Analyzing algebraic varieties from the point of view of motivic homotopy theory

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Algebraic geometry, one of the oldest branches of mathematics, is at its core concerned with the study of systems of polynomial equations in many variables. The solutions to such systems give rise to algebraic varieties, which are fundamental objects of study in algebraic geometry. Algebraic topology is the study of systematically attaching algebraic invariants (e.g., numbers or abstract algebraic structures) to spaces; these invariants should not depend on the way a space is pulled or twisted without tearing it. When algebraic varieties have a spatial structure, it is natural to try to analyze them using the tools of algebraic topology. However, when algebraic varieties do not have an obvious spatial structure, e.g., if they arise in arithmetic settings, then a new approach is required. The focus of this project is to analyze algebraic invariants of algebraic varieties using the framework of the Morel-Voevodsky A^1-homotopy theory. This theory allows one to apply the full power of techniques of algebraic topology to objects of interest in algebraic geometry – one may treat spaces having complicated arithmetic structure, but a priori limited geometric structure, in essentially the same way as more classical spaces. The current project seeks to analyze certain classical algebraic and arithmetic questions using these new techniques and to provide a better understanding of specific systems of algebraic equations, which is fundamental to many areas of mathematics. More specifically, the PI will study problems in linear algebra over commutative unital rings, for example, the theory of projective modules and decompositions of matrices. These structures lie in the domain of algebraic K-theory and are related to topological and arithmetic questions by means of Morel-Voeovdsky A^1-homotopy theory. Among others, the PI will investigate the following concrete (and classical) question: given a complex algebraic variety, which topological vector bundles admit algebraic structures? By their very nature, such problems draw together several branches of mathematics and thus illustrate the fundamental unity of the subject.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.