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

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The investigator will continue his longstanding project with Henry King of the University of Maryland to formulate topological characterizations of manifolds which can support the structure of real algebraic varieties. The subgroup of algebraic cycles within the homology figures prominently in this work, and surprisingly, this year, they have shown that certain obstructions vanish if one allows singular varieties. A second direction for the investigator's research involves gauge theory and its role in the study of 4-manifolds. A final direction concerns the question of Nash: do all closed smooth manifolds have rational structure?