Motivic homotopy theory

Northeastern University, Boston MA

Investigators

Abstract

In this project, the PI intends to use motivic homotopy theory to create new tools for the study of problems in algebraic geometry. The PI plans to transfer classical obstruction theory to the motivic setting, with the specific goal of understanding the obstructions to finding sections to algebraic fiber bundles over an algebraically closed field. The PI plans to study algebraic cobordism, an algebraic versions of the topological theory of complex cobordism, and to further examine its connection with Donaldson-Thomas theory. Additionally, the PI plans a further study of the Deligne-Goncharov motivic fundamental group. Finally, the PI plans a further study of the motivic Postnikov tower, with the goal of gaining a better understanding of this tower for a variety of interesting generalized cohomology theories on algebraic varieties, as well as for the motives of smooth projective varieties. Homotopy theory is a branch of topology, which deals with fundamental properties of curves, surfaces and shapes of higher dimension. Algebraic geometry, on the other hand, tries to understand the properties of solutions of equations, even when one cannot actually solve the equation explicitly. Creating analogies between the seemingly unrelated fields of algebra and topology has often been a fruitful approach to solving difficult problems in both fields. Morel and Voevodsky have transferred an entire branch of topology, called stable homotopy theory, to the algebraic setting, making ideas from stable homotopy theory applicable to problems in algebra and number theory. The PI plans to take a number of specific constructions from homotopy theory, adapt them to this new setting, and use these constructions to solve problems in algebraic geometry.