The PI proposes to further pursue his investigation of modular representation theory, highlighting insights from algebraic geometry and a focus on the role of p-nilpotent operators, using a new perspective on support varieties for modules which has led to finer "local invariants", new computational tools, and interesting classes of representations. The techniques developed by the PI and his coauthors have led to "global invariants" in the form of algebraic vector bundles and the proposed research is to delve deeper into the significance of these invariants for finite group schemes, and to extend their applicability to other finite dimensional algebras as well as to rational representations of algebraic groups. The PI also proposes to further investigate algebraic cycles and algebraic vector bundles on algebraic varieties. This includes investigating the relationship between spaces of regular maps and spaces of continuous maps from projective smooth varieties to homogeneous varieties, revisiting earlier "semi-topological" constructions in order to offer insights into some of the most challenging questions of classical algebraic geometry. Techniques to be employed will come from abstract algebraic geometry, algebraic K-theory, and homotopy theory. The PI also proposes to study questions of real algebraic geometry using "stable methods" of morphic cohomology and semi-topological K-theory.
Goals of the proposed research are to find new structures and new relationships for mathematical objects which are classical, familiar, and fundamental. His research at times is concrete and calculational, at times abstract and theoretical. The PI proposes to augment his recent ``elementary" constructions in representation theory and his "semi-topological" approach to algebraic geometry with new constructions, foundational results, explicit examples, and general results. In representation theory, the PI proposes to investigate the actions of group-like structures on vector spaces which are not part of the classical framework but which are highly important to many aspects of mathematics. Results obtained from this approach to modular representation theory may provide enlightening examples related to difficult conjectures in algebraic geometry. Motivation for this approach arises in part from the explicit nature of the actions which enable the construction of examples for important, but very difficult geometric structures. In algebraic geometry, the PI will continue to search for techniques which might illuminate some of the most fundamental challenges which arise in the geometric study of solutions to polynomial equations.
