Catherine Buell (Fitchburg State)

Biography: Dr. Catherine Buell is a Professor in the Mathematics Department at Fitchburg State University, located in Fitchburg, Massachusetts. She completed her Ph.D at North Carolina State University in 2011 under the supervision of Dr. Aloysius Helminck. Dr. Buell’s research focuses on algebraic group theory, symmetric spaces, Lie groups and Lie algebras, generalized symmetric spaces (studying the structure and orbit decomposition of symmetric spaces for SL_n(F_q)), and visual stylometry (integrating mathematics, philosophy, computer science, and art to classify artwork). Additionally, she researches mathematics education and social justice/equity/ethics in math, including creating inquiry-based labs for the calculus I and calculus II sequence, democratizing access to authentic mathematics for all students, and advancing anti-oppressive mathematics education and the incorporation of social justice into mathematics classrooms.

Title: Fixed-Point Orbits of Unipotent Elements in Symmetric k-Varieties and Finding your Research Orbit

Abstract: Symmetric k-varieties were introduced in the late 1980s as generalizations of Riemannian symmetric spaces of Lie groups defined over the reals or complex numbers to linear algebraic groups defined over general fields. The study of non-Riemannian symmetric spaces and generalizations of these spaces to other base fields has led to exciting applications in many areas, including representation theory and singularity theory. These applications often come down to questions concerning orbit decompositions in the symmetric space (ie. cosets). In this talk, we will define these objects for an undergraduate audience using the special linear group, and explore the questions and techniques used to characterize the fixed-point orbits of elements in the generalized symmetric spaces (ie. a quotient group) by considering an element’s decomposition into a semi-simple and unipotent piece.  We will also talk a bit about paths, inspirations, and considerations for research during and after graduate study.