Complexity in Cauchy-Riemann Geometry

柯西-黎曼几何的复杂性

• 批准号: 1362337
• 负责人: Jiri Lebl
• 金额: $ 14.1万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362337&HistoricalAwards=false
• 关键词: Complexity; Cauchy; Riemann; Geometry

The mathematical field of complex analysis has found diverse application in applied and pure sciences such as engineering and theoretical physics; complex analysis is also of foundational importance in certain subfields of mathematics. For example, physical phenomena can often be modeled with differential equations, and certain of these equations are best understood using techniques issuing from the study of complex numbers. Another example is provided by string theory in theoretical physics. Aspects of this model of high energy physics require a deep understanding of the geometry of complex numbers. This project is focussed on the geometry of spaces where complex and real variables interact. The spaces studied with this geometry have a certain complexity that is not yet well understood but is of great interest in the mathematical community. The PI will study this complexity and work towards building proper foundations for the field.The PI will study the CR complexity of holomorphic mappings and real submanifolds or real singular subvarieties. The key method is to study the Hermitian matrix of coefficients of the defining equations. This approach links the complexity of a real submanifold to a CR mapping to a hyperquadric, that is a hypersurface defined by a Hermitian form. Hyperquadrics are the model submanifolds in CR geometry. The greater the complexity of a submanifold, the greater the dimension that the target hyperquadric must live in, and the greater the complexity of the CR mapping involved. In particular the PI is interested in further refining the results on the singular set of Levi-flat hypersurfaces, that is sets which have in some sense the smallest CR complexity. The proposed method is to refine the Segre variety approach that was successfully used in the PI's previous work on the subject. To study the complexity of the set of the mappings involved, it is necessary to also understand the complexity of mappings between hyperquadrics themselves. The PI will work to refine the quantitative results in previous work by applying a mix of techniques from commutative algebra and analysis that has proved successful before. This work will then be applied to more general domains.

复分析的数学领域在应用科学和纯科学中有着广泛的应用，如工程和理论物理;复分析在数学的某些子领域也具有基础性的重要性。例如，物理现象通常可以用微分方程来建模，其中某些方程最好使用来自复数研究的技术来理解。另一个例子是理论物理学中的弦理论。这种高能物理模型的各个方面需要对复数几何的深刻理解。这个项目的重点是空间的几何复杂和真实的变量相互作用。 用这种几何学研究的空间具有一定的复杂性，尚未得到很好的理解，但在数学界引起了极大的兴趣。 PI将研究这种复杂性并致力于为该领域建立适当的基础。PI将研究全纯映射和真实的子流形或真实的奇异子流形的CR复杂性。 其关键方法是研究定义方程系数的厄米特矩阵。 这种方法将真实的子流形的复杂性与超二次曲面的CR映射联系起来，超二次曲面是由埃尔米特形式定义的超曲面。 超二次曲面是CR几何中的模型子流形。 子流形的复杂性越大，目标超二次曲面必须存在的维数越大，涉及的CR映射的复杂性也越大。特别是PI感兴趣的是进一步完善的结果奇异的列维平坦超曲面集，这是集在某种意义上有最小的CR复杂性。 所提出的方法是完善塞格雷品种的方法，成功地用于PI的以前的工作的主题。 为了研究所涉及的映射集合的复杂性，也有必要理解超二次曲面本身之间映射的复杂性。 PI将通过应用交换代数和分析的混合技术来改进以前工作中的定量结果，这些技术在以前已经被证明是成功的。 这项工作将被应用到更广泛的领域。