Abramovich will continue studying problems in moduli theory, including several aspects of Gromov--Witten theory, in particular (1) generalized moduli of relative stable maps; (2) comparison of virtual fundamental classes on moduli of relative and orbifold stable maps; and (3) further study of moduli of very twisted curves and their maps. Abramovich will continue studying problems in birational geometry, in particular (1) Campana constellations, their maps and arithmetic properties; (2) Birational geometry of and moduli of higher dimensional orbifolds, and their application in the compactification of moduli of higher dimensional varieties.
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.
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. Algebraic geometry has significant applications in coding, industrial control, and computation. But the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as a piece 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 just a metaphor but a rigorous and quite useful fact. 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. During this project, Abramovich worked with his collaborators and students on a number of problems in algebraic geometry. Major effort was put into generalizing the moduli the spaces of relative stable maps to the orbifold and logarithmic cases. In practical terms, this work enhances computational tools for calculating those important numbers, called Gromov-Witten invariants, that arise in mathematical physics and enumerative geometry. When developing such an enhancement, it is important to verify that its results are compatible with earlier computations. This turns out to require significant work, which was also achieved during this grant period. In addition, two project in birational geometry aimed at a better understanding of the fine structure of birational equivalence. Abramovich's work had broader impact primarily in training students and young researchers. Abramovich's trainees were involved in the project in all its facets, with major contributions to moduli of logarithmic stable maps, logarithmic geometry in general, and the structure of moduli spaces. Three PhD theses were completed and three started with funding of this project. In addition, Abramovich was involved in organizing a number of conferences and workshops where lectures were aimed for trainees. Especially successful was an MRC workshop on Birational Geometry and Moduli. Finally, a chapter on Logarithmic Geometry and Moduli completed and published during this grant period will hopefully be a useful introductory text to the subject.
