CAREER: Vector bundles, rational points and homotopy theory

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The Principal Investigator will study problems regarding projective modules over commutative unital algebras over a field, and existence of rational points on special classes of algebraic varieties by transposing ideas from the classical algebro-topological theory of obstructions to the context of algebraic geometry by means of the Morel-Voevodsky A^1-homotopy theory. Together with J. Fasel, he will, for example, attempt to study splitting problems for projective modules of rank below the dimension by better understanding A^1-homotopy sheaves of punctured affine spaces. Together with C. Haesemeyer, he will attempt to produce computable new obstructions to existence of a rational point on smooth algebraic varieties over a field, again using A^1-homotopy theory. By their very nature, these problems draw together several branches of mathematics and thus illustrate the fundamental unity of the subject. To make the techniques used in the above study and related areas accessible to a new generation of researchers, the PI will organize a series of summer schools devoted to introducing graduate students to A^1-homotopy theory and related subjects. Algebraic geometry, one of the oldest branches of mathematics, is at its core concerned with the study of algebraic varieties, i.e., solutions to systems of polynomial equations in many variables. Algebraic topology studies the problem of attaching invariants, e.g., numbers or, more generally, abstract algebraic structures, to spaces in a manner independent of the way they are pulled and twisted (without being torn). Solutions to systems of polynomial equations are not obviously amenable to the tools of algebraic topology, especially when such systems arise from arithmetic. Nevertheless, the relatively recently introduced subject of A^1-homotopy theory provides a framework in which one may apply the full power of techniques of algebraic topology to objects of interest in algebraic geometry---one can treat spaces having deep arithmetic structure, but a priori limited geometric structure, on equal footing with spaces that are more inherently geometric. This project aims to study two general classes of problems by transplanting a very successful method from algebraic topology (the theory of obstructions) to the setting of algebraic geometry by means of A^1-homotopy theory. With the goal of making some ideas involved in the above pursuits accessible to a broader audience, the PI will collaborate with artist Lun-Yi Tsai to organize public ``dialogues" between artists and mathematicians to explore some common themes between the subjects.This award is cofunded by the Algebra and Number Theory and the Topology programs.