PHYS 5311 provides students with a comprehensive understanding of mathematical tools essential for advanced studies in physics. The course emphasizes both algebraic structures and analytical techniques, preparing students to tackle complex problems in various areas of physics. Material covered includes algebraic and analytical methods used in modern physics. Algebra: matrices, groups, and tensors, with application to quantum mechanics, the solid state, and special relativity. Analysis: vector calculus, ordinary and partial differential equations, with applications to electromagnetic and seismic wave propagation.