Applications of homotopy theory to algebraic geometry and physics

同伦理论在代数几何和物理学中的应用

• 批准号: 2305373
• 负责人: Michael Hopkins
• 金额: $ 55万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305373&HistoricalAwards=false
• 关键词: Applications; homotopy; theory; algebraic; geometry

The field of homotopy theory is the study of mathematical structures that are insensitive to deformations. It is applicable whenever one is interested in studying qualitative aspects of a system, emergent properties of a statistical ensemble, or whenever there might be imprecision in the specification of the state of a system. In recent years the methods of homotopy theory have found use in fields as diverse as the physics of materials and the study of algebraic equations. This project aims to bolster these relationships with new tools from homotopy theory, and is focused on applications to classical algebraic geometry and the theory of manifolds. Broader impacts of this project include work with students and postdoctoral researchers.In more specialized terms, the work in this proposal involves two main themes of study. One describes advances in motivic homotopy theory that achieve the goals of a longstanding vision concerning an interface between complex analysis, algebraic geometry and algebraic topology. It exhibits a very close alignment between the topological construction of vector bundles over smooth algebraic varieties and the much more challenging construction of algebraic ones. Another aspect of motivic homotopy theory studied is the structure of cohomology rings of Eilenberg-MacLane spaces in that setting. The other theme offers a new approach to the "Immersion Conjecture," which is one of the most famous theorems about general manifolds, and promises to simplify and clarify the notoriously complicated existing proof. Part of this line on the Immersion Conjecture involves spaces which are potentially models for spaces BO/I_n constructed by Brown and Peterson that are understood only weakly.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.

同伦理论的领域是研究对变形不敏感的数学结构。 它适用于任何有兴趣研究系统的定性方面，统计系综的涌现特性，或者任何系统状态的规范可能不精确的时候。 近年来，同伦理论的方法在材料物理学和代数方程研究等不同领域得到了应用。 该项目旨在用同伦理论的新工具来支持这些关系，并专注于经典代数几何和流形理论的应用。这个项目的更广泛的影响包括与学生和博士后研究人员的工作。在更专业的术语，在这个建议中的工作涉及两个主要的研究主题。 一个描述了动机同伦理论的进展，实现了一个长期的愿景，关于复杂的分析，代数几何和代数拓扑之间的接口的目标。 它表现出一个非常紧密的对齐之间的拓扑结构的向量丛光滑代数簇和更具挑战性的建设代数。 动机同伦理论研究的另一个方面是Eilenberg-MacLane空间的上同调环的结构。另一个主题提供了一种新的方法来“浸入猜想”，这是关于一般流形的最著名的定理之一，并承诺简化和澄清众所周知的复杂的现有证明。 浸没猜想的一部分涉及到布朗和彼得森所构造的BO/I_n空间的潜在模型。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而被认为值得支持。