Applications of homotopy theory to algebraic geometry and physics

同伦理论的领域是研究对变形不敏感的数学结构。当人们对研究系统的定性方面、统计系综的涌现性质感兴趣时，或者在系统状态的说明中可能存在不精确的时候，它都是适用的。近年来，同伦理论的方法在材料物理和代数方程研究等领域得到了广泛的应用。这个项目旨在用同伦理论中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。用更专业的术语来说，这个提案中的工作涉及两个主要研究主题。其中一个描述了动机同伦理论的进展，这些进展实现了关于复分析、代数几何和代数拓扑之间的接口的长期愿景的目标。它显示了光滑代数簇上向量丛的拓扑构造与更具挑战性的代数丛的构造之间的非常密切的一致。动机同伦理论研究的另一个方面是Eilenberg-MacLane空间的上同调环在该环境下的结构。另一个主题提供了一种新的方法来解释“沉浸猜想”，它是关于一般流形的最著名的定理之一，并承诺简化和澄清臭名昭著的复杂的现有证明。沉浸猜想的这条线的一部分涉及到一些空间，这些空间是布朗和彼得森建造的BO/I_n空间的潜在模型，人们对这些空间的理解很弱。这一裁决反映了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，被认为值得支持。

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.