Pevtsova intends to develop a theory that relates representations of finite group schemes to vector bundles on projective varieties, thereby establishing a novel connection between representation theory and algebraic geometry. The proposed research takes its roots in previous work of the PI and collaborators in representations and cohomology of finite dimensional algebras. The unique perspective of the proposed development has a potential for significant applications in both algebraic geometry and representation theory, ranging from construction of new examples of vector bundles to classification results. Other specific areas of the proposed research include computations of important examples of support varieties, constructions of new invariants for small quantum groups, and advances to the theory of modules of constant Jordan type. In addition, Pevtsova proposes to make contributions to the ``triangular geometry", a new geometric theory that encodes the structure of triangulated categories. Bringing the projects on finite group schemes and triangular geometry to a meeting point, Pevtsova is seeking to compare derived categories associated to different algebraic objects via their geometry.
Representation theory studies actions of groups and other algebraic structures on vector spaces. It takes its origins in the study of symmetries and has emerged as a subject on its own about hundred years ago in the work of Frobenius and Schur. In its current stage of development, representation theory has been discovered to be intimately intertwined with numerous brunches of mathematics, such as geometry, topology and combinatorics, as well as physics. Pevtsova is particularly interested in connections with geometry. She has always been fascinated by the beautiful interplay between algebra and geometry that transcends many areas of mathematical research. In her CAREER project, she seeks to develop a new algebra-geometric connection, building a bridge between algebraic descriptions of representations which can be presented in terms of matrices and certain geometric invariants which can be thought of as spaces. In that endeavor she hopes to develop techniques that will hold a potential to shed light on some longstanding problems in both representation theory and geometry.
The PI also has an extensive educational program. She intends to supervise both undergraduate and graduate research and to develop a course on problem solving in connection with the training of the Putnam team at the University of Washington. Building on her existing outreach experience the PI intends to create a network of ``Math challenge" afterschool programs for gifted elementary and middle school students in Seattle Public schools. She also proposes to organize a summer school for young researchers in Representation Theory and related areas in the Summer of 2012. By creating these opportunities for talented K-12 students, undergraduates and graduate students, Pevtsova intends to attract more qualified candidates to careers in Mathematical Sciences.
