RUI: Interactions Between Homotopy Theory and Commutative Algebra

近年来，在同伦理论和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着这一追求。首先，通过代数不变量来研究拓扑空间或结构谱，例如同伦或(余)同调，可以通过结构的模空间从代数面的全局角度来研究。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，将与Paul Goerss合作，将Hopkins-Miller-Mahowald-Goerss关于拓扑模形式的研究从实现Lubin-Tate代数ASE_\无限环谱并计算其自同态的拓扑模形式扩展到更一般的Shimura簇。同样，这项研究将试图通过与David Blanc和Mark Johnson的合作，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，来澄清这个余切复形的变化如何控制将\PI-代数的图实现为空间图的问题。最后，本研究将寻求将Grothendieck的光滑格式分类程序扩展到更一般的导出代数几何设置下的光滑格式，其驱动机制是将余切复形作为Kahler微分的性质类似并相应地进行推广。这将为在代数几何环境中使用单纯分解(如M.Andre和D.Quillen最终阐明的)刻画交换代数的方法奠定基础。纵观理论物理中弦理论的历史，在纯数学的几个领域之间已经建立了令人惊讶的联系--特别是数论和拓扑学。在费马最后定理的A.Wiles分解中重要的椭圆曲线和模形式的算法与拓扑学中的上同调理论之间已经建立了一种这样的联系-这种联系通过椭圆属的概念与弦理论相关。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伴随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关于拓扑中的交换代数和同伦结构如何相互作用的问题。这项研究的目标是从两个方面进一步了解这种相互作用。第一种是集体地研究一个谱上所有可能的乘法结构，它具有一个指定的代数数据，即模空间的概念。另一种方法利用一种称为堆栈的全局几何装置，在代数几何的另一种同伦概念的背景下共同理解这些乘性光谱。这项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，这些上同调理论产生于更一般的算术形式，如自同构形式。这个项目的目的也是刻画这个同伦代数几何中对象的基本类型，并绘制与最近代数的同调刻画的联系。

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.