The Principal Investigators will join in a collaborative effort to investigate fundamental questions in algebraic geometry using modern homotopical techniques; a unifying thread in these questions is the importance of various classes of invariants ranging from purely algebro-geometric to purely topological. First, the PIs propose to investigate the structure of morphism spaces between real algebraic varieties, especially unstable and stable homotopy types of spaces of "real algebraic" morphisms. Second, the PIs will examine the cohomology of various discrete and arithmetic groups, including algebraic versions of homotopy invariance for cohomology and the related Friedlander-Milnor conjecture. Third, the PIs propose to investigate invariants of singularities arising from methods involving the cdh-topology, continuing the recent flurry of activity in this subject. Finally, motivated by comparisons between algebro-geometric and topological invariants, the PIs will investigate semi-topological or morphic invariants of algebraic varieties, which lie partway between the worlds of algebraic geometry and topology.
Algebraic geometry, one of the oldest branches of mathematics, has at its heart the goal of studying the structure of solutions to systems of polynomial equations; these collections of solutions are called algebraic varieties. Homotopy theory, sometimes called rubber sheet geometry, attempts to study those aspects of geometric objects that are independent of the way they are pulled or twisted; one way to do this is to attach "invariants," e.g., numbers (or more general algebraic structures), to these objects. Algebraic varieties arising from equations with real or complex coefficients can be studied by means of homotopy theory, and the invariants that arise are necessarily somewhat restricted. The goal of this project is to study classical questions in algebraic geometry using invariants of algebraic varieties arising from homotopy theory. A major aim of this project is to convey some of the enthusiasm, techniques, and mathematical goals of the principal investigators to the next generation of mathematicians represented by graduate students and postdoctoral fellows. Methods to recruit and involve early career mathematicians will include the organization of a large international conference, the running of several workshops, the sharing of travel funds, and activities involving visitors from other institutions.
