li-chung chen and mark haiman studied a family of symmetric functions called catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase $(n-1, \dots, 1, 0)$ (of which there are catalan many) and a composition weight of length $n$. they include the schur functions, the hall-littlewood polynomials and their parabolic generalizations. they can be defined by a demazure-operator formula, and are equal to the $gl$-equivariant euler characteristics of vector bundles on the flag variety by the borel-weil-bott theorem. we have discovered various properties of catalan functions, providing new insight on the existing theorems and conjectures inspired by the macdonald positivity conjecture. a key discovery in our work is an elegant set of ideals of roots whose associate catalan functions are $k$-schur functions, proving that graded $k$-schur functions are $gl$-equivariant euler characteristics of vector bundles on the flag variety, settling a conjecture of chen-haiman. we exposed a new shift invariance property of the graded $k$-schur functions and resolved the schur positivity and $k$-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. we conjectured that catalan functions with a partition weight are $k$-schur positive which strengthens the schur positivity of the catalan function conjecture by chen-haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems. this is joint work with jonah blasiak, jennifer morse, and daniel summers.