Mathematical Sciences: Presidential Young Investigator

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The main thrust of this research is to study three dimensional algebraic varieties (threefolds). The Principal Investigator's past work will be continued on questions on algebraic fibre spaces together with the applications to the classification of threefolds. The current work also concerns the structure of birational mappings between threefolds, especially the codimension-two algebraic surgerys called "flips" and "flops" which are now understood to be a central feature of birational maps in dimension three. A potential application of all of this is to produce a geometrically meaningful compactification of moduli spaces of surfaces via the study of the fibration of the threefold obtained from a one-parameter family of surfaces, especially curvature properties of that fibration.