h-vectors: A tour of simplicial complexes, shellability and Stanley--Reisner theory

The h-vector of a simplicial complex is a wonderful statistic which tells us much about the structure of its associated complex. This vector gets its strength through its multitude of different interpretations. These interpretations let us understand various properties of h-vectors, and, through them, properties of simplicial complexes. We will highlight three of these interpretations using f-vectors, shellability and Stanley--Reisner rings along the way! This talk will be expository; previous exposure to simplicial complexes is welcome but not required.