I will introduce and study (generic) determinantal varieties: the loci of matrices of rank bounded from above.

In addition to being a central topic in both Commutative Algebra and Algebraic Geometry,
these varieties have several connections with Invariant Theory, Representation Theory and Combinatorics.

For example, many classical constructions such as rational normal curves, Veronese manifolds, Segre manifolds, rational normal scrolls, fall into the class of determinant varieties.

I will describe their fundamental properties (such as normality, irreducibility, singular locus, etc) and calculate their basic invariants (degree, dimension, topological Euler number, etc).
To do so, I will establish the first and the second fundamental theorems of invariant theory as one of the main tools.