Derived Geometry, Elliptic Cohomology, and Loop Stacks

导出几何、椭圆上同调和循环堆栈

• 批准号: 1714273
• 负责人: Jeremy Miller
• 金额: $ 18.54万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-06-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714273&HistoricalAwards=false
• 关键词: Derived; Geometry; Elliptic; Cohomology; Loop

Algebraic topology is the study of topological spaces via algebraic methods. The principal investigator will study various naturally-occurring topological objects from the perspective of algebraic geometry. A fundamental goal of the project is to place mathematical concepts in reach of computational methods. The investigator plans to use recent theoretical advances in algebraic geometry and topology in order to develop calculational tools to explore a relationship between quantum field theory and the elusive notion of elliptic object. Generalized cohomology theories are arguably the most useful and important tool in modern algebraic topology. In addition to ordinary cohomology, examples such as K-theory, elliptic cohomology, and complex cobordism admit surprisingly close connections to algebraic geometry via the theory of formal groups. While the formal groups associated to ordinary cohomology and K-theory are additive and multiplicative, respectively, elliptic curves carry much more complicated group structures (in fact, their relative intractability has been exploited in real world applications such as public key cryptography). The advantage of formal groups that arise from global geometric objects, such as the multiplicative group or elliptic curves, is that their corresponding cohomology theories are very highly structured. For instance, K-theory is structurally similar to ordinary representation theory, whereas the analogous "elliptic representation theory," while related to diverse fields such as arithmetic geometry and mathematical physics, remains a mystery. Important work over the past few decades has led to a construction of a universal elliptic cohomology theory, a topological refinement of the classical theory of modular forms. Exploiting its structure allows one to associate to a topological stack an algebro-geometric object over the moduli stack of elliptic curves. While the derived ring of functions on this object is technically the elliptic cohomology of the original topological object, elliptic curves are not affine objects, meaning that this passage from geometry to algebra loses significant information. The more fundamental structure is that which exists on the algebro-geometric level itself, before affinization, and one retains considerably more conceptual and calculational control by manipulating these objects directly. An interesting twist is that, while there is no a priori understanding of elliptic cohomology classes (in stark contrast to K-theory, where cocycles correspond to formal differences of vector bundles), it may be possible to gain insight into this fundamentally important problem by interpreting calculations in several key examples.

代数拓扑是通过代数方法研究拓扑空间的研究。主要研究者将从代数几何形状的角度研究各种天然拓扑对象。该项目的基本目标是将数学概念置于计算方法的范围内。研究者计划使用代数几何学和拓扑的最新理论进步，以开发计算工具，以探索量子场理论与椭圆对象难以捉摸的概念之间的关系。广义的同胞理论可以说是现代代数拓扑中最有用，最重要的工具。除了普通的共同学之外，诸如K理论，椭圆形的共同体学和复杂的恢复诸如诸如通过正式群体理论与代数几何的联系令人惊讶地接近与代数几何的联系。尽管与普通同胞和K理论相关的正式群体分别具有加性和乘法性，但椭圆形曲线具有更复杂的组结构（实际上，它们的相对可行性已在现实世界中的应用中（如公共密钥密码学）中得到了利用）。由全局几何对象（例如乘法群或椭圆曲线）产生的形式组的优点在于它们相应的共同体理论非常结构化。例如，K理论在结构上与普通表示理论相似，而类似的“椭圆形表示理论”，尽管与诸如算术几何和数学物理学之类的不同领域有关，但仍然是一个谜。在过去的几十年中，重要的工作导致了通用椭圆的共同体学理论的构建，这是对模块化形式的经典理论的拓扑改进。利用其结构使人们可以在椭圆曲线的模量堆栈上与拓扑堆栈相关联。从技术上讲，虽然该对象上的函数环是原始拓扑对象的椭圆形共同体，但椭圆曲线不是仿射对象，这意味着从几何形状到代数的段落失去了重要信息。更基本的结构是在亲密关系之前本身存在于代数几何水平上的结构，并且通过直接操纵这些对象来保留更大的概念和计算控制。一个有趣的转折是，尽管对椭圆形的共同体学类别没有先验的理解（与K理论形成鲜明对比的是，共生对应于矢量捆绑包的形式差异），但通过在几个关键示例中解释计算，可能有可能深入了解这一根本重要的问题。

项目成果

K-theoretic obstructions to bounded t-structures

• DOI: 10.1007/s00222-018-00847-0
• 发表时间: 2016-10
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Benjamin Antieau;David Gepner;J. Heller

∞-Operads as Analytic Monads

-作为分析单子的操作

• DOI: 10.1093/imrn/rnaa332
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Gepner, David;Haugseng, Rune;Kock, Joachim
• 通讯作者: Kock, Joachim

Brauer groups and Galois cohomology of commutative ring spectra

交换环谱的布劳尔群和伽罗瓦上同调

• DOI: 10.1112/s0010437x21007065
• 发表时间: 2021
• 期刊: Compositio Mathematica
• 影响因子: 1.8
• 作者: Gepner, David;Lawson, Tyler
• 通讯作者: Lawson, Tyler