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.
Friedlander has furthered national priorities by contributing original mathematical research, educating and mentoring early career mathematicians, and actively participating in advancing U.S. excellence in science. His activities have been undertaken as the Noyes Professor of Mathematics at Northwestern University (NU), the Dean's Professor Mathematics at the University of Southern California (USC), and the President of the American Mathematical Society. During the period of this grant, Friedlander with coauthors has published 12 research articles. These articles introduce surprising connections between algebra and geometry, and develop techniques for further progress. Highlights of this research include the identification of a special type of action (``modules of constant Jordan type") and new numerical invariants (``maximal Jordan type") which further the study of the very classical subject of finite groups and their representations. These articles also introduce for the first time certain important geometric structures (algebraic vector bundles) into the study of group representations. A long-term goal is to not only use these vector bundles to study the actions which led to their formulation but to also to provide a method to produce new vector bundles, one of the central problems in algebraic geometry. Friedlander's Ph.D students, current and former, have played an important role in the activities of this grant. Two of Friedlander's former Ph.D students (Christian Haesemeyer at UCLA and Julia Pevtsova at the University of Washington) were coauthors on some of the newly published papers. Two of Friedlander's students (Chenghao Chu at NU) and Paul Sobaje (at USC) completed their Ph.D dissertations during the period of this grant, and 3 other students (Tylan Bilal, Joseph Timmer, and H. Jared Warner) are working towards their Ph.D under Friedlander's direction. Two other former Ph.D. students (Jeremiah Heller and Mircea Voineagu) are pursuing a research project with Friedlander which relates to the goals of this grant. Friedlander regularly advises young mathematicians both informally and formerly. Currently, Friedlander is actively mentoring as well as collaborating with USC postdoctoral fellow Joseph Ross, and is the "official mentor" of his younger colleague Aravind Asok. Friedlander has served on panels at Northwestern, USC, and the national AMS meeting advising early career mathematicians about job prospects. He helped to establish the AMS-Simons grants for mathematicians recently completing their Ph.D's and was the primary instigator of the new AMS program of student chapters. During the period of this grant, Friedlander has co-edited 4 volumes of mathematical proceedings as part of his general service to the mathematical community. Additional service includes editorships on various journals as well as referee work for other journals. As President of the AMS, Friedlander has promoted the centrality of mathematics in science research and education. He has been alert to the changing climate of science publishing, especially developments concerning open access. He continues to participate in efforts to insure the long-term leadership of the American mathematics.
