Elliptic Cohomology, Geometry, and Physics

在过去的四十年里，理论物理学的思想经常推动纯数学的重要进展。故事通常从看似无关的物理和数学理论的计算开始，这些计算巧合地给出了相同的答案。仔细的调查揭示了这些学科之间的深层联系。这些发现开辟了新的前景，促进了新的研究方向。这种现象的一个新生的例子将某种类型的量子场论与代数拓扑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，期望的是，答案将阐明数学和物理的几个分支的重要对象。特别是，一个决议应该提供洞察弦理论的基本问题，同时也揭示椭圆上同的几何意义。这些项目将开发新的工具来研究这个问题。该奖项支持与PI一起工作的研究生，他们的研究将为这一领域做出贡献。PI将继续指导和建议活动，同时他还通过伊利诺斯州的教育正义项目参与在押人员的数学教育。本工作的主要目标是利用2等变椭圆上同调，将关键问题简化为几何和量子场论中一些结构的研究。这些方法利用场理论的过滤，模拟色过滤，允许人们将较大的问题分解成更小、更易于管理的部分。过滤较低阶段的成功已经提供了几个期待已久的椭圆上同调的几何应用，以及量子场论的新变形不变量。先前的工作表明，滤波的第一步提供了复数上的等变椭圆上同的几何模型。提出的研究建立在这个现有的模型之上，并超越了特征零。该奖项反映了美国国家科学基金会的法定使命，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。

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.