Algebraic stacks and their applications

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The research project is focused on a broad range of questions concerning algebraic stacks and their applications. The project is divided into two parts. The first part concerns foundational problems which have arisen in recent important applications of stacks such as the geometric Langland's program, the theory of moduli of stable maps, and the construction of moduli spaces for higher dimensional varieties. The second part of the program concerns applications of the theory. In particular new applications of the relationship between log geometry in the sense of Fontaine and Illusie and stacks discovered by the PI in earlier work, generalizations of the theory of twisted stable maps, as well as applications to the construction and study of moduli spaces for varieties of general type, abelian varieties, and vector bundles on curves.The notion of stack is a tool used to deal with internal symmetries of mathematical objects, as well as actions of groups. For example, when trying to classify geometric objects one is naturally forced to deal with the symmetries of the objects in question. In recent years, the theory of stacks has come to play an important role in almost every part of algebraic geometry, arithmetic geometry, and mathematical physics and a great number of exciting new applications of stacks have been found. This is not surprising considering the importance of symmetries in mathematics and other fields of science. This great interest in stacks has brought to light a number of important problems about stacks. The research project aims to broaden our understanding of both the foundational aspects of the theory of stacks and as well as the many applications.