RUI: Interactions Between Homotopy Theory and Commutative Algebra

近年来，人们在同伦理论与交换代数之间发现了许多深刻的联系。这项研究在几个这样的方向上继续这一追求。首先，通过代数不变量（如同伦或共同）研究拓扑空间或结构谱，可以通过结构的模空间从代数侧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。这项研究的一部分，与Paul Goerss共同工作，将寻求扩展Hopkins-Miller-Mahowald-Goerss本体模形式的研究，该研究实现了Lubin-Tate代数asE＿ \infinity环谱并计算了它们的自同态，到更一般的Shimura变体。与此类似，本研究将与David Blanc和Mark Johnson合作，通过寻找将上同调障碍与Toda括号之类的东西联系起来的方法，试图澄清这种余切复调的变化如何控制将\Pi -代数图作为空间图实现的问题。最后，本研究将寻求扩展Grothendieck的程序分类光滑方案到更一般的方案在派生代数几何的设置。驱动机制是将余切复合体作为Kahler微分的适当类比并进行相应的推广。这将构建在代数几何环境中使用简单分辨率（最终由M. Andre和D. Quillen阐述）表征交换代数的方法。纵观理论物理学中弦理论的历史，在纯数学的几个领域——特别是数论和拓扑学——之间建立了令人惊讶的联系。在椭圆曲线的算法和模形式（在A. Wiles对费马最后定理的解析中很重要）和拓扑中的上同调理论（通过椭圆属的概念与弦理论相关）之间建立了这样的联系。从拓扑学家的观点来看，所寻求的一些相关性质可以用上同调的相关谱是否具有合适的乘法结构来表示。实现这种结构的问题可能导致有关拓扑中交换代数和同调结构相互作用方式的问题。本研究的目的是进一步了解这种相互作用的两种方式。第一种是通过模空间的概念，在给定代数数据的谱上，集体研究所有可能的乘法结构。另一种方法是利用一种称为堆栈的全局几何装置，在代数几何的同调概念的背景下共同理解这些乘法谱。本研究的目的将集中在研究与上同调理论相关的其他可能的乘法谱，这些谱来自更一般的算术形式，如自同构形式。在这个项目中，还涉及到在这个同调代数几何中描述对象的基本类型，并绘制与最近代数同调特征的联系。

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.