Many fundamental properties of projective varieties are encoded by their Hilbert functions, which record dimensions of the graded pieces of their coordinate rings. A variety is called Hilbertian if its Hilbert function is a polynomial. We will introduce these notions from the ground up before explaining the ubiquity of Hilbertian varieties in combinatorial commutative algebra via Stanley-Reisner theory. If time permits we will also discuss possible generalizations of these ideas to multigraded settings.
