Abstract:   In this talk I will discuss some joint work with Sami
Assaf on the k-Schur functions.    Historically, the Schur functions
have been an important basis of the ring of symmetric functions.
About 10 years ago, Lascoux, Lapointe and Morse introduced a new
analog of Schur functions which they call k-Schur functions.   There
are many interesting connections between the k-Schur functions,
Schubert varieties, Gromov-Witten invariants, affine Grassmannians,
and tableaux combinatorics.    For example,  Lam showed that these
elements represent Schubert classes in the cohomology ring of the
affine Grassmannian when the indeterminate t=1.    It is conjectured
that the Macdonald polynomials expand positively into the basis of
k-Schurs for certain k.  In this talk, we will discuss the final step
toward proving that the k-Schur functions are Schur postive via a
computer proof and Assaf's D-graph machinery.    The required
computation is a halting problem with a termination condition but no
upper bound on the number of steps required to stop.     Along the
way, I will include several  ideas for projects in SAGE that would be
useful in this area of research.