Degenerations and moduli

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Valery Alexeev will continue his work on a wide variety of projects in algebraic geometry, many of them revolving around the construction and study of the moduli spaces of stable pairs and maps, the higher-dimensional generalizations of Deligne-Mumford-Knudson-Kontsevich's moduli spaces of stable curves and maps, as well as questions related to the Minimal Model Program, and to varieties with group action. The grant will contribute to training of graduate students and postdocs, and will support an active program in algebraic geometry at the University of Georgia.Algebraic geometry arose in the ancient times from the study of polynomial equations. It now employs a dazzling array of sophisticated tools and methods, and has connections and applications to most other fields of mathematics, as well as to physics and engineering. The study of moduli spaces is at the heart of algebraic geometry and answers questions such as: What is the totality of algebraic objects of a given type? Does it have a special structure? What happens to the objects when they "degenerate", for example when they are stretched to their physical limits?