Finite group schemes and semi-topological theories

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One aspect of Friedlander's proposed research is to advance the understanding of how finite groups and their generalizations act on vector spaces. The elementary example of Z/p x Z/p is one of the first examples encountered by beginning students of abstract algebra, yet its representation theory is ``wild" so that no listing of all finite dimensional representations is possible. Friedlander has introduced constructive techniques which apply to this and other specific examples yet extend to very general situations. Friedlander proposes to continue his study of representations of arbitrary finite group schemes using insights and techniques from algebraic geometry as well as more traditional techniques of algebra. One goal is to contribute to the understanding of specific examples; a second goal is to sketch a general theory which incorporates these examples; and a third goal is to utilize certain special actions to study the algebraic K-theory of certain singular projective varieties associated to finite group schemes. A second aspect of Friedlander's proposed research is the investigtion of algebraic cycles on algebraic varieties. This is one of the most fundamental and challenging topics of algebraic geometry, much studied in the past hundred years. Friedlander's focus will be on algebraic equivalence classes of cycles, influenced by insights from the better understood analogue in algebraic topology. Applications are envisioned to algebraic K-theory as well as algebraic geometry. How can finite groups of symmetries act on vector spaces over finite fields or over even more general fields? How does the consideration of more general algebraic objects (finite group schemes) reflect on the original problem, especially in basic, familiar examples? How does the geometry, at first unrecognized, constrain the possibilities and lead to concrete examples? Can the explicit nature of these examples give structures in abstract contexts? These are some of the questions Friedlander proposes to investigate with several collaborators. In addition, he proposes to study solution sets of polynomial equations (algebraic geometry) using techniques developed in algebraic topology (theory of shapes). Friedlander plans to encourage younger mathematicians (including his past, present, and future students) in his quest. He also plans to continue his active roles in publishing mathematics, organizing mathematical events, and serving the national mathematical community.