In recent years, there have been many deep connections found between homotopy theory and commutative algebra. This research continues this pursuit in several such directions. First, the study of topological spaces or structured spectra through algebraic invariants, such as homotopy or (co-)homology, can be studied from the global perspective on the algebraic side via a moduli space of structures. A fine invariant for studying such a moduli space is the cotangent complex associated to the given algebra. One part of this research, in joint work with Paul Goerss, will look to extend the study of Hopkins-Miller-Mahowald-Goerss on topological modular forms from, which realized Lubin-Tate algebras as E_infinity ring spectra and calculated their endomorphisms, to more general Shimura varieties. In a similar vein, this research will seek to clarify, in joint work with David Blanc and Mark Johnson, how a variation of this cotangent complex controls the question of realizing a diagram of Pi-algebras as a diagram of spaces, by looking for ways of connecting cohomological obstructions to things like Toda brackets. Finally, this research will seek to extend Grothendieck's program for classifying smooth schemes to more general schemes in the setting of derived algebraic geometry. The driving mechanism will be to take the cotangent complex to be the proper analogue of the Kahler differentials and generalize accordingly. This would frame the approach of characterizing commutative algebras using the simplicial resolutions (as ultimately articulated by M. Andre and D. Quillen) in an algebraic geometric setting.
Throughout the history of string theory in theoretical physics, surprising connections have been made between several areas of pure mathematics - in particular, number theory and topology. One such connection has been made between the arithmetic of elliptic curves and modular forms, important in A. Wiles resolution of Fermat's Last Theorem, and cohomology theories in topology - the connection being relevant to string theory via the concept of elliptic genera. From the topologist's standpoint, some of the relevant properties being sought can be expressed in terms of whether the cohomology's associated spectrum possesses a suitable multiplicative structure. The question of realizing such structures can lead to questions regarding ways in which commutative algebras and homotopical structures in topology interact. The goal of this research is to further understand such interactions in two ways. The first is to collectively study all possible multiplicative structures on a spectrum with a prescribed algebraic data via 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 a more homotopical notion of algebraic geometry. The aim of this research will then focus on studying other possible multiplicative spectra associated to cohomology theories which arise from more general arithmetic forms, such as automorphic forms. Involved in this project is also the aim to characterize basic types of objects in this homotopical algebraic geometry and drawing connections to recent homological characterizations of algebras.
