Studies in Moduli Theory and Birational Geometry

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The investigator and his colleagues study problems in moduli theory and birational geometry. In moduli theory they study questions related to the moduli stacks of twisted stable maps into a tame Deligne-Mumford stack. Questions addressed include constructions towards Gromov-Witten invariants of stacks, generalizations of twisted stable maps to higher dimensions and to wild stacks, and applications to the theory of admissible covers, level structures, spin structures and higher dimensional varieties. The investigator and colleagues will also study problems in birational geometry: the strong factorization conjecture, which says that a birational map of smooth projective varieties should factor as a sequence of smooth blowings up followed by a sequence of smooth blowings down; and the toroidalization conjecture, which says that, given a surjective morphism of complex projective manifolds, there is a way to blow up both source and target along smooth centers so that the resulting map is a toroidal morphism.The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. While algebraic geometry has contributed applications in coding, industrial control, and computation, the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as components of the fine structure of our universe. This is especially true with the first topic, moduli theory. This theory studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety, called a moduli space, in its own right. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not a just a metaphor but a rigorous and quite useful fact. Sometimes a collection of algebraic varieties manifests itself as a slightly more general object, called a stack, rather than a variety. Such stacks are a central object of study of this project. The other topic studied in this project is birational geometry, which is devoted to a certain abstract relationship, called birational equivalence, among algebraic varieties, which lies at the foundation of algebraic geometry. A goal of this project is to understand the connection between this abstract notion, and a much more concrete relationship given by an algebraic "surgery" procedure called "blowing up".