[JONGWON KIM], DIMACS REU [2016]

We seek for new partition identities and proofs. We want to explore their relationship with affine Weyl groups.

Week 1:: Read many papers that introduce parition identities. We can approach these identities in many different ways, analytically, combinatorially, algebraically, etc. We are mostly interested in the Rogers-Ramanujan type identities.
Week 2:: Wrote IdentityFinder in C. Rediscovered most of the identities from the papers I read including the Rogers-Ramanujan, and Alladi's. IdentityFinder found some potential new identities.
Week 3:: Came up with a conjecture. Tried a couple of methods and refinements to prove it. Meanwhile, continued reading about root systems and Weyl group, and Lie algebras from books by Humphreys.
Week 4:: Proved the identity. Started reading for another project related to Bud's thesis. Learned about the semisimplicity and decomposition of sl2 and where root systems come into play in the theory of Lie algebras.
Week 5:: Learned how affine weyl group of sl2 and coxeter system were employed in the motivated proof of Rogers-Ramanujan. Came up with the shelf picture and established the empirical hypothesis for an overpartition analogue of Gordon-Andrews identities.
Week 6:: From the weighted words proof of the new identity, I was able to generate many more identities of a similar shape.