FRG: Collaborative Research: New Birational Invariants

代数簇是由多项式方程组的解集定义的形状。它们自然地出现在不同的科学和工程领域，包括物理学，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，即两个代数簇称为双有理，如果它们在某些低维轨迹之外相等。在这个提议中，PI将研究新的双有理不变量。这些不变量将为双有理分类问题提供新的思路。主要研究人员将带来新的想法，从微分方程，范畴理论，镜像对称和共形场理论，以实现这一目标。本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Hodge结构变异的扩展理论。它将基于一个新的奇异性理论的朗道-金兹伯格模型和一个非交换细化的概念谱的量子乘法算子。这些新的非交换谱将为Fano簇的合理性和等变合理性提供自然的障碍。此外，PI将研究非对易谱和共形场论的R-电荷之间的联系。这将导致更强的双有理不变量，以及几何和其他数学分支之间的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-流形的拓扑不变量和复杂曲面的非交换谱之间的联系;代数簇的非交换谱的连续性和目标在这类簇上的sigma模型上的RG流以及Kaehler-Ricci流与RG-该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.