Chromatic homotopy theory, algebraic K-theory, and L-functions

Algebraic topology is a subject that studies topological spaces via associated algebraic entities. One of the most important such entities, or invariants, is the homotopy groups of a topological space, which measure whether or not the space can be continuously deformed into a single point. The computation of homotopy groups of spheres has been a driving force in the development of algebraic topology with wide applications to many other fields of mathematics, including number theory. The overarching goal of this project is to further explore the profound connections between number theory and homotopy theory. Broader impacts of this project focus on graduate curriculum development in chromatic homotopy theory, student mentoring, and conference organization. The PI plans to investigate various aspects of the connections between chromatic homotopy theory, algebraic K-theory, and L-functions. Building on his previous work, the PI will explore generalizations of the Quillen-Lichtenbaum Conjecture to various L-functions associated to abelian characters of Galois representations, using tools from equivariant algebraic K-theory. Algebraic K-theory spectra have a coherent multiplicative structure. The PI plans to give explicit algebraic formulas for this structure and investigate its implications in zeta functions. Another question the PI will study is how to relate Eisenstein series and Hida’s L-functions of two variables to power operations on elliptic cohomology. Finally, the PI plans to use tools from arithmetic geometry to investigate the Vanishing Conjecture and Picard groups in chromatic homotopy theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.