Elliptic Cohomology, Geometry, and Physics

椭圆上同调、几何和物理

• 批准号: 2205835
• 负责人: Daniel Berwick Evans
• 金额: $ 24.07万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205835&HistoricalAwards=false
• 关键词: Elliptic; Cohomology; Geometry; Physics

In the last forty years, ideas from theoretical physics have frequently driven important advances in pure mathematics. The story often begins with calculations in seemingly unrelated physical and mathematical theories that by coincidence give the same answer. Careful investigations then reveal a deep structure linking the disciplines. Such discoveries open new vistas and prompt novel lines of research. A nascent example of this phenomenon relates a certain type of quantum field theory to an object in algebraic topology called elliptic cohomology. The deep structure explaining the relationship between these areas remains elusive. However, the expectation is that the answer will illuminate important objects in several branches of mathematics and physics. In particular, a resolution should provide insight into foundational questions in string theory, while also revealing the geometric meaning of elliptic cohomology. The projects will develop new tools to study this problem. The award supports graduate students working with the PI whose research will contribute to this area. The PI will continue mentoring and advising activities along with his involvement in mathematics education for incarcerated people through the Education Justice Project in Illinois.The main goal of the work is to leverage 2-equivariant elliptic cohomology to reduce the key questions to the study of a few structures in geometry and quantum field theory. The methods utilize a filtration on field theories mimicking the chromatic filtration, allowing one to break the larger problem into smaller and more manageable pieces. Success in lower stages of the filtration already offers several long-anticipated geometric applications of elliptic cohomology, as well as new deformation invariants of quantum field theories. Prior work shows that the first step of the filtration affords a geometric model for equivariant elliptic cohomology over the complex numbers. The proposed research builds upon this existing model and moves beyond characteristic zero.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.

在过去的40年里，理论物理学的思想经常推动纯数学的重要进展。故事往往始于看似无关的物理和数学理论的计算，巧合地给出了相同的答案。仔细的研究揭示了一个连接学科的深层结构。这些发现开辟了新的前景，并促进了新的研究方向。这种现象的一个新生的例子是将某种类型的量子场论与代数拓扑学中称为椭圆上同调的对象联系起来。解释这些区域之间关系的深层结构仍然难以捉摸。然而，人们期望答案能够阐明数学和物理学多个分支中的重要对象。特别是，一个解决方案应该提供对弦理论基础问题的洞察，同时也揭示了椭圆上同调的几何意义。这些项目将开发新的工具来研究这一问题。该奖项支持与PI合作的研究生，他们的研究将有助于这一领域。PI将继续指导和咨询活动沿着他参与数学教育的监禁的人通过教育正义项目在伊利诺伊州。工作的主要目标是利用2-等变椭圆上同调减少的关键问题的研究的几个结构的几何和量子场论。该方法利用场理论的过滤，模仿色过滤，允许将较大的问题分解为较小的和更易于管理的部分。在较低阶段的过滤的成功已经提供了几个期待已久的几何应用椭圆上同调，以及新的变形不变量的量子场论。先前的工作表明，过滤的第一步提供了一个几何模型的等变椭圆上同调的复数。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。