Equivariant birational geometry

该奖项的重点是在许多变量的非线性代数方程系统的研究。特别感兴趣的是简单的参数化的解决方案和隐藏的对称性，这样的解决方案空间的存在有关的问题。除了固有的重要性，代数几何领域，这项研究有潜在的应用理论计算机科学，并因此在密码学，信息处理和管理的大型数据结构的问题。此外，它是潜在的适用于理论物理，其中非线性系统发挥重要作用。此外，它刺激了发展的有效算法计算的离散不变量的系统，并提供了许多具体的问题和例子，为下一代的几何。PI将继续培训研究生，并从事推广活动，使广大观众的数学意识。具体来说，拟议中的研究将联合收割机以新颖的方式结合算术和几何的见解，以显着提高我们的理解合理性和稳定的合理性在小尺寸，以及线性化和稳定的线性化有限群的行动代数簇。合理性建构通常涉及纤维化的研究，因此涉及基的函数域（一个非封闭域）上的合理性研究。反过来，非闭域上的几何与等变双有理几何紧密相连，因为绝对伽罗瓦群的作用与自同构的作用之间有很强的相似性。探索几何、算术和群论之间的这些联系是这个提议的主要推动力。长期目标之一是获得一个完整的分类，这些行动的理性品种的维度多达三个。另一个目标是探索最近发现的不变量在双有理几何中的适用范围，在有限群，体积形式和其他结构的作用下。第三个目标是在等变和orbifold的背景下发展通用torsors的理论，并将其应用于产生稳定的birationalities的新例子。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This award is focused on the study of systems of nonlinear algebraic equations in many variables. Of particular interest are problems concerning the existence of simple parametrizations of solutions and of hidden symmetries of such solution spaces. Apart from intrinsic importance to the field of algebraic geometry, this research has potential applications to theoretical computer science, and as a consequence to problems in cryptography, information processing, and management of large data structures. Furthermore, it is potentially applicable to theoretical physics, where nonlinear systems play an important role. Moreover, it stimulates the development of efficient algorithms for the computation of discrete invariants of such systems, and provides many concrete problems and examples for the next generation of geometers. The PI will continue to train graduate students and engage in outreach activities bringing mathematical awareness to a broad audience. Specifically, the proposed research would combine in novel ways arithmetic and geometric insights to significantly advance our understanding of rationality and stable rationality in small dimensions, as well as linearizability and stable linearizability of actions of finite groups on algebraic varieties. Rationality constructions often involve the study of fibrations and thus the study of rationality over the function field of the base, a nonclosed field. In turn, geometry over nonclosed fields is tightly linked to equivariant birational geometry, as there are strong parallels between the action of the absolute Galois group and the action of automorphisms. Exploring these connections between geometry, arithmetic, and group theory is a major thrust of this proposal. One of the long-term goals is to obtain a full classification of such actions on rational varieties in dimensions up to three. Another goal is to explore the range of applicability of recently discovered invariants in birational geometry, in presence of actions of finite groups, volume forms, and other structures. A third goal is to develop the theory of universal torsors in the equivariant and orbifold context, and to apply it to produce new examples of stable birationalities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.