Mathematical Sciences: Topology of Real Algebraic Sets and 4-Manifolds

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This research project consists of two main parts, concerning (1) the topology of real algebraic sets, and (2) the topology of smooth 4-manifolds. The first part will build on the investigator's success (together with Henry King of the University of Maryland) in settling an old conjecture of John Nash. The second part will focus on the construction of fake smooth structures on 4-manifolds, particularly their detection by means of such new invariants as Floer homology and the still newer invariants of Turaev-Reshetikhin-Witten coming from quantum groups. Both parts of this project concern themselves with very natural objects which are ubiquitous in mathematics and mathematical physics. Four-manifolds are cosmologically important, since we live in a manifold of three space dimensions plus one time dimension.