This advanced class combines a traditional abstract algebra curriculum with topics in linear algebra and geometry. Topics include: introductory group theory; Sylow theorems; isometries of the Euclidean plane; symmetries of wallpaper patterns; isometries of the hyperbolic plane and Kleinian groups; elementary Lie groups and their discrete subgroups; Lie algebras; spectral theorems and decomposition theorems for finite-dimensional vector spaces; multilinear algebra; Galois theory; rings and modules; and introductory representation theory. Exploration of these topics will extend over both the Fall and the Spring term. There will be opportunity for students to pursue individual research questions.