Equivariant birational geometry

等变双有理几何

• 批准号: 2301983
• 负责人: Yuri Tschinkel
• 金额: $ 32万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301983&HistoricalAwards=false
• 关键词: Equivariant; birational; geometry

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.

该奖项的重点是许多变量中非线性代数方程式系统的研究。特别令人感兴趣的是有关解决方案空间的解决方案和隐藏对称性的简单参数化的问题。除了对代数几何形状领域的内在重要性外，这项研究还对理论计算机科学有潜在的应用，并因此对密码学，信息处理和大型数据结构的管理中的问题进行了应用。此外，它可能适用于理论物理学，非线性系统起着重要作用。此外，它刺激了该系统离散不变的有效算法的开发，并为下一代的几何体提供了许多具体的问题和示例。 PI将继续培训研究生，并从事外展活动，将数学意识带给广泛的受众。具体而言，拟议的研究将以新颖的方式结合算术和几何见解，以显着提高我们对小维度中有限群体对代数品种的有限群体作用的线性性和稳定性的理解理性和稳定的合理性的理解。理性构建体通常涉及纤维化的研究，从而研究基础功能场（一个未透明场）的合理性研究。反过来，对非公开磁场上的几何形状与模棱两可的异性几何形状紧密相关，因为绝对Galois组的作用与自动形态的作用之间存在很强的相似之处。探索几何，算术和群体理论之间的这些联系是该提议的主要作用。长期目标之一是在多达三个方面的理性品种上进行此类行动的完整分类。另一个目标是在有限群，体积形式和其他结构的作用下探索最近发现的不变几何不变的范围。第三个目标是在模棱两可的和Orbifold的背景下发展普遍扭转的理论，并将其应用于产生新的稳定典范的例子。该奖项反映了NSF的法定使命，并被认为是值得通过基金会的知识分子来评估的。和更广泛的影响审查标准。