CAREER: Stacks, moduli spaces, and log geometry

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The PI studies both foundational questions and applications of stacks and logarithmic geometry. The project roughly breaks up into two parts. The first part concerns the study of moduli spaces and log geometry. Since log geometry and logarithmic structures are closely related to degenerations, log geometry plays a natural role in the study of compactifications of moduli spaces. The PI studies several problems related to log geometry and moduli spaces, and also studies a logarithmic Grothendieck-Riemann-Roch theorem, generalizing the classical theorem for schemes. The second part of the project concerns sheaves on algebraic stacks. A central theme in this part of the project is trace formulas and their generalizations to stacks. Central to all parts of the project is the close relationship between algebraic stacks and log geometry discovered in earlier work of the PI.Given a system of polynomial equations defining an algebraic variety, there are two natural ways to study the system. One approach is to study the symmetries of the equations, and a second approach is to vary the coefficients of the equations ("moduli") and "degenerate" them to a simpler (though possibly singular) system. Stacks, introduced by Artin, Deligne, Grothendieck, and Mumford in the late 1960's, are the main tool used in algebraic geometry to study spaces with additional symmetries. 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. For example, stacks play a central role in the study of Brauer groups and in the rich interaction between string theory and algebraic geometry. For the study of degenerations, a key tool is the theory of logarithmic geometry developed by Kato, Fontaine, and Illusie in the late 1980's. As mentioned above, in earlier work the PI related this theory to stacks, and the ongoing projects use this relationship to further our understanding of degenerations and the moduli of algebraic varieties.