Studies in moduli theory and birational geometry

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Abramovich will continue studying problems inmoduli theory, in particular (1) the moduli stacks of twisted stablemaps, reductions of moduli of principal bundles in characteristic p,Gromov-Witten theory of stacks, and (2) moduli spaces of Bridgeland-Douglas semistable objects in the derived category of a variety. Abramovich will continue studying problems in birational geometry, in particular the strong factorization conjecture and the toroidalization conjecture. The area of study of this project lies within algebraic geometry, thebranch of mathematics devoted to geometric shapes called algebraicvarieties, defined by polynomial equations. While algebraic geometryhas contributed applications in coding, industrial control, andcomputation, the topics of this project are more closely related toapplications in theoretical physics, where physicists consideralgebraic varieties as components of the fine structure of ouruniverse. This is especially true with the first topic, modulitheory. This theory studies a remarkable phenomenon in which thecollection of all algebraic varieties of the same type is oftenmanifested as an algebraic variety, called a moduli space, in its ownright. Thus in algebraic geometry, the metaphor of thinking about acommunity of "organisms" as itself being an "organism" is not a just ametaphor but a rigorous and quite useful fact. Sometimes a collectionof algebraic varieties manifests itself as a slightly more generalobject, called a stack, rather than a variety. Such stacks are acentral object of study of this project. The other topic studied inthis project is birational geometry, which is devoted to a certainabstract relationship, called birational equivalence, among algebraicvarieties, which lies at the foundation of algebraic geometry.