Abstract: Matroids were introduced by Hassler Whitney in 1935 in his paper ``On the abstract properties of linear dependence''. Whitney defined a matroid $(E, {\cal I})$ to be a finite set and a non-empty hereditary collection ${\cal I}$ of subsets of $E$ such that if $I$ and $J$ are in ${\cal I}$ and $|I| < |J|$, then $I \cup \{j\}$ is in ${\cal I}$ for some $j$ in $J-I$. Matroids arise naturally in numerous algebraic and combinatorial contexts. In particular, if $E$ is any finite subset of a vector space $V$ over a field $F$ and ${\cal I}$ is the collection of linearly independent subsets of $E$, then $(E, {\cal I})$ is a matroid. Such a matroid is called $F$-representable and much of the focus of matroid theory has been directed towards characterizing such matroids for certain particular choices of the field $F$. This talk will survey results in this area culminating with some exciting recent developments.\par
This talk will be accessible to graduate students.