### General Information

Student: Terence Coelho TBD Rutgers University terence dot coelho at rutgers dot edu Partition Identities, Lie Theory

### Project Description

The project is threefold- we experimentally search for partition identity conjectures, try to prove existing conjectures and look for motivated proofs of existing partition identities, and find algebraic interpretations of existing proofs.

### Weekly Log

Week 1:
Wrote code to compute paritions quickly in java with difference at distance conditions then spit out the product side in order to look for patterns. Started learning about Weyl Groups and following the motivated proof of Gordon-Gollnitz. Met with Dr. Russell 3 times during the week to discuss progress.
Week 2:
Continued adding code to restrict partition identities. Tried some quick searches for partition identities with no luck. Partner (Jongwon) found one that we thought was new, but eventually realized it was buried in a paper published just 17 years ago! Started learning the motivated proof of Andrews-Bressoud written by Dr. Russell, Dr. Kanade, and Dr. Lepowsky
Week 3:
Jongwon conjectured a new identity that we've checked up to 200! We are now in the process of trying to prove it and are convinced that we can!
Week 4:
Started learning Lie Algebra/Representation theory to extend Dr. Coulson's thesis to an overpartition identity. Wrote an almost-proof of Jongwon's identity (hereafter J) that has a final step that needs to be checked further (by expanding many (but finite) terms and checking some base cases) but must be true. We found an analogue to J; started trying to prove it and find similar analogues of other identities.
Week 5:
Discovered that Jongwon's identity, along with several other known identities, have a remarkable property in common that I believe can easily be proven by analyzing a newer proof of one of the other identities. Once this conjecture is proven, it links Jongwon's and the other identities together in a manner that makes them all corollaries of each other (which must be unknown by the world or else J would have been discovered) and explains all the strange initial conditions. It also immediately proves the analogues of J discovered last week and leads to a whole new class of analogues of J and gives the analogues of the other theorems. Once the proof is complete, I will be less cryptic here.

• My Mentors

### Project Description

My project is on discovering new properties of Lie Algebraic structures and ultimately relating these to Vertex Operator Algebras.

### Weekly Log

Week 1:
Mostly just read through Humphreys/Vertex Operator Algebras and the Monster
Week 2:
Came across an interesting (possibly new) property of sl(3); trying to understand it in terms of the known theory.
Week 3:
Started reading through chapters 5+6 of Vertex Operator Algebras and the Monster. Met several times with advisor to clarify the material
Week 4:
Polished results on sl(3), continued reading through Humphreys to try to figure out how to generalize the result to other families of Lie algebras.
Week 5:
Tried to improve the result for sl(3) by abstracting it further. Didn't make an improvement but the work looks promising.
Week 6:
Lie algebra theory finally clicked; figured out why it should generalize to other families of Lie Algebras. Started working on a proof
Week 7:
Generalized theory fully to any semi-simple Lie Algebra. Gave presentation and wrote up results. Next steps will be to continue to try to improve the results of sl(3) and the general case.