Birational Geometry and Algebraic Dynamics

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This project focuses on the role of symmetry in algebraic geometry, an area of mathematics concerned with the study of the solution sets of systems of polynomial equations. This topic has long played a central role in mathematics, and it lies at the intersection of many fields of mathematics: the problems to be studied relate to wide variety of areas of mathematics, including topology, algebra, and number theory. The project will investigate a variety of situations in which unexpected symmetries of equations make it possible to explore otherwise difficult-to-understand geometric phenomena and to test a wide variety of geometric conjectures in new settings.The particular emphasis of this project will be on the connections between algebraic geometry and algebraic dynamics, the study of iterated rational maps, and the results will include applications of each field to the other. On one hand, the investigator will work to understand how ideas from dynamics can be used to reveal a variety of otherwise obscure phenomena in higher-dimensional algebraic geometry, and he aims to demonstrate that varieties admitting dynamically interesting self-maps provide a fertile source of examples (and counterexamples). Conversely, the investigator will also work to understand how the methods of algebraic geometry (and the minimal model program in particular) can be employed to understand the geometric properties of algebraic varieties with very large groups of symmetries.