RUI: Interactions Between Homotopy Theory and Commutative Algebra

近年来，人们发现同伦论和交换代数之间有许多深刻的联系。这项研究在几个这样的方向上继续了这一追求。首先，通过代数不变量（例如同伦或（共）同调）对拓扑空间或结构化谱的研究，可以通过结构的模空间从代数方面的全局角度进行研究。研究这种模空间的一个很好的不变量是与给定代数相关的余切复形。这项研究的一部分是与 Paul Goerss 合作，希望将 Hopkins-Miller-Mahowald-Goerss 本体拓扑模形式的研究扩展到更一般的 Shimura 簇，该形式将 Lubin-Tate 代数实现为 E_\无穷环谱并计算其自同态。同样，本研究将与 David Blanc 和 Mark Johnson 合作，通过寻找将同调障碍物与 Toda 括号之类的东西连接起来的方法，来阐明这种余切复形的变体如何控制将 \Pi 代数图实现为空间图的问题。 最后，本研究将寻求将格洛腾迪克的平滑方案分类程序扩展到派生代数几何设置中更一般的方案。驱动机制是将余切复形作为卡勒微分的正确模拟并进行相应的推广。这将构建在代数几何环境中使用单纯解析（最终由 M. Andre 和 D. Quillen 阐明）来表征交换代数的方法。纵观理论物理学中弦理论的历史，纯数学的几个领域——特别是数论和拓扑——之间已经建立了令人惊讶的联系。在椭圆曲线和模形式的算术之间建立了这样一种联系，这对于费马大定理的 A. Wiles 解析和拓扑中的上同调理论很重要 - 这种联系通过椭圆属的概念与弦理论相关。 从拓扑学家的角度来看，所寻求的一些相关性质可以用上同调的关联谱是否具有合适的乘法结构来表达。实现这种结构的问题可能会导致关于拓扑中交换代数和同伦结构相互作用的方式的问题。这项研究的目标是通过两种方式进一步了解这种相互作用。第一个是通过模空间的概念，用规定的代数数据集体研究谱上所有可能的乘法结构。另一种方法利用称为堆栈的全局几何设备，在代数几何的同伦概念的背景下共同理解这些乘法谱。这项研究的目的将集中于研究与上同调理论相关的其他可能的乘法谱，这些乘法谱源自更一般的算术形式，例如自守形式。该项目的目的还在于表征同伦代数几何中对象的基本类型，并与最近的代数同调表征建立联系。

In recent years, there have been many deep connections found betweenhomotopy theory and commutative algebra. This research continues thispursuit in several such directions. First, the study of topological spacesor structured spectra through algebraic invariants, such as homotopy or(co-)homology, can be studied from the global perspective on thealgebraic side via a moduli space of structures. A fine invariant forstudying such a moduli space is the cotangent complex associated to thegiven algebra. One part of this research, in joint work with Paul Goerss,will look to extend the study of Hopkins-Miller-Mahowald-Goerss ontopological modular forms from, which realized Lubin-Tate algebras asE_\infinity ring spectra and calculated their endomorphisms, to moregeneral Shimura varieties. In a similar vein, this research willseek to clarify, in joint work with David Blanc and Mark Johnson, how avariation of this cotangent complex controls the question of realizing a diagram of\Pi-algebras as a diagram of spaces, by looking for ways of connectingcohomological obstructions to things like Toda brackets. Finally, thisresearch will seek to extend Grothendieck's program for classifying smoothschemes to more general schemes in the setting of derived algebraic geometry.The driving mechanism will be to take the cotangent complex to be the properanalogue of the Kahler differentials and generalize accordingly. This wouldframe the approach of characterizing commutative algebras using thesimplicial resolutions (as ultimately articulated by M. Andre and D. Quillen) inan algebraic geometric setting.Throughout the history of string theory in theoretical physics, surprising connectionshave been made between several areas of pure mathematics - in particular, numbertheory and topology. One such connection has been made between the arithmetic ofelliptic curves and modular forms, important in A. Wiles resolution of Fermat's LastTheorem, and cohomology theories in topology - the connectionbeing relevant to string theory via the concept of elliptic genera. From the topologist'sstandpoint, some of the relevant properties being sought can be expressed in terms ofwhether the cohomology's associated spectrum possesses a suitable multiplicative structure.The question of realizing such structures can lead to questions regarding ways in whichcommutative algebras and homotopical structures in topology interact. The goal of thisresearch is to further understand such interactions in two ways. The first is to collectivelystudy all possible multiplicative structures on a spectrum with a prescribed algebraic datavia the concept of a moduli space. The other approach utilizes a global geometric device,called a stack, to understand these multiplicative spectra collectively in the context of amore homotopical notion of algebraic geometry. The aim of this research will then focuson studying other possible multiplicative spectra associated to cohomology theories which arisefrom more general arithmetic forms, such as automorphic forms. Involved in this projectis also the aim to characterize basic types of objects in this homotopical algebraic geometryand drawing connections to recent homological characterizations of algebras.