Topics in Algebraic Combinatorics
This MIT OpenCourseWare course surveys advanced topics in algebraic combinatorics, taught through a graduate-level curriculum of lecture notes and problem sets. Students work through the matrix-tree theorem and other applications of linear algebra to counting problems, then move into commutative and exterior algebra techniques for analyzing the face structure of simplicial complexes. A further unit applies algebraic methods to tiling problems, connecting combinatorial questions to polynomial rings and algebraic invariants. Materials include full lecture notes covering each topic in detail, following MIT's standard OCW format of free downloadable course content with no enrollment or certificate. The course assumes strong prior background in linear algebra and abstract algebra, and suits students who already have a foundation in combinatorics and want to see how algebraic tools sharpen counting arguments across several distinct problem areas.