3-Manifolds and Applications to Geometry

Barnard College, New York NY

Investigators

Abstract

Proposal: DMS-0083097 PI: Walter D. Neumann Despite the venerable history of the algebraic geometry, many difficult problems remain even in low dimensions. Methods of low dimensional topology have proved very helpful here, and part of this project concerns a continuation of the investigator's successful application of such methods to the study of singularities of complex surfaces and the topology of plane affine curves. One issue to be addressed is explicit analytic description of singularities with given topology. The general question is notoriously difficult, but the investigator and J. Wahl have recently made a series of conjectures on realising rationally Gorenstein singularities with rational homology sphere links as explicit quotients of complete intersections and are making significant progress towards their resolution. 3-manifold topology is also being applied to the study of the global topology and analytic classification issues for two-variable polynomials. These issues are of intrinsic interest as well as having potential to contribute to the resolution of the famous Jacobian Conjecture. In a rather different direction, the investigator will study invariants of 3-dimensional manifolds that relate to number theory and algebra, namely the so-called character and eigenvalue varieties and the Bloch invariant. Connections are emerging between these invariants that will inform both topology and algebra/number theory. Algebraic Geometry, which is essentially the study of the zero sets of families of polynomials, has always been an important area of fundamental research, and is also important in such diverse applications as control theory and communications. The project will apply topological methods to algebraic geometry, to obtain results on how to realise and classify families of polynomials that lead to particular topologies. The methods of 3-dimensional topology are of particular importance to this project, and there is also feedback from algebra and number-theory to topology, as well as some interaction with theoretical physics and cosmology. The project also studies new insights that these connections bring to all these fields.