FRG: Collaborative Research: New Birational Invariants

FRG：合作研究：新的双理性不变量

• 批准号: 2245171
• 负责人: Ludmil Katzarkov
• 金额: $ 50万
• 依托单位: University of Miami
• 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=2245171&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; New; Birational

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.

代数品种是由多项式方程系统的溶液集定义的形状。它们自然出现在科学和工程的不同领域，包括物理，密码学，控制理论，机器人技术，计算机视觉等。几何学中的一个基本问题是代数品种的分类，因为它有助于我们更好地了解它们之间的结构及其关系。分类的第一步称为Birational分类，即，如果两个代数品种在某些较低维的基因座之外相等，则称为Birational。在此提案中，PI将调查新的异性不变。这些不变的人将为男性分类问题提供新的启示。主要研究人员将从微分方程，类别理论，镜像对称性和共形场理论中带来新的想法，以实现这一目标。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出一种扩展的非交流性Hodge结构的变化理论。它将基于Landau-Ginzburg模型的新奇异性理论，并基于对量子乘法频谱概念的非共同改进。这些新的非共同光谱将为FANO品种的合理性和模棱两可的合理性提供自然的障碍。此外，PI将研究非共同光谱与共形场理论的R型电源之间的联系。这将导致更强大的异性不变，以及几何学与其他数学分支之间的新的意外桥梁，包括：在顶点操作员代数中，Steenbrink Spectra与共同重量光谱之间的新联系； 3个manifolds的拓扑不变性与复杂表面的非共同光谱之间的联系；代数品种的非交通谱和Sigma模型上的RG-FLOWS的半持续性，具有此类品种的靶标；这一奖项反映了NSF的法定任务，并通过基金会的知识分子优点和更广泛的影响标准，这反映了NSF的法定任务，这反映了NSF的法定任务。