Birational Geometry and Algebraic Dynamics

这个项目的重点是对称性在代数几何中的作用，这是一个与研究多项式方程组的解集有关的数学领域。 这个主题长期以来在数学中发挥着核心作用，它位于许多数学领域的交叉点：要研究的问题涉及到各种各样的数学领域，包括拓扑学，代数和数论。 该项目将调查各种情况下，意外对称的方程，使人们有可能探索，否则难以理解的几何现象和测试各种各样的几何结构在新的设置。该项目的特别重点将是代数几何和代数动力学之间的联系，研究迭代有理映射，结果将包括每个领域对另一个领域的应用。 一方面，研究人员将努力了解如何从动力学的想法可以用来揭示各种其他模糊的现象在高维代数几何，他的目标是证明品种承认动态有趣的自映射提供了一个肥沃的来源的例子（和反例）。 相反，研究人员也将努力了解如何代数几何的方法（特别是最小模型程序）可以用来理解具有非常大的对称群的代数簇的几何性质。

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.