Cohomology and Support Varieties

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Representation theory studies symmetries of linear spaces. For such tractable example as a two-dimensional plane one can study symmetries via periodic tessellations they generate. Beautiful examples and even complete lists of such tessellations are frequently implemented via artistic means, as illustrated in Escher's famous symmetry drawings or wallpaper patterns in the Alhambra in Spain. In fact, it is known that there are exactly seventeen essentially different tessellations of the plane. Enumerating symmetries of more complicated and higher dimensional objects is often a daunting if not outright impossible task. Yet, having a grasp on such complicated symmetries proves to be important not only in representation theory but in many other areas, both pure and applied, including geometry, topology, algebra, as well as physics and chemistry. In this project, the PI will investigate certain invariants of symmetries that come in the form of interesting geometric shapes. Her goal is to investigate how much data about the original symmetries these geometric objects contain, shaping them to be a useful tool in the study of symmetries themselves. Founded more than a hundred years ago, representation theory is now a thriving field that exhibits many deep connections with other areas of mathematics. Modular representation theory draws methods and motivation from a host of other areas, including algebraic geometry and topology. In 1971, Quillen laid the foundations for algebraic geometry applications to group cohomology, opening a new chapter in the study of modular representation theory that is being actively explored to this day. This research project has its roots in Quillen's work, seeking to develop the theory of support varieties in several different, but interrelated contexts. This involves solving several fundamental problems concerning the structure of representations and the cohomology of finite dimensional algebras. The applications will provide new information on the global structure of various triangulated categories associated to representations of finite supergroup schemes, Schur algebras, Nichols algebras, Frobenius kernels of reductive groups, and Lie superalgebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.