### General Information

Student: |
Terence Coelho |

Office: |
TBD |

School: |
Rutgers University |

E-mail: |
terence dot coelho at rutgers dot edu |

Project: |
Partition Identities |

### 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.

### Coworkers

### Additional Information