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.
This grant’s over-arching theme has been to promote the relatively new and very promising mathematical field of homotopical algebraic geometry. The objects of study are typically geometric structures described by polynomial equations, a major focus of many of the world’s leading mathematicians. In recent years, homotopical algebraic geometry has been enhanced by deepening connections with more established fields such as algebraic geometry, algebraic K-Theory, and algebraic topology. The breadth and complexity of these connections have made it difficult for early career mathematicians to fully participate in further developments. A major goal of this grant was to foster such participation by funding conferences and seminars in order to bring together leading research mathematicians, post-doctoral fellows, and graduate students. The success of this grant can be partially measured by activities it funded: two conferences at USC (in 2013 and 2014) which brought together senior researchers from the U.S. and abroad, postdoctoral researchers, and graduate students; a weekly seminar held jointly with faculty at UCLA which involved once again a good mix of established mathematicians from USC and UCLA, visiting mathematicians whose travel was often funded by the grant, postdoctoral fellows partially funded by the grant, and graduate students. The grant also partially funded the two PI’s (Asok and Friedlander) as well as three postdoctoral fellows (Murfet, Ross, and Williams) and four graduate students (Bilal, Ejder, Erickson, and Warner). The grant funded small ``workshops" between the participating institutions allowing all the researchers involved, their graduate students and also postdoctoral fellows to participate. These small workshops, together with the weekly joint seminar mentioned above fostered some very vibrant cross- institution collaborations. The grant also funded travel by the two PIs and the postdoctoral fellows to disseminate research results at a number of conferences/workshops and seminars, both domestically and internationally. Reflecting on the activities funded by this grant, we are particularly pleased by the active participation of early career mathematicians, including several successful female mathematicians. Another feature is the success achieved in published research papers which apply new and still developing methods to geometric problems.
