New Frontiers in Homotopy Theory

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The field of homotopy theory is the study of mathematical invariants that are insensitive to deformations. It is applicable whenever one is interested in studying qualitative aspects of a system, emergent properties of a statistical ensemble of structures, or whenever there might be imprecision in the specification of the state of a system. In recent years the methods of homotopy theory have found use in fields as diverse as condensed matter physics and the foundations of mathematics. This project aims to bolster these relationships with new tools from algebraic topology, and to apply them to these and other areas of mathematics and science. There are applications of this work to condensed matter physics, classical algebraic geometry, algebraic topology and categorical logic.The scope of this project involves several interrelated areas of study. One of these, on algebraic vector bundles, depicts a new interface between complex analysis and algebraic topology, and is intended to get at the obstruction to topological vector bundles having algebraic structures. Another explores the prospect of investigating the aggregate of all models of a given physical system, with the idea that the phases of physically measurable quantities occur as topological invariants of the space of models. The two principal investigators will continue their joint work on strict units in chromatic homotopy theory. This project has produced many striking analogies between chromatic homotopy theory and higher category theory. Six further projects involve new stuctures in homotopy theory and its applications. One investigates new formulas and expressions of duality in chromatic homotopy theory, another explores the "transchromatic" version of the hierarchy of limits expressed by the principal investigator's earlier work on "ambidexterity," and a third offers a new explanation for the ubiquity of Lie algebras occurring in stable homotopy theory. Other projects involve applications of homotopy theory to categorical logic, to the characterization of higher categories of interest in mathematical physics, and to a new construction of the famous de Rham-Witt complex used in number theory and algebraic geometry.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.