Algebraic geometry is the study of geometric shapes defined by polynomial equations. This project studies motivic homotopy theory, a relatively new subject that allows us to utilize algebraic topology methods to understand the objects of interest in algebraic geometry. The motivic theory has had several spectacular successes in resolving deep mathematical problems. In this project, we develop tools that are well suited for studying not just algebraic varieties but also their symmetries. Spheres are simple yet essential objects of study in geometry, and they are considered in the realm of algebraic geometry. One of the central questions is the classification of all possible mappings of a high-dimensional sphere onto a lower dimension sphere.
This research project lies at the interface of algebraic geometry and homotopy theory, two pillars of contemporary mathematics. Algebraic geometry has ancient origins with many connections to real-world problems. Its goal is to understand the geometry of algebraic varieties or solutions sets of polynomial equations. Homotopy theory on the other hand is developed more recently and aims to study a general notion of shape; such invariants do not depend on the way a space is pulled or twisted, and is less rigid than the idea of shape studied in algebraic geometry. Perfectly shaped higher dimensional spheres are simple yet important building blocks for geometric structures. It turns out that spheres of different dimensions can fit together in only certain combinations to create more complicated geometric objects. The computation of stable homotopy groups of the sphere is among the most fundamental problems in homotopy theory. By importing the notion of spheres to the study of algebraic varieties one arrives at the modern subject of motivic homotopy theory. This idea provides novel ways for qualitatively describing the shapes of equations, and has in the past twenty-five years enjoyed spectacular successes in resolving open problems. Several arithmetic and geometric phenomena which appear delicate are invariant under motivc homotopies. The project addresses directly the heart of motivic homotopy theory by computing invariants like numbers, groups, rings, and sheaves to understand shapes. Motivic homotopy theory heavily utilizes the fact that the stable motivic homotopy category behaves like a derived category of modules. The motivic sphere behaves as a brave new system of numbers which has much deeper structures than numbers we use in our daily lives. Computing the symmetries of the motivic sphere is one of the overarching themes in the subject. This project aims to broaden and deepen our knowledge in this fundamental area of mathematics.