have led to deep breakthroughs relating the topology of an algebraic surface to its geometry. These methods have also led to progress in the understanding of four-dimensional objects with a geometry connected to classical mechanics, which are called symplectic 4-manifolds.