Derived Geometry, Elliptic Cohomology, and Loop Stacks

代数拓扑学是通过代数方法研究拓扑空间的学科。首席研究员将从代数几何的角度研究各种自然发生的拓扑对象。该项目的一个基本目标是将数学概念引入计算方法。研究人员计划使用代数几何和拓扑学的最新理论进展，以开发计算工具来探索量子场论与椭圆物体难以捉摸的概念之间的关系。广义上同调理论可以说是现代代数拓扑学中最有用和最重要的工具。除了普通的上同调，例如K-理论、椭圆上同调和复配边等例子都承认通过形式群理论与代数几何有着惊人的密切联系。虽然与普通上同调和K-理论相关的形式群分别是加法和乘法的，但椭圆曲线具有更复杂的群结构（事实上，它们的相对棘手性已在真实的世界应用中得到利用，如公钥密码学）。从全局几何对象（如乘法群或椭圆曲线）中产生的形式群的优点是它们对应的上同调理论是非常高度结构化的。例如，K理论在结构上类似于普通的表示理论，而类似的“椭圆表示理论”虽然与算术几何和数学物理等不同领域有关，但仍然是一个谜。在过去的几十年里，重要的工作导致了一个普遍的椭圆上同调理论的建设，一个拓扑细化的经典理论的模形式。利用它的结构允许一个拓扑堆栈的代数几何对象的模堆栈的椭圆曲线。虽然这个对象上的函数的导出环在技术上是原始拓扑对象的椭圆上同调，但椭圆曲线不是仿射对象，这意味着从几何到代数的这一过程丢失了重要信息。更基本的结构是在仿射化之前存在于代数几何层次本身的结构，人们通过直接操纵这些对象来保留更多的概念和计算控制。一个有趣的转折是，虽然没有对椭圆上同调类的先验理解（与K理论形成鲜明对比，其中上循环对应于向量丛的形式差异），但通过解释几个关键例子中的计算，可能会深入了解这个根本重要的问题。

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.