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.
. This is a new area of mathematics, whose foundations and goals lie in the combination of two of the major areas of mathematics of the twentieth century, algebraic geometry and algebraic topology. Algebraic geometry is the study of solutions of polynomial equations, whereas algebraic topology deals with topological spaces (so-called ``rubber-sheet geometry") and how they relate to each other. In the 1990's Morel and Voevodsky developed a framework for combining these two fields in a new way, creating the field of motivic homotopy theory. My specific results in this area were partly foundational, settling questions within the field that would hopefully be of use for further developments, and partly applications to other fields. In the first area, I extended constructions of Bondarko and others to give a theory of smooth motives over a base, and answered a question of Deligne and Goncharov on the motivic first homotopy group of a variety. As for applications, together with R. Pandharipande, we used the theory of algebraic cobordism (developed a few years ago by myself and F. Morel) to settle some conjectures in an area of algebraic geometry, namely, the study of Donaldson-Thomas invariants.
