Collaborative Research: New Birational Invariants

合作研究：新的双有理不变量

• 批准号: 2245099
• 负责人: Tony Yue Yu
• 金额: $ 48.14万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2245099&HistoricalAwards=false
• 关键词: Collaborative; Research; New; Birational; Invariants

Algebraic varieties are shapes defined by solution sets of systems of polynomial equations. They appear naturally in different fields of science and engineering, including physics, cryptography, control theory, robotics, computer vision, etc,. A fundamental problem in geometry is the classification of algebraic varieties, as it helps us gain a better understanding of the structures and relations between them. The first step in classification is called birational classification, i.e. two algebraic varieties are called birational if they are equal outside some lower-dimensional loci. In this proposal, the PIs will investigate new birational invariants. These invariants will shed new light on the birational classification problem. The Principal Investigators will bring new ideas from differential equations, category theory, mirror symmetry and conformal field theory for achieving this goal. This project will provide research training opportunities for graduate students and early-career researchers.More concretely, the Principal Investigators will develop an extended theory of variations of non-commutative Hodge structures. It will be based on a new singularity theory of Landau-Ginzburg models and a non-commutative refinement of the notion of spectrum of quantum multiplication operators. These new non-commutative spectra will provide natural obstructions to rationality and equivariant rationality of Fano varieties. Additionally the PIs will investigate the connection between non-commutative spectra and R-charges of conformal field theories. This will lead to even stronger birational invariants, as well as to new unexpected bridges between geometry and other branches of mathematics, including: a new connection between Steenbrink spectra and the spectra of conformal weights in vertex operator algebras; a connection between topological invariants of 3-manifolds and non-commutative spectra of complex surfaces; semicontinuity of non-commutative spectra of algebraic varieties and the RG-flows on sigma-models with targets on such varieties; and a relation between the Kaehler-Ricci flow and the RG-flow.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.

代数簇是由多项式方程组的解集定义的形状。它们自然地出现在科学和工程的不同领域，包括物理学、密码学、控制理论、机器人、计算机视觉等。几何中的一个基本问题是代数簇的分类，因为它有助于我们更好地理解它们之间的结构和关系。分类的第一步称为双有理分类，即如果两个代数簇在某些低维轨迹之外相等，则称为双有理。在本提案中，PI 将研究新的双有理不变量。这些不变量将为双理性分类问题提供新的思路。首席研究员将从微分方程、范畴论、镜像对称和共形场论中带来新的想法来实现这一目标。该项目将为研究生和早期职业研究人员提供研究培训机会。更具体地说，主要研究人员将开发非交换霍奇结构变体的扩展理论。它将基于朗道-金兹堡模型的新奇点理论和量子乘法算子谱概念的非交换细化。这些新的非交换谱将为 Fano 簇的理性和等变理性提供天然障碍。此外，PI 还将研究非交换谱与共形场论的 R 电荷之间的联系。这将导致更强的双有理不变量，以及几何学和其他数学分支之间新的意想不到的桥梁，包括：Steenbrink 谱和顶点算子代数中的共形权谱之间的新联系； 3-流形的拓扑不变量与复杂曲面的非交换谱之间的联系；代数簇的非交换谱的半连续性以及以此类簇为目标的 sigma 模型上的 RG 流；该奖项反映了 NSF 的法定使命，并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，被认为值得支持。