RUI: Interactions Between Homotopy Theory and Commutative Algebra

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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.