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。它们自然而然地出现在科学和工程的不同领域，包括物理学、密码学、控制论、机器人学、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了解它们的结构和它们之间的关系。分类的第一步称为二元分类，即如果两个代数簇在某些低维轨迹外相等，则称它们为二元分类。在这项提议中，PI将研究新的出生不变量。这些不变量将为两国分类问题提供新的线索。首席调查者将从微分方程式、范畴理论、镜像对称和保形场理论中带来新的想法来实现这一目标。这个项目将为研究生和早期研究人员提供研究培训机会。更具体地说，首席研究人员将发展非对易Hodge结构变化的扩展理论。它将基于Landau-Ginzburg模型的新奇点理论和对量子乘法算符频谱概念的非对易精化。这些新的非对易谱将为Fano变种的合理性和等变合理性提供自然障碍。此外，PI还将研究共形场理论中非对易光谱和R电荷之间的联系。这将导致更强的双子不变量，以及几何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-流形的拓扑不变量和复杂曲面的非对易谱之间的联系；代数簇的非对易谱和具有此类簇上目标的sigma模型上的RG流的半连续性；以及Kaehler-Ricci流和RG流之间的关系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，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.