Birational Geometry and Algebraic Dynamics

这个项目主要关注对称性在代数几何中的作用，代数几何是一个研究多项式方程组的解集的数学领域。这个主题长期以来在数学中扮演着核心角色，它位于数学的许多领域的交叉点：要研究的问题涉及数学的各个领域，包括拓扑学、代数和数论。这个项目将调查各种情况，在这些情况下，方程的意外对称使探索原本难以理解的几何现象成为可能，并在新的环境中测试各种几何猜想。本项目的重点将是代数几何和代数动力学之间的联系，对迭代有理映射的研究，结果将包括每个领域在另一个领域的应用。一方面，研究人员将致力于理解如何使用动力学的想法来揭示高维代数几何中各种原本鲜为人知的现象，他的目标是证明，允许动态有趣的自我映射的种类提供了丰富的例子(和反例)。相反，研究人员还将致力于了解如何使用代数几何的方法(特别是最小模型程序)来理解具有非常大的对称性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.