In this project, the investigator will study questions in algebraic geometry that require the use of algebraic stacks. One aspect of this research will study how in the context of the study of Gromov-Witten invariants, the somewhat exotic structure of Deligne-Mumford stack can be related to the more familiar resolutions of singularities which have long been a staple of algebraic geometry. A second project will focus on bringing to bear the machinery that has been developed in the theory of algebraic stacks on the study of Donaldson type invariants.
Algebraic geometry is among the oldest branches of mathematics and is concerned with studying the solutions of systems of polynomial equations. Modern applications of this theory often place a strong emphasis on understanding the symmetries of such systems. The most powerful framework for understanding geometry in conjunction with symmetries is the theory of algebraic stacks. This point of view has acquired extra significance due to a surprising connection to theoretical physics, where attempts to unify quantum field theory and general relativity have led to a host of interesting mathematical problems concerning the symmetries of various geometric structures. This interaction provides a rich source of problems where the language of stacks, which has a reputation for being excessively abstract and technical, can be used to study very concrete questions about simple geometric objects.
