RUI: Interactions Between Homotopy Theory and Commutative Algebra

近年来，同伦理论与交换代数之间有着许多深刻的联系。这项研究在几个这样的方向上继续这种追求。首先，通过代数不变量（如同伦或（上）同调）研究拓扑空间或结构谱，可以通过结构的模空间从代数方面的全局视角进行研究。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。本研究的一部分是与Paul Goerss合作，将Hopkins-Miller-Mahowald-Goerss关于拓扑模形式的研究从把Lubin-Tate代数实现为E_\无穷环谱并计算它们的自同态扩展到更一般的Shimura簇。同样，本研究将与大卫布兰克和马克约翰逊共同努力，通过寻找将上同调障碍物连接到户田括号之类的东西的方法，来澄清这种余切复形的变化如何控制将π-代数的图实现为空间图的问题。 最后，本研究将寻求扩展Grothendieck的程序分类光滑计划，以更一般的计划在设置派生代数几何.驱动机制将采取余切复是适当的模拟的Kahler微分和推广相应的.这将构成用单纯分解刻画交换代数的方法（最终由M. Andre和D.在理论物理学中的弦理论的整个历史中，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。在椭圆曲线的算术和模形式之间已经建立了这样一种联系，这在A.怀尔斯解决费马的最后定理，和上同调理论的拓扑-连接相关的弦理论通过概念的椭圆属。 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以引出关于交换代数和拓扑中同伦结构相互作用的方式的问题。本研究的目的是从两个方面进一步了解这种相互作用。第一种方法是通过模空间的概念，在给定的代数数据下，研究谱上所有可能的乘法结构。另一种方法利用一个全球性的几何设备，称为堆栈，理解这些乘法谱集体在代数几何的同伦概念的上下文中。本研究的目的，然后将集中在研究其他可能的乘法谱相关的上同调理论arisefrom更一般的算术形式，如自守形式。参与这个项目的目的也是为了表征基本类型的对象在这个同伦代数几何和绘图连接到最近的同调表征代数。

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.