NSF-BSF: CHROMATIC HOMOTOPY THEORY, ALGEBRAIC K-THEORY, AND POWER OPERATIONS

The PI will explore new structures in higher algebra, in collaboration with Dr. Shachar Carmeli at the Weizmann Institute of Science, Israel. This is an emerging field that combines ideas from classical algebra and modern homotopy theory. Classical algebra studies systems like the real numbers, with operations such as addition and multiplication that satisfy several rules (such as associative and distributive laws). Higher algebra studies structures in which these equalities are replaced by coherent witnesses, called homotopies. Over the past several decades, mathematicians have discovered that many important algebraic structures can be refined in this way, leading to many applications to other disciplines, such as mathematical physics and foundations of computer science. The project will study such higher structures in the subfield of stable homotopy theory. Moreover, the project will support the training and development of junior mathematicians in the field. The project aims to use methods from algebraic K-theory and power operations to study chromatic homotopy theory. Chromatic homotopy theory studies questions in stable homotopy theory (e.g., stable homotopy groups of spheres) via tools arising from the algebraic geometry of formal groups. Recently, categorical and K-theoretic techniques play an increasing role in the subject. The PI and collaborator intend to study the chromatic localizations of K-theory and other invariants of ring spectra, and relate them with recent advances in p-adic geometry such as the new theory of prismatization. This collaborative US/Israel project is supported by the Division of Mathematical Sciences of the US National Science Foundation and by the Israeli Binational Science Foundation. 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.